5.1: Central Tendency

5.1.1: Mean: The Average

Learning Objective

Define the average and distinguish between arithmetic, geometric, and harmonic means.

Key Points

• An average is a measure of the “middle” or “typical” value of a data set.
• The three most common averages are the Pythagorean means – the arithmetic mean, the geometric mean, and the harmonic mean.
• The arithmetic mean is the sum of a collection of numbers divided by the number of numbers in the collection.
• The geometric mean is a type of mean or average which indicates the central tendency, or typical value, of a set of numbers by using the product of their values. It is defined as the
  th root (where
  is the count of numbers) of the product of the numbers.
• The harmonic mean
  of the positive real numbers
  is defined to be the reciprocal of the arithmetic mean of the reciprocals of
  . It is typically appropriate for situations when the average of rates is desired.

Key Terms

average

any measure of central tendency, especially any mean, the median, or the mode

arithmetic mean

the measure of central tendency of a set of values computed by dividing the sum of the values by their number; commonly called the mean or the average

central tendency

a term that relates the way in which quantitative data tend to cluster around some value

Example

• The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value
  such that
  . One may find that
  . Switching the order of 2 and 8 to read 8 and 2 does not change the resulting value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list for which we want an average, we get, for example, that the arithmetic mean of 2, 8, and 11 is found by solving for the value of
   in the equation
  . One finds that
  .

The term central tendency relates to the way in which quantitative data tend to cluster around some value. A measure of central tendency is any of a variety of ways of specifying this “central value”. Central tendency is contrasted with statistical dispersion (spread), and together these are the most used properties of distributions. Statistics that measure central tendency can be used in descriptive statistics as a summary statistic for a data set, or as estimators of location parameters of a statistical model.

In the simplest cases, the measure of central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context. An average is a measure of the “middle” or “typical” value of a data set. In the most common case, the data set is a list of numbers. The average of a list of numbers is a single number intended to typify the numbers in the list. If all the numbers in the list are the same, then this number should be used. If the numbers are not the same, the average is calculated by combining the numbers from the list in a specific way and computing a single number as being the average of the list.

The term mean has three related meanings:

1. The arithmetic mean of a sample,
2. The expected value of a random variable, or
3. The mean of a probability distribution

The Pythagorean Means

The three most common averages are the Pythagorean means – the arithmetic mean, the geometric mean, and the harmonic mean.

The Arithmetic Mean

When we think of means, or averages, we are typically thinking of the arithmetic mean. It is the sum of a collection of numbers divided by the number of numbers in the collection. The collection is often a set of results of an experiment, or a set of results from a survey of a subset of the public. In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation’s population.

Suppose we have a data set containing the values
. The arithmetic mean is defined via the expression:

If the data set is a statistical population (i.e., consists of every possible observation and not just a subset of them), then the mean of that population is called the population mean. If the data set is a statistical sample (a subset of the population) we call the statistic resulting from this calculation a sample mean. If it is required to use a single number as an estimate for the values of numbers, then the arithmetic mean does this best. This is because it minimizes the sum of squared deviations from the estimate.

The Geometric Mean

The geometric mean is a type of mean or average which indicates the central tendency, or typical value, of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean applies only to positive numbers. The geometric mean is defined as the
th root (where
is the count of numbers) of the product of the numbers.

For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is
. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2; that is
.

A geometric mean is often used when comparing different items – finding a single “figure of merit” for these items – when each item has multiple properties that have different numeric ranges. The use of a geometric mean “normalizes” the ranges being averaged, so that no range dominates the weighting, and a given percentage change in any of the properties has the same effect on the geometric mean.

For example, the geometric mean can give a meaningful “average” to compare two companies which are each rated at 0 to 5 for their environmental sustainability, and are rated at 0 to 100 for their financial viability. If an arithmetic mean was used instead of a geometric mean, the financial viability is given more weight because its numeric range is larger – so a small percentage change in the financial rating (e.g. going from 80 to 90) makes a much larger difference in the arithmetic mean than a large percentage change in environmental sustainability (e.g. going from 2 to 5).

The Harmonic Mean

The harmonic mean is typically appropriate for situations when the average of rates is desired. It may (compared to the arithmetic mean) mitigate the influence of large outliers and increase the influence of small values.

The harmonic mean
of the positive real numbers
is defined to be the reciprocal of the arithmetic mean of the reciprocals of
. For example, the harmonic mean of 1, 2, and 4 is:

The harmonic mean is the preferable method for averaging multiples, such as the price/earning ratio in Finance, in which price is in the numerator. If these ratios are averaged using an arithmetic mean (a common error), high data points are given greater weights than low data points. The harmonic mean, on the other hand, gives equal weight to each data point.

5.1.2: The Average and the Histogram

Learning Objective

Demonstrate the effect that the shape of a distribution has on measures of central tendency.

Key Points

• Histograms tend to form shapes, which when measured can describe the distribution of data within a dataset.
• A key feature of the normal distribution is that the mode, median and mean are the same and are together in the center of the curve.
• A key feature of the skewed distribution is that the mean and median have different values and do not all lie at the center of the curve.
• Skewed distributions with two or more modes are known as bi-modal or multimodal, respectively.

Key Terms

normal distribution

A family of continuous probability distributions such that the probability density function is the normal (or Gaussian) function.

bell curve

In mathematics, the bell-shaped curve that is typical of the normal distribution.

histogram

A representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval.

As discussed, a histogram is a bar graph displaying tabulated frequencies. Histograms tend to form shapes, which when measured can describe the distribution of data within a dataset. The shape of the distribution can assist with identifying other descriptive statistics, such as which measure of central tendency is appropriate to use.

The distribution of data item values may be symmetrical or asymmetrical. Two common examples of symmetry and asymmetry are the “normal distribution” and the “skewed distribution. “

Central Tendency and Normal Distributions

In a symmetrical distribution the two sides of the distribution are a mirror image of each other. A normal distribution is a true symmetric distribution of data item values. When a histogram is constructed on values that are normally distributed, the shape of columns form a symmetrical bell shape. This is why this distribution is also known as a “normal curve” or “bell curve. ” is an example of a normal distribution:

If represented as a ‘normal curve’ (or bell curve) the graph would take the following shape (where
 is the mean and
 is the standard deviation):

A key feature of the normal distribution is that the mode, median and mean are the same and are together in the center of the curve.

Also, there can only be one mode (i.e. there is only one value which is most frequently observed). Moreover, most of the data are clustered around the center, while the more extreme values on either side of the center become less rare as the distance from the center increases (i.e. about 68% of values lie within one standard deviation (
) away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This is known as the empirical rule or the 3-sigma rule).

Central Tendency and Skewed Distributions

In an asymmetrical distribution the two sides will not be mirror images of each other. Skewness is the tendency for the values to be more frequent around the high or low ends of the
-axis. When a histogram is constructed for skewed data it is possible to identify skewness by looking at the shape of the distribution. For example, a distribution is said to be positively skewed when the tail on the right side of the histogram is longer than the left side. Most of the values tend to cluster toward the left side of the
-axis (i.e, the smaller values) with increasingly fewer values at the right side of the
-axis (i.e. the larger values).

A distribution is said to be negatively skewed when the tail on the left side of the histogram is longer than the right side. Most of the values tend to cluster toward the right side of the
-axis (i.e. the larger values), with increasingly less values on the left side of the
-axis (i.e. the smaller values).

A key feature of the skewed distribution is that the mean and median have different values and do not all lie at the center of the curve.

There can also be more than one mode in a skewed distribution. Distributions with two or more modes are known as bi-modal or multimodal, respectively. The distribution shape of the data in is bi-modal because there are two modes (two values that occur more frequently than any other) for the data item (variable).

5.1.3: The Root-Mean-Square

Learning Objective

Compute the root-mean-square and express its usefulness.

Key Points

• The root-mean-square is especially useful when a data set includes both positive and negative numbers.
• Its name comes from its definition as the square root of the mean of the squares of the values.
• The process of computing the root mean square is to: 1) Square all of the values 2) Compute the average of the squares 3) Take the square root of the average.
• The root-mean-square is always greater than or equal to the average of the unsigned values.

Key Term

root mean square

the square root of the arithmetic mean of the squares

The root-mean-square, also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity, or set of numbers. It can be calculated for a series of discrete values or for a continuously varying function. Its name comes from its definition as the square root of the mean of the squares of the values.

This measure is especially useful when a data set includes both positive and negative numbers. For example, consider the set of numbers
. Computing the average of this set of numbers wouldn’t tell us much because the negative numbers cancel out the positive numbers, resulting in an average of zero. This gives us the “middle value” but not a sense of the average magnitude.

One possible method of assigning an average to this set would be to simply erase all of the negative signs. This would lead us to compute an average of 5.6. However, using the RMS method, we would square every number (making them all positive) and take the square root of the average. Explicitly, the process is to:

1. Square all of the values
2. Compute the average of the squares
3. Take the square root of the average

In our example:

The root-mean-square is always greater than or equal to the average of the unsigned values. Physical scientists often use the term “root-mean-square” as a synonym for standard deviation when referring to the square root of the mean squared deviation of a signal from a given baseline or fit. This is useful for electrical engineers in calculating the “AC only” RMS of an electrical signal. Standard deviation being the root-mean-square of a signal’s variation about the mean, rather than about 0, the DC component is removed (i.e. the RMS of the signal is the same as the standard deviation of the signal if the mean signal is zero).

5.1.4: Which Average: Mean, Mode, or Median?

Learning Objective

Assess various situations and determine whether the mean, median, or mode would be the appropriate measure of central tendency.

Key Points

• In symmetrical, unimodal distributions, such as the normal distribution (the distribution whose density function, when graphed, gives the famous “bell curve”), the mean (if defined), median and mode all coincide.
• If elements in a sample data set increase arithmetically, when placed in some order, then the median and arithmetic mean are equal. For example, consider the data sample
  . The mean is 2.5, as is the median.
• While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers (values that are very much larger or smaller than most of the values).
• The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data is contaminated, the median will not give an arbitrarily large result.
• Unlike mean and median, the concept of mode also makes sense for “nominal data” (i.e., not consisting of numerical values in the case of mean, or even of ordered values in the case of median).

Key Terms

Mode

the most frequently occurring value in a distribution

breakdown point

the number or proportion of arbitrarily large or small extreme values that must be introduced into a batch or sample to cause the estimator to yield an arbitrarily large result

median

the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half

Example

• The mode is the value that appears most often in a set of data. For example, the mode of the sample
  is 6. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one (e.g., the median of
   is 5).

The Mode

The mode is the value that appears most often in a set of data. For example, the mode of the sample
is 6. Like the statistical mean and median, the mode is a way of expressing, in a single number, important information about a random variable or a population.

The mode is not necessarily unique, since the same maximum frequency may be attained at different values. Given the list of data
the mode is not unique – the dataset may be said to be bimodal, while a set with more than two modes may be described as multimodal. The most extreme case occurs in uniform distributions, where all values occur equally frequently.

For a sample from a continuous distribution, the concept is unusable in its raw form. No two values will be exactly the same, so each value will occur precisely once. In order to estimate the mode, the usual practice is to discretize the data by assigning frequency values to intervals of equal distance, as with making a histogram, effectively replacing the values with the midpoints of the intervals they are assigned to. The mode is then the value where the histogram reaches its peak.

The Median

The median is the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one (e.g., the median of
is 5). If there is an even number of observations, then there is no single middle value. In this case, the median is usually defined to be the mean of the two middle values.

The median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers (e.g., because there may be measurement errors).

Which to Use?

In symmetrical, unimodal distributions, such as the normal distribution (the distribution whose density function, when graphed, gives the famous “bell curve”), the mean (if defined), median and mode all coincide. For samples, if it is known that they are drawn from a symmetric distribution, the sample mean can be used as an estimate of the population mode.

If elements in a sample data set increase arithmetically, when placed in some order, then the median and arithmetic mean are equal. For example, consider the data sample
. The mean is 2.5, as is the median. However, when we consider a sample that cannot be arranged so as to increase arithmetically, such as
, the median and arithmetic mean can differ significantly. In this case, the arithmetic mean is 6.2 and the median is 4. In general the average value can vary significantly from most values in the sample, and can be larger or smaller than most of them.

While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers (values that are very much larger or smaller than most of the values). Notably, for skewed distributions, such as the distribution of income for which a few people’s incomes are substantially greater than most people’s, the arithmetic mean may not be consistent with one’s notion of “middle,” and robust statistics such as the median may be a better description of central tendency.

The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data is contaminated, the median will not give an arbitrarily large result. Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normally distributed. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from parametric distributions.

Unlike median, the concept of mean makes sense for any random variable assuming values from a vector space. For example, a distribution of points in the plane will typically have a mean and a mode, but the concept of median does not apply.

Unlike mean and median, the concept of mode also makes sense for “nominal data” (i.e., not consisting of numerical values in the case of mean, or even of ordered values in the case of median). For example, taking a sample of Korean family names, one might find that “Kim” occurs more often than any other name. Then “Kim” would be the mode of the sample. In any voting system where a plurality determines victory, a single modal value determines the victor, while a multi-modal outcome would require some tie-breaking procedure to take place.

5.1.5: Averages of Qualitative and Ranked Data

Learning Objective

Categorize levels of measurement and identify the appropriate measures of central tendency.

Key Points

• Qualitative data can be defined as either nominal or ordinal.
• The nominal scale differentiates between items or subjects based only on their names and/or categories and other qualitative classifications they belong to.
• The mode is allowed as the measure of central tendency for nominal data.
• The ordinal scale allows for rank order by which data can be sorted, but still does not allow for relative degree of difference between them. The median and the mode are allowed as the measure of central tendency; however, the mean as the measure of central tendency is not allowed.
• The median and the mode are allowed as the measure of central tendency for ordinal data; however, the mean as the measure of central tendency is not allowed.

Key Terms

quantitative

of a measurement based on some quantity or number rather than on some quality

qualitative

of descriptions or distinctions based on some quality rather than on some quantity

dichotomous

dividing or branching into two pieces

Levels of Measurement

In order to address the process for finding averages of qualitative data, we must first introduce the concept of levels of measurement. In statistics, levels of measurement, or scales of measure, are types of data that arise in the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his typology in a 1946 Science article entitled “On the Theory of Scales of Measurement. ” In that article, Stevens claimed that all measurement in science was conducted using four different types of scales that he called “nominal”, “ordinal”, “interval” and “ratio”, unifying both qualitative (which are described by his “nominal” type) and quantitative (to a different degree, all the rest of his scales).

Nominal Scale

The nominal scale differentiates between items or subjects based only on their names and/or categories and other qualitative classifications they belong to. Examples include gender, nationality, ethnicity, language, genre, style, biological species, visual pattern, and form.

The mode, i.e. the most common item, is allowed as the measure of central tendency for the nominal type. On the other hand, the median, i.e. the middle-ranked item, makes no sense for the nominal type of data since ranking is not allowed for the nominal type.

Ordinal Scale

The ordinal scale allows for rank order (1st, 2nd, 3rd, et cetera) by which data can be sorted, but still does not allow for relative degree of difference between them. Examples include, on one hand, dichotomous data with dichotomous (or dichotomized) values such as “sick” versus “healthy” when measuring health, “guilty” versus “innocent” when making judgments in courts, or “wrong/false” versus “right/true” when measuring truth value. On the other hand, non-dichotomous data consisting of a spectrum of values is also included, such as “completely agree,” “mostly agree,” “mostly disagree,” and “completely disagree” when measuring opinion .

The median, i.e. middle-ranked, item is allowed as the measure of central tendency; however, the mean (or average) as the measure of central tendency is not allowed. The mode is also allowed.

In 1946, Stevens observed that psychological measurement, such as measurement of opinions, usually operates on ordinal scales; thus means and standard deviations have no validity, but they can be used to get ideas for how to improve operationalization of variables used in questionnaires.

4.1: Frequency Distributions for Quantitative Data

4.1.1: Guidelines for Plotting Frequency Distributions

Learning Objective

Define statistical frequency and illustrate how it can be depicted graphically.

Key Points

• Frequency distributions can be displayed in a table, histogram, line graph, dot plot, or a pie chart, just to name a few.
• A histogram is a graphical representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval.
• There is no “best” number of bins, and different bin sizes can reveal different features of the data.
• Frequency distributions can be displayed in a table, histogram, line graph, dot plot, or a pie chart, to just name a few.

Key Terms

frequency

number of times an event occurred in an experiment (absolute frequency)

histogram

a representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval

In statistics, the frequency (or absolute frequency) of an event is the number of times the event occurred in an experiment or study. These frequencies are often graphically represented in histograms. The relative frequency (or empirical probability) of an event refers to the absolute frequency normalized by the total number of events. The values of all events can be plotted to produce a frequency distribution.

A histogram is a graphical representation of tabulated frequencies , shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval. The height of a rectangle is also equal to the frequency density of the interval, i.e., the frequency divided by the width of the interval. The total area of the histogram is equal to the number of data. An example of the frequency distribution of letters of the alphabet in the English language is shown in the histogram in .

A histogram may also be normalized displaying relative frequencies. It then shows the proportion of cases that fall into each of several categories, with the total area equaling 1. The categories are usually specified as consecutive, non-overlapping intervals of a variable. The categories (intervals) must be adjacent, and often are chosen to be of the same size. The rectangles of a histogram are drawn so that they touch each other to indicate that the original variable is continuous.

There is no “best” number of bins, and different bin sizes can reveal different features of the data. Some theoreticians have attempted to determine an optimal number of bins, but these methods generally make strong assumptions about the shape of the distribution. Depending on the actual data distribution and the goals of the analysis, different bin widths may be appropriate, so experimentation is usually needed to determine an appropriate width.

4.1.2: Outliers

Learning Objective

Discuss outliers in terms of their causes and consequences, identification, and exclusion.

Key Points

• Outliers can occur by chance, by human error, or by equipment malfunction.
• Outliers may be indicative of a non-normal distribution, or they may just be natural deviations that occur in a large sample.
• Unless it can be ascertained that the deviation is not significant, it is not wise to ignore the presence of outliers.
• There is no rigid mathematical definition of what constitutes an outlier; thus, determining whether or not an observation is an outlier is ultimately a subjective experience.

Key Terms

skewed

Biased or distorted (pertaining to statistics or information).

standard deviation

a measure of how spread out data values are around the mean, defined as the square root of the variance

interquartile range

The difference between the first and third quartiles; a robust measure of sample dispersion.

What is an Outlier?

In statistics, an outlier is an observation that is numerically distant from the rest of the data. Outliers can occur by chance in any distribution, but they are often indicative either of measurement error or of the population having a heavy-tailed distribution. In the former case, one wishes to discard the outliers or use statistics that are robust against them. In the latter case, outliers indicate that the distribution is skewed and that one should be very cautious in using tools or intuitions that assume a normal distribution.

In most larger samplings of data, some data points will be further away from the sample mean than what is deemed reasonable. This can be due to incidental systematic error or flaws in the theory that generated an assumed family of probability distributions, or it may be that some observations are far from the center of the data. Outlier points can therefore indicate faulty data, erroneous procedures, or areas where a certain theory might not be valid. However, in large samples, a small number of outliers is to be expected, and they typically are not due to any anomalous condition.

Outliers, being the most extreme observations, may include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low. However, the sample maximum and minimum are not always outliers because they may not be unusually far from other observations.

Interpretations of statistics derived from data sets that include outliers may be misleading. For example, imagine that we calculate the average temperature of 10 objects in a room. Nine of them are between 20° and 25° Celsius, but an oven is at 175°C. In this case, the median of the data will be between 20° and 25°C, but the mean temperature will be between 35.5° and 40 °C. The median better reflects the temperature of a randomly sampled object than the mean; however, interpreting the mean as “a typical sample”, equivalent to the median, is incorrect. This case illustrates that outliers may be indicative of data points that belong to a different population than the rest of the sample set. Estimators capable of coping with outliers are said to be robust. The median is a robust statistic, while the mean is not.

Causes for Outliers

Outliers can have many anomalous causes. For example, a physical apparatus for taking measurements may have suffered a transient malfunction, or there may have been an error in data transmission or transcription. Outliers can also arise due to changes in system behavior, fraudulent behavior, human error, instrument error or simply through natural deviations in populations. A sample may have been contaminated with elements from outside the population being examined. Alternatively, an outlier could be the result of a flaw in the assumed theory, calling for further investigation by the researcher.

Unless it can be ascertained that the deviation is not significant, it is ill-advised to ignore the presence of outliers. Outliers that cannot be readily explained demand special attention.

Identifying Outliers

There is no rigid mathematical definition of what constitutes an outlier. Thus, determining whether or not an observation is an outlier is ultimately a subjective exercise. Model-based methods, which are commonly used for identification, assume that the data is from a normal distribution and identify observations which are deemed “unlikely” based on mean and standard deviation. Other methods flag observations based on measures such as the interquartile range (IQR). For example, some people use the
rule. This defines an outlier to be any observation that falls
below the first quartile or any observation that falls
above the third quartile.

Working With Outliers

Deletion of outlier data is a controversial practice frowned on by many scientists and science instructors. While mathematical criteria provide an objective and quantitative method for data rejection, they do not make the practice more scientifically or methodologically sound — especially in small sets or where a normal distribution cannot be assumed. Rejection of outliers is more acceptable in areas of practice where the underlying model of the process being measured and the usual distribution of measurement error are confidently known. An outlier resulting from an instrument reading error may be excluded, but it is desirable that the reading is at least verified.

Even when a normal distribution model is appropriate to the data being analyzed, outliers are expected for large sample sizes and should not automatically be discarded if that is the case. The application should use a classification algorithm that is robust to outliers to model data with naturally occurring outlier points. Additionally, the possibility should be considered that the underlying distribution of the data is not approximately normal, but rather skewed.

4.1.3: Relative Frequency Distributions

Learning Objective

Define relative frequency and construct a relative frequency distribution.

Key Points

• To find the relative frequencies, divide each frequency by the total number of data points in the sample.
• Relative frequencies can be written as fractions, percents, or decimals. The column should add up to 1 (or 100%).
• The only difference between a relative frequency distribution graph and a frequency distribution graph is that the vertical axis uses proportional or relative frequency rather than simple frequency.
• Cumulative relative frequency (also called an ogive) is the accumulation of the previous relative frequencies.

Key Terms

cumulative relative frequency

the accumulation of the previous relative frequencies

relative frequency

the fraction or proportion of times a value occurs

histogram

a representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval

What is a Relative Frequency Distribution?

A relative frequency is the fraction or proportion of times a value occurs. To find the relative frequencies, divide each frequency by the total number of data points in the sample. Relative frequencies can be written as fractions, percents, or decimals.

How to Construct a Relative Frequency Distribution

Constructing a relative frequency distribution is not that much different than from constructing a regular frequency distribution. The beginning process is the same, and the same guidelines must be used when creating classes for the data. Recall the following:

• Each data value should fit into one class only (classes are mutually exclusive).
• The classes should be of equal size.
• Classes should not be open-ended.
• Try to use between 5 and 20 classes.

Create the frequency distribution table, as you would normally. However, this time, you will need to add a third column. The first column should be labeled Class or Category. The second column should be labeled Frequency. The third column should be labeled Relative Frequency. Fill in your class limits in column one. Then, count the number of data points that fall in each class and write that number in column two.

Next, start to fill in the third column. The entries will be calculated by dividing the frequency of that class by the total number of data points. For example, suppose we have a frequency of 5 in one class, and there are a total of 50 data points. The relative frequency for that class would be calculated by the following: 

You can choose to write the relative frequency as a decimal (0.10), as a fraction (
), or as a percent (10%). Since we are dealing with proportions, the relative frequency column should add up to 1 (or 100%). It may be slightly off due to rounding.

Relative frequency distributions is often displayed in histograms and in frequency polygons. The only difference between a relative frequency distribution graph and a frequency distribution graph is that the vertical axis uses proportional or relative frequency rather than simple frequency.

Cumulative Relative Frequency Distributions

Just like we use cumulative frequency distributions when discussing simple frequency distributions, we often use cumulative frequency distributions when dealing with relative frequency as well. Cumulative relative frequency (also called an ogive) is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row.

4.1.4: Cumulative Frequency Distributions

Learning Objective

Define cumulative frequency and construct a cumulative frequency distribution.

Key Points

• To create a cumulative frequency distribution, start by creating a regular frequency distribution with one extra column added.
• To complete the cumulative frequency column, add all the frequencies at that class and all preceding classes.
• Cumulative frequency distributions are often displayed in histograms and in frequency polygons.

Key Terms

histogram

a representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval

frequency distribution

a representation, either in a graphical or tabular format, which displays the number of observations within a given interval

What is a Cumulative Frequency Distribution?

A cumulative frequency distribution is the sum of the class and all classes below it in a frequency distribution. Rather than displaying the frequencies from each class, a cumulative frequency distribution displays a running total of all the preceding frequencies.

How to Construct a Cumulative Frequency Distribution

Constructing a cumulative frequency distribution is not that much different than constructing a regular frequency distribution. The beginning process is the same, and the same guidelines must be used when creating classes for the data. Recall the following:

• Each data value should fit into one class only (classes are mutually exclusive).
• The classes should be of equal size.
• Classes should not be open-ended.
• Try to use between 5 and 20 classes.

Create the frequency distribution table, as you would normally. However, this time, you will need to add a third column. The first column should be labeled Class or Category. The second column should be labeled Frequency. The third column should be labeled Cumulative Frequency. Fill in your class limits in column one. Then, count the number of data points that falls in each class and write that number in column two.

Next, start to fill in the third column. The first entry will be the same as the first entry in the Frequency column. The second entry will be the sum of the first two entries in the Frequency column, the third entry will be the sum of the first three entries in the Frequency column, etc. The last entry in the Cumulative Frequency column should equal the number of total data points, if the math has been done correctly.

Graphical Displays of Cumulative Frequency Distributions

There are a number of ways in which cumulative frequency distributions can be displayed graphically. Histograms are common , as are frequency polygons . Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms, but are especially helpful in comparing sets of data.

4.1.5: Graphs for Quantitative Data

Learning Objective

Identify common plots used in statistical analysis.

Key Points

• Graphical procedures such as plots are used to gain insight into a data set in terms of testing assumptions, model selection, model validation, estimator selection, relationship identification, factor effect determination, or outlier detection.
• Statistical graphics give insight into aspects of the underlying structure of the data.
• Graphs can also be used to solve some mathematical equations, typically by finding where two plots intersect.

Key Terms

histogram

a representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval

plot

a graph or diagram drawn by hand or produced by a mechanical or electronic device

scatter plot

A type of display using Cartesian coordinates to display values for two variables for a set of data.

A plot is a graphical technique for representing a data set, usually as a graph showing the relationship between two or more variables. Graphs of functions are used in mathematics, sciences, engineering, technology, finance, and other areas where a visual representation of the relationship between variables would be useful. Graphs can also be used to read off the value of an unknown variable plotted as a function of a known one. Graphical procedures are also used to gain insight into a data set in terms of:

• testing assumptions,
• model selection,
• model validation,
• estimator selection,
• relationship identification,
• factor effect determination, or
• outlier detection.

Plots play an important role in statistics and data analysis. The procedures here can broadly be split into two parts: quantitative and graphical. Quantitative techniques are the set of statistical procedures that yield numeric or tabular output. Some examples of quantitative techniques include:

• hypothesis testing,
• analysis of variance,
• point estimates and confidence intervals, and
• least squares regression.

There are also many statistical tools generally referred to as graphical techniques which include:

• scatter plots ,
• histograms,
• probability plots,
• residual plots,
• box plots, and
• block plots.

Below are brief descriptions of some of the most common plots:

Scatter plot: This is a type of mathematical diagram using Cartesian coordinates to display values for two variables for a set of data. The data is displayed as a collection of points, each having the value of one variable determining the position on the horizontal axis and the value of the other variable determining the position on the vertical axis. This kind of plot is also called a scatter chart, scattergram, scatter diagram, or scatter graph.

Histogram: In statistics, a histogram is a graphical representation of the distribution of data. It is an estimate of the probability distribution of a continuous variable or can be used to plot the frequency of an event (number of times an event occurs) in an experiment or study.

Box plot: In descriptive statistics, a boxplot, also known as a box-and-whisker diagram, is a convenient way of graphically depicting groups of numerical data through their five-number summaries (the smallest observation, lower quartile (Q1), median (Q2), upper quartile (Q3), and largest observation). A boxplot may also indicate which observations, if any, might be considered outliers.

4.1.6: Typical Shapes

Learning Objective

Evaluate the shapes of symmetrical and asymmetrical frequency distributions.

Key Points

• A normal distribution is a symmetric distribution in which the mean and median are equal. Most data are clustered in the center.
• An asymmetrical distribution is said to be positively skewed (or skewed to the right) when the tail on the right side of the histogram is longer than the left side.
• An asymmetrical distribution is said to be negatively skewed (or skewed to the left) when the tail on the left side of the histogram is longer than the right side.
• Distributions can also be uni-modal, bi-modal, or multi-modal.

Key Terms

skewness

A measure of the asymmetry of the probability distribution of a real-valued random variable; is the third standardized moment, defined as where is the third moment about the mean and is the standard deviation.

empirical rule

That a normal distribution has 68% of its observations within one standard deviation of the mean, 95% within two, and 99.7% within three.

standard deviation

a measure of how spread out data values are around the mean, defined as the square root of the variance

Distribution Shapes

In statistics, distributions can take on a variety of shapes. Considerations of the shape of a distribution arise in statistical data analysis, where simple quantitative descriptive statistics and plotting techniques, such as histograms, can lead to the selection of a particular family of distributions for modelling purposes.

Symmetrical Distributions

In a symmetrical distribution, the two sides of the distribution are mirror images of each other. A normal distribution is an example of a truly symmetric distribution of data item values. When a histogram is constructed on values that are normally distributed, the shape of the columns form a symmetrical bell shape. This is why this distribution is also known as a “normal curve” or “bell curve. ” In a true normal distribution, the mean and median are equal, and they appear in the center of the curve. Also, there is only one mode, and most of the data are clustered around the center. The more extreme values on either side of the center become more rare as distance from the center increases. About 68% of values lie within one standard deviation (σ) away from the mean, about 95% of the values lie within two standard deviations, and about 99.7% lie within three standard deviations . This is known as the empirical rule or the 3-sigma rule.

Asymmetrical Distributions

In an asymmetrical distribution, the two sides will not be mirror images of each other. Skewness is the tendency for the values to be more frequent around the high or low ends of the x-axis. When a histogram is constructed for skewed data, it is possible to identify skewness by looking at the shape of the distribution.

A distribution is said to be positively skewed (or skewed to the right) when the tail on the right side of the histogram is longer than the left side. Most of the values tend to cluster toward the left side of the x-axis (i.e., the smaller values) with increasingly fewer values at the right side of the x-axis (i.e., the larger values). In this case, the median is less than the mean .

A distribution is said to be negatively skewed (or skewed to the left) when the tail on the left side of the histogram is longer than the right side. Most of the values tend to cluster toward the right side of the x-axis (i.e., the larger values), with increasingly less values on the left side of the x-axis (i.e., the smaller values). In this case, the median is greater than the mean .

When data are skewed, the median is usually a more appropriate measure of central tendency than the mean.

Other Distribution Shapes

A uni-modal distribution occurs if there is only one “peak” (or highest point) in the distribution, as seen previously in the normal distribution. This means there is one mode (a value that occurs more frequently than any other) for the data. A bi-modal distribution occurs when there are two modes. Multi-modal distributions with more than two modes are also possible.

4.1.7: Z-Scores and Location in a Distribution

Learning Objective

Define $z$-scores and demonstrate how they are converted from raw scores

Key Points

• A positive
  -score represents an observation above the mean, while a negative
  -score represents an observation below the mean.
• We obtain a
  -score through a conversion process known as standardizing or normalizing.
• 
  -scores are most frequently used to compare a sample to a standard normal deviate (standard normal distribution, with
   and
  ).
• While
  -scores can be defined without assumptions of normality, they can only be defined if one knows the population parameters.
• 
  -scores provide an assessment of how off-target a process is operating.

Key Terms

Student’s t-statistic

a ratio of the departure of an estimated parameter from its notional value and its standard error

z-score

The standardized value of observation $x$ from a distribution that has mean $\mu$ and standard deviation $\sigma$.

raw score

an original observation that has not been transformed to a $z$-score

A
-score is the signed number of standard deviations an observation is above the mean of a distribution. Thus, a positive
-score represents an observation above the mean, while a negative
-score represents an observation below the mean. We obtain a
-score through a conversion process known as standardizing or normalizing.

-scores are also called standard scores,
-values, normal scores or standardized variables. The use of “
” is because the normal distribution is also known as the “
distribution.”
-scores are most frequently used to compare a sample to a standard normal deviate (standard normal distribution, with
 and
).

While
-scores can be defined without assumptions of normality, they can only be defined if one knows the population parameters. If one only has a sample set, then the analogous computation with sample mean and sample standard deviation yields the Student’s
-statistic.

Calculation From a Raw Score

A raw score is an original datum, or observation, that has not been transformed. This may include, for example, the original result obtained by a student on a test (i.e., the number of correctly answered items) as opposed to that score after transformation to a standard score or percentile rank. The
-score, in turn, provides an assessment of how off-target a process is operating.

The conversion of a raw score,
, to a
-score can be performed using the following equation:

where
 is the mean of the population and
is the standard deviation of the population. The absolute value of
represents the distance between the raw score and the population mean in units of the standard deviation.
is negative when the raw score is below the mean and positive when the raw score is above the mean.

A key point is that calculating
requires the population mean and the population standard deviation, not the sample mean nor sample deviation. It requires knowing the population parameters, not the statistics of a sample drawn from the population of interest. However, in cases where it is impossible to measure every member of a population, the standard deviation may be estimated using a random sample.

3.1: The Histogram

3.1.1: Cross Tabulation

Learning Objective

Demonstrate how cross tabulation provides a basic picture of the interrelation between two variables and helps to find interactions between them.

Key Points

• Crosstabs are heavily used in survey research, business intelligence, engineering, and scientific research.
• Crosstabs provide a basic picture of the interrelation between two variables and can help find interactions between them.
• Most general-purpose statistical software programs are able to produce simple crosstabs.

Key Term

cross tabulation

a presentation of data in a tabular form to aid in identifying a relationship between variables

Cross tabulation (or crosstabs for short) is a statistical process that summarizes categorical data to create a contingency table. It is used heavily in survey research, business intelligence, engineering, and scientific research. Moreover, it provides a basic picture of the interrelation between two variables and can help find interactions between them.

In survey research (e.g., polling, market research), a “crosstab” is any table showing summary statistics. Commonly, crosstabs in survey research are combinations of multiple different tables. For example, combines multiple contingency tables and tables of averages.

Contingency Tables

A contingency table is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. A crucial problem of multivariate statistics is finding the direct dependence structure underlying the variables contained in high dimensional contingency tables. If some of the conditional independences are revealed, then even the storage of the data can be done in a smarter way. In order to do this, one can use information theory concepts, which gain the information only from the distribution of probability. Probability can be expressed easily from the contingency table by the relative frequencies.

As an example, suppose that we have two variables, sex (male or female) and handedness (right- or left-handed). Further suppose that 100 individuals are randomly sampled from a very large population as part of a study of sex differences in handedness. A contingency table can be created to display the numbers of individuals who are male and right-handed, male and left-handed, female and right-handed, and female and left-handed .

The numbers of the males, females, and right-and-left-handed individuals are called marginal totals. The grand total–i.e., the total number of individuals represented in the contingency table– is the number in the bottom right corner.

The table allows us to see at a glance that the proportion of men who are right-handed is about the same as the proportion of women who are right-handed, although the proportions are not identical. If the proportions of individuals in the different columns vary significantly between rows (or vice versa), we say that there is a contingency between the two variables. In other words, the two variables are not independent. If there is no contingency, we say that the two variables are independent.

Standard Components of a Crosstab

• Multiple columns – each column refers to a specific sub-group in the population (e.g., men). The columns are sometimes referred to as banner points or cuts (and the rows are sometimes referred to as stubs).
• Significance tests – typically, either column comparisons–which test for differences between columns and display these results using letters– or cell comparisons–which use color or arrows to identify a cell in a table that stands out in some way (as in the example above).
• Nets or netts – which are sub-totals.
• One or more of the following: percentages, row percentages, column percentages, indexes, or averages.
• Unweighted sample sizes (i.e., counts).

Most general-purpose statistical software programs are able to produce simple crosstabs. Creation of the standard crosstabs used in survey research, as shown above, is typically done using specialist crosstab software packages, such as:

1. New Age Media Systems (EzTab)
2. SAS
3. Quantum
4. Quanvert
5. SPSS Custom Tables
6. IBM SPSS Data Collection Model programs
7. Uncle
8. WinCross
9. Q
10. SurveyCraft
11. BIRT

3.1.2: Drawing a Histogram

Learning Objective

Outline the steps involved in creating a histogram.

Key Points

• There is no “best” number of bars, and different bar sizes may reveal different features of the data.
• A convenient starting point for the first interval is a lower value carried out to one more decimal place than the value with the most decimal places.
• To calculate the width of the intervals, subtract the starting point from the ending value and divide by the number of bars.

Key Term

histogram

a representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval

To construct a histogram, one must first decide how many bars or intervals (also called classes) are needed to represent the data. Many histograms consist of between 5 and 15 bars, or classes. One must choose a starting point for the first interval, which must be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places.

For example, if the value with the most decimal places is 6.1, and this is the smallest value, a convenient starting point is 6.05 (
). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is 1.495 (
). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is 0.9995 (
). If all the data happen to be integers and the smallest value is 2, then a convenient starting point is 1.5 (
). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary.

Consider the following data, which are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data since height is measured.

60; 60.5; 61; 61; 61.5; 63.5; 63.5; 63.5; 64; 64; 64; 64; 64; 64; 64; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69.5; 69.5; 69.5; 69.5; 69.5; 70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71; 72; 72; 72; 72.5; 72.5; 73; 73.5; 74

The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, and so on are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point. The starting point, then, is 59.95.

The largest value is 74.
is the ending value.

Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Note that there is no “best” number of bars, and different bar sizes can reveal different features of the data. Some theoreticians have attempted to determine an optimal number of bars, but these methods generally make strong assumptions about the shape of the distribution. Depending on the actual data distribution and the goals of the analysis, different bar widths may be appropriate, so experimentation is usually needed to determine an appropriate width.

Suppose, in our example, we choose 8 bars. The bar width will be as follows:

We will round up to 2 and make each bar or class interval 2 units wide. Rounding up to 2 is one way to prevent a value from falling on a boundary. The boundaries are:

59.95, 61.95, 63.95, 65.95, 67.95, 69.95, 71.95, 73.95, 75.95

So that there are 2 units between each boundary.

The heights 60 through 61.5 inches are in the interval 59.95 – 61.95. The heights that are 63.5 are in the interval 61.95 – 63.95. The heights that are 64 through 64.5 are in the interval 63.95 – 65.95. The heights 66 through 67.5 are in the interval 65.95 – 67.95. The heights 68 through 69.5 are in the interval 67.95 – 69.95. The heights 70 through 71 are in the interval 69.95 – 71.95. The heights 72 through 73.5 are in the interval 71.95 – 73.95. The height 74 is in the interval 73.95 – 75.95.

displays the heights on the x-axis and relative frequency on the y-axis.

3.1.3: Recognizing and Using a Histogram

Learning Objective

Indicate how frequency and probability distributions are represented by histograms.

Key Points

• First introduced by Karl Pearson, a histogram is an estimate of the probability distribution of a continuous variable.
• If the distribution of
  is continuous, then
  is called a continuous random variable and, therefore, has a continuous probability distribution.
• An advantage of a histogram is that it can readily display large data sets (a rule of thumb is to use a histogram when the data set consists of 100 values or more).

Key Terms

frequency

number of times an event occurred in an experiment (absolute frequency)

histogram

a representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval

probability distribution

A function of a discrete random variable yielding the probability that the variable will have a given value.

A histogram is a graphical representation of the distribution of data. More specifically, a histogram is a representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval. First introduced by Karl Pearson, it is an estimate of the probability distribution of a continuous variable.

A histogram has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency. The graph will have the same shape with either label. An advantage of a histogram is that it can readily display large data sets (a rule of thumb is to use a histogram when the data set consists of 100 values or more). The histogram can also give you the shape, the center, and the spread of the data.

The categories of a histogram are usually specified as consecutive, non-overlapping intervals of a variable. The categories (intervals) must be adjacent and often are chosen to be of the same size. The rectangles of a histogram are drawn so that they touch each other to indicate that the original variable is continuous.

Frequency and Probability Distributions

In statistical terms, the frequency of an event is the number of times the event occurred in an experiment or study. The relative frequency (or empirical probability) of an event refers to the absolute frequency normalized by the total number of events:

Put more simply, the relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample.

The height of a rectangle in a histogram is equal to the frequency density of the interval, i.e., the frequency divided by the width of the interval. A histogram may also be normalized displaying relative frequencies. It then shows the proportion of cases that fall into each of several categories, with the total area equaling one.

As mentioned, a histogram is an estimate of the probability distribution of a continuous variable. To define probability distributions for the simplest cases, one needs to distinguish between discrete and continuous random variables. In the discrete case, one can easily assign a probability to each possible value. For example, when throwing a die, each of the six values 1 to 6 has the probability 1/6. In contrast, when a random variable takes values from a continuum, probabilities are nonzero only if they refer to finite intervals. For example, in quality control one might demand that the probability of a “500 g” package containing between 490 g and 510 g should be no less than 98%.

Intuitively, a continuous random variable is the one which can take a continuous range of values — as opposed to a discrete distribution, where the set of possible values for the random variable is, at most, countable. If the distribution of
is continuous, then
is called a continuous random variable and, therefore, has a continuous probability distribution. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.

3.1.4: The Density Scale

Learning Objective

Describe how density estimation is used as a tool in the construction of a histogram.

Key Points

• The unobservable density function is thought of as the density according to which a large population is distributed. The data are usually thought of as a random sample from that population.
• A probability density function, or density of a continuous random variable, is a function that describes the relative likelihood for a random variable to take on a given value.
• Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel.

Key Terms

quartile

any of the three points that divide an ordered distribution into four parts, each containing a quarter of the population

density

the probability that an event will occur, as a function of some observed variable

interquartile range

The difference between the first and third quartiles; a robust measure of sample dispersion.

Density Estimation

Histograms are used to plot the density of data, and are often a useful tool for density estimation. Density estimation is the construction of an estimate based on observed data of an unobservable, underlying probability density function. The unobservable density function is thought of as the density according to which a large population is distributed. The data are usually thought of as a random sample from that population.

A probability density function, or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value. The probability for the random variable to fall within a particular region is given by the integral of this variable’s density over the region .

The above image depicts a probability density function graph against a box plot. A box plot is a convenient way of graphically depicting groups of numerical data through their quartiles. The spacings between the different parts of the box help indicate the degree of dispersion (spread) and skewness in the data and to identify outliers. In addition to the points themselves, box plots allow one to visually estimate the interquartile range.

A range of data clustering techniques are used as approaches to density estimation, with the most basic form being a rescaled histogram.

Kernel Density Estimation

Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel. To see this, we compare the construction of histogram and kernel density estimators using these 6 data points:

,
,
,
,
, 

For the histogram, first the horizontal axis is divided into sub-intervals, or bins, which cover the range of the data. In this case, we have 6 bins, each having a width of 2. Whenever a data point falls inside this interval, we place a box of height
. If more than one data point falls inside the same bin, we stack the boxes on top of each other .

For the kernel density estimate, we place a normal kernel with variance 2.25 (indicated by the red dashed lines) on each of the data points
. The kernels are summed to make the kernel density estimate (the solid blue curve). Kernel density estimates converge faster to the true underlying density for continuous random variables thus accounting for their smoothness compared to the discreteness of the histogram.

3.1.5: Types of Variables

Learning Objective

Distinguish between quantitative and categorical, continuous and discrete, and ordinal and nominal variables.

Key Points

• Numeric (quantitative) variables have values that describe a measurable quantity as a number, like “how many” or “how much”.
• A continuous variable is an observation that can take any value between a certain set of real numbers.
• A discrete variable is an observation that can take a value based on a count from a set of distinct whole values.
• Categorical variables have values that describe a “quality” or “characteristic” of a data unit, like “what type” or “which category”.
• An ordinal variable is an observation that can take a value that can be logically ordered or ranked.
• A nominal variable is an observation that can take a value that is not able to be organized in a logical sequence.

Key Terms

continuous variable

a variable that has a continuous distribution function, such as temperature

discrete variable

a variable that takes values from a finite or countable set, such as the number of legs of an animal

variable

a quantity that may assume any one of a set of values

What Is a Variable?

A variable is any characteristic, number, or quantity that can be measured or counted. A variable may also be called a data item. Age, sex, business income and expenses, country of birth, capital expenditure, class grades, eye colour and vehicle type are examples of variables. Variables are so-named because their value may vary between data units in a population and may change in value over time.

What Are the Types of Variables?

There are different ways variables can be described according to the ways they can be studied, measured, and presented. Numeric variables have values that describe a measurable quantity as a number, like “how many” or “how much. ” Therefore, numeric variables are quantitative variables.

Numeric variables may be further described as either continuous or discrete. A continuous variable is a numeric variable. Observations can take any value between a certain set of real numbers. The value given to an observation for a continuous variable can include values as small as the instrument of measurement allows. Examples of continuous variables include height, time, age, and temperature.

A discrete variable is a numeric variable. Observations can take a value based on a count from a set of distinct whole values. A discrete variable cannot take the value of a fraction between one value and the next closest value. Examples of discrete variables include the number of registered cars, number of business locations, and number of children in a family, all of of which measured as whole units (i.e., 1, 2, 3 cars).

Categorical variables have values that describe a “quality” or “characteristic” of a data unit, like “what type” or “which category. ” Categorical variables fall into mutually exclusive (in one category or in another) and exhaustive (include all possible options) categories. Therefore, categorical variables are qualitative variables and tend to be represented by a non-numeric value.

Categorical variables may be further described as ordinal or nominal. An ordinal variable is a categorical variable. Observations can take a value that can be logically ordered or ranked. The categories associated with ordinal variables can be ranked higher or lower than another, but do not necessarily establish a numeric difference between each category. Examples of ordinal categorical variables include academic grades (i.e., A, B, C), clothing size (i.e., small, medium, large, extra large) and attitudes (i.e., strongly agree, agree, disagree, strongly disagree).

A nominal variable is a categorical variable. Observations can take a value that is not able to be organized in a logical sequence. Examples of nominal categorical variables include sex, business type, eye colour, religion and brand.

3.1.6: Controlling for a Variable

Learning Objective

Discuss how controlling for a variable leads to more reliable visualizations of probability distributions.

Key Points

• Variables refer to measurable attributes, as these typically vary over time or between individuals.
• Temperature is an example of a continuous variable, while the number of legs of an animal is an example of a discrete variable.
• In causal models, a distinction is made between “independent variables” and “dependent variables,” the latter being expected to vary in value in response to changes in the former.
• While independent variables can refer to quantities and qualities that are under experimental control, they can also include extraneous factors that influence results in a confusing or undesired manner.
• The essence of controlling is to ensure that comparisons between the control group and the experimental group are only made for groups or subgroups for which the variable to be controlled has the same statistical distribution.

Key Terms

correlation

One of the several measures of the linear statistical relationship between two random variables, indicating both the strength and direction of the relationship.

control

a separate group or subject in an experiment against which the results are compared where the primary variable is low or nonexistence

variable

a quantity that may assume any one of a set of values

Histograms help us to visualize the distribution of data and estimate the probability distribution of a continuous variable. In order for us to create reliable visualizations of these distributions, we must be able to procure reliable results for the data during experimentation. A method that significantly contributes to our success in this matter is the controlling of variables.

Defining Variables

In statistics, variables refer to measurable attributes, as these typically vary over time or between individuals. Variables can be discrete (taking values from a finite or countable set), continuous (having a continuous distribution function), or neither. For instance, temperature is a continuous variable, while the number of legs of an animal is a discrete variable.

In causal models, a distinction is made between “independent variables” and “dependent variables,” the latter being expected to vary in value in response to changes in the former. In other words, an independent variable is presumed to potentially affect a dependent one. In experiments, independent variables include factors that can be altered or chosen by the researcher independent of other factors.

There are also quasi-independent variables, which are used by researchers to group things without affecting the variable itself. For example, to separate people into groups by their sex does not change whether they are male or female. Also, a researcher may separate people, arbitrarily, on the amount of coffee they drank before beginning an experiment.

While independent variables can refer to quantities and qualities that are under experimental control, they can also include extraneous factors that influence results in a confusing or undesired manner. In statistics the technique to work this out is called correlation.

Controlling Variables

In a scientific experiment measuring the effect of one or more independent variables on a dependent variable, controlling for a variable is a method of reducing the confounding effect of variations in a third variable that may also affect the value of the dependent variable. For example, in an experiment to determine the effect of nutrition (the independent variable) on organism growth (the dependent variable), the age of the organism (the third variable) needs to be controlled for, since the effect may also depend on the age of an individual organism.

The essence of the method is to ensure that comparisons between the control group and the experimental group are only made for groups or subgroups for which the variable to be controlled has the same statistical distribution. A common way to achieve this is to partition the groups into subgroups whose members have (nearly) the same value for the controlled variable.

Controlling for a variable is also a term used in statistical data analysis when inferences may need to be made for the relationships within one set of variables, given that some of these relationships may spuriously reflect relationships to variables in another set. This is broadly equivalent to conditioning on the variables in the second set. Such analyses may be described as “controlling for variable
” or “controlling for the variations in
“. Controlling, in this sense, is performed by including in the experiment not only the explanatory variables of interest but also the extraneous variables. The failure to do so results in omitted-variable bias.

3.1.7: Selective Breeding

Learning Objective

Illustrate how controlled experiments have allowed human beings to selectively breed domesticated plants and animals.

Key Points

• Unwittingly, humans have carried out evolution experiments for as long as they have been domesticating plants and animals.
• More recently, evolutionary biologists have realized that the key to successful experimentation lies in extensive parallel replication of evolving lineages as well as a larger number of generations of selection.
• Because of the large number of generations required for adaptation to occur, evolution experiments are typically carried out with microorganisms such as bacteria, yeast, or viruses.

Key Terms

breeding

the process through which propagation, growth, or development occurs

evolution

a gradual directional change, especially one leading to a more advanced or complex form; growth; development

stochastic

random; randomly determined

Experimental Evolution and Selective Breeding

Experimental evolution is a field in evolutionary and experimental biology that is concerned with testing hypotheses and theories of evolution by using controlled experiments. Evolution may be observed in the laboratory as populations adapt to new environmental conditions and/or change by such stochastic processes as random genetic drift.

With modern molecular tools, it is possible to pinpoint the mutations that selection acts upon, what brought about the adaptations, and to find out how exactly these mutations work. Because of the large number of generations required for adaptation to occur, evolution experiments are typically carried out with microorganisms such as bacteria, yeast, or viruses.

History of Selective Breeding

Unwittingly, humans have carried out evolution experiments for as long as they have been domesticating plants and animals. Selective breeding of plants and animals has led to varieties that differ dramatically from their original wild-type ancestors. Examples are the cabbage varieties, maize, or the large number of different dog breeds .

One of the first to carry out a controlled evolution experiment was William Dallinger. In the late 19^th century, he cultivated small unicellular organisms in a custom-built incubator over a time period of seven years (1880–1886). Dallinger slowly increased the temperature of the incubator from an initial 60 °F up to 158 °F. The early cultures had shown clear signs of distress at a temperature of 73 °F, and were certainly not capable of surviving at 158 °F. The organisms Dallinger had in his incubator at the end of the experiment, on the other hand, were perfectly fine at 158 °F. However, these organisms would no longer grow at the initial 60 °F. Dallinger concluded that he had found evidence for Darwinian adaptation in his incubator, and that the organisms had adapted to live in a high-temperature environment .

More recently, evolutionary biologists have realized that the key to successful experimentation lies in extensive parallel replication of evolving lineages as well as a larger number of generations of selection. For example, on February 15, 1988, Richard Lenski started a long-term evolution experiment with the bacterium E. coli. The experiment continues to this day, and is by now probably the largest controlled evolution experiment ever undertaken. Since the inception of the experiment, the bacteria have grown for more than 50,000 generations.

2.1: Observational Studies

2.1.1: What are Observational Studies?

Learning Objective

Identify situations in which observational studies are necessary and the challenges that arise in their interpretation.

Key Points

• An observational study is in contrast with experiments, such as randomized controlled trials, where each subject is randomly assigned to a treated group or a control group.
• Variables may be uncontrollable because 1) a randomized experiment would violate ethical standards, 2) the investigator may simply lack the requisite influence, or 3) a randomized experiment may be impractical.
• Observational studies can never identify causal relationships because even though two variables are related both might be caused by a third, unseen, variable.
• A major challenge in conducting observational studies is to draw inferences that are acceptably free from influences by overt biases, as well as to assess the influence of potential hidden biases.
• A major challenge in conducting observational studies is to draw inferences that are acceptably free from influences by overt biases, as well as to assess the influence of potential hidden biases.

Key Terms

causality

the relationship between an event (the cause) and a second event (the effect), where the second event is understood as a consequence of the first

observational study

a study drawing inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator

A common goal in statistical research is to investigate causality, which is the relationship between an event (the cause) and a second event (the effect), where the second event is understood as a consequence of the first. There are two major types of causal statistical studies: experimental studies and observational studies. An observational study draws inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator. This is in contrast with experiments, such as randomized controlled trials, where each subject is randomly assigned to a treated group or a control group. In other words, observational studies have no independent variables — nothing is manipulated by the experimenter. Rather, observations have the equivalent of two dependent variables.

In an observational study, the assignment of treatments may be beyond the control of the investigator for a variety of reasons:

1. A randomized experiment would violate ethical standards: Suppose one wanted to investigate the abortion – breast cancer hypothesis, which postulates a causal link between induced abortion and the incidence of breast cancer. In a hypothetical controlled experiment, one would start with a large subject pool of pregnant women and divide them randomly into a treatment group (receiving induced abortions) and a control group (bearing children), and then conduct regular cancer screenings for women from both groups. Needless to say, such an experiment would run counter to common ethical principles. The published studies investigating the abortion–breast cancer hypothesis generally start with a group of women who already have received abortions. Membership in this “treated” group is not controlled by the investigator: the group is formed after the “treatment” has been assigned.
2. The investigator may simply lack the requisite influence: Suppose a scientist wants to study the public health effects of a community-wide ban on smoking in public indoor areas. In a controlled experiment, the investigator would randomly pick a set of communities to be in the treatment group. However, it is typically up to each community and/or its legislature to enact a smoking ban. The investigator can be expected to lack the political power to cause precisely those communities in the randomly selected treatment group to pass a smoking ban. In an observational study, the investigator would typically start with a treatment group consisting of those communities where a smoking ban is already in effect.
3. A randomized experiment may be impractical: Suppose a researcher wants to study the suspected link between a certain medication and a very rare group of symptoms arising as a side effect. Setting aside any ethical considerations, a randomized experiment would be impractical because of the rarity of the effect. There may not be a subject pool large enough for the symptoms to be observed in at least one treated subject. An observational study would typically start with a group of symptomatic subjects and work backwards to find those who were given the medication and later developed the symptoms

Usefulness and Reliability of Observational Studies

Observational studies can never identify causal relationships because even though two variables are related both might be caused by a third, unseen, variable. Since the underlying laws of nature are assumed to be causal laws, observational findings are generally regarded as less compelling than experimental findings.

Observational studies can, however:

1. Provide information on “real world” use and practice
2. Detect signals about the benefits and risks of the use of practices in the general population
3. Help formulate hypotheses to be tested in subsequent experiments
4. Provide part of the community-level data needed to design more informative pragmatic clinical trials
5. Inform clinical practice

A major challenge in conducting observational studies is to draw inferences that are acceptably free from influences by overt biases, as well as to assess the influence of potential hidden biases.

2.1.2: The Clofibrate Trial

Learning Objective

Outline how the use of placebos in controlled experiments leads to more reliable results.

Key Points

• Clofibrate was one of four lipid-modifying drugs tested in an observational study known as the Coronary Drug Project.
• Placebo-controlled studies are a way of testing a medical therapy in which, in addition to a group of subjects that receives the treatment to be evaluated, a separate control group receives a sham “placebo” treatment which is specifically designed to have no real effect.
• The purpose of the placebo group is to account for the placebo effect — that is, effects from treatment that do not depend on the treatment itself.
• Appropriate use of a placebo in a clinical trial often requires, or at least benefits from, a double-blind study design, which means that neither the experimenters nor the subjects know which subjects are in the “test group” and which are in the “control group. “.
• The use of placebos is a standard control component of most clinical trials which attempt to make some sort of quantitative assessment of the efficacy of medicinal drugs or treatments.

Key Terms

regression to the mean

the phenomenon by which extreme examples from any set of data are likely to be followed by examples which are less extreme; a tendency towards the average of any sample

placebo

an inactive substance or preparation used as a control in an experiment or test to determine the effectiveness of a medicinal drug

placebo effect

the tendency of any medication or treatment, even an inert or ineffective one, to exhibit results simply because the recipient believes that it will work

Clofibrate (tradename Atromid-S) is an organic compound that is marketed as a fibrate. It is a lipid-lowering agent used for controlling the high cholesterol and triacylglyceride level in the blood. Clofibrate was one of four lipid-modifying drugs tested in an observational study known as the Coronary Drug Project. Also known as the World Health Organization Cooperative Trial on Primary Prevention of Ischaemic Heart Disease, the study was a randomized, multi-center, double-blind, placebo-controlled trial that was intended to study the safety and effectiveness of drugs for long-term treatment of coronary heart disease in men.

Placebo-Controlled Observational Studies

Placebo-controlled studies are a way of testing a medical therapy in which, in addition to a group of subjects that receives the treatment to be evaluated, a separate control group receives a sham “placebo” treatment which is specifically designed to have no real effect. Placebos are most commonly used in blinded trials, where subjects do not know whether they are receiving real or placebo treatment.

The purpose of the placebo group is to account for the placebo effect — that is, effects from treatment that do not depend on the treatment itself. Such factors include knowing one is receiving a treatment, attention from health care professionals, and the expectations of a treatment’s effectiveness by those running the research study. Without a placebo group to compare against, it is not possible to know whether the treatment itself had any effect.

Appropriate use of a placebo in a clinical trial often requires, or at least benefits from, a double-blind study design, which means that neither the experimenters nor the subjects know which subjects are in the “test group” and which are in the “control group. ” This creates a problem in creating placebos that can be mistaken for active treatments. Therefore, it can be necessary to use a psychoactive placebo, a drug that produces physiological effects that encourage the belief in the control groups that they have received an active drug.

Patients frequently show improvement even when given a sham or “fake” treatment. Such intentionally inert placebo treatments can take many forms, such as a pill containing only sugar, a surgery where nothing is actually done, or a medical device (such as ultrasound) that is not actually turned on. Also, due to the body’s natural healing ability and statistical effects such as regression to the mean, many patients will get better even when given no treatment at all. Thus, the relevant question when assessing a treatment is not “does the treatment work? ” but “does the treatment work better than a placebo treatment, or no treatment at all? “

Therefore, the use of placebos is a standard control component of most clinical trials which attempt to make some sort of quantitative assessment of the efficacy of medicinal drugs or treatments.

Results of The Coronary Drug Project

Those in the placebo group who adhered to the placebo treatment (took the placebo regularly as instructed) showed nearly half the mortality rate as those who were not adherent. A similar study of women found survival was nearly 2.5 times greater for those who adhered to their placebo. This apparent placebo effect may have occurred because:

1. Adhering to the protocol had a psychological effect, i.e. genuine placebo effect.
2. People who were already healthier were more able or more inclined to follow the protocol.
3. Compliant people were more diligent and health-conscious in all aspects of their lives.

The Coronary Drug Project found that subjects using clofibrate to lower serum cholesterol observed excess mortality in the clofibrate-treated group despite successful cholesterol lowering (47% more deaths during treatment with clofibrate and 5% after treatment with clofibrate) than the non-treated high cholesterol group. These deaths were due to a wide variety of causes other than heart disease, and remain “unexplained”.

Clofibrate was discontinued in 2002 due to adverse affects.

2.1.3: Confounding

Learning Objective

Break down why confounding variables may lead to bias and spurious relationships and what can be done to avoid these phenomenons.

Key Points

• A perceived relationship between an independent variable and a dependent variable that has been misestimated due to the failure to account for a confounding factor is termed a spurious relationship.
• Confounding by indication – the most important limitation of observational studies – occurs when prognostic factors cause bias, such as biased estimates of treatment effects in medical trials.
• Confounding variables may also be categorised according to their source: such as operational confounds, procedural confounds or person confounds.
• A reduction in the potential for the occurrence and effect of confounding factors can be obtained by increasing the types and numbers of comparisons performed in an analysis.
• Moreover, depending on the type of study design in place, there are various ways to modify that design to actively exclude or control confounding variables.

Key Terms

peer review

the scholarly process whereby manuscripts intended to be published in an academic journal are reviewed by independent researchers (referees) to evaluate the contribution, i.e. the importance, novelty and accuracy of the manuscript’s contents

placebo effect

the tendency of any medication or treatment, even an inert or ineffective one, to exhibit results simply because the recipient believes that it will work

prognostic

a sign by which a future event may be known or foretold

confounding variable

an extraneous variable in a statistical model that correlates (positively or negatively) with both the dependent variable and the independent variable

Example

• In risk assessments, factors such as age, gender, and educational levels often have impact on health status and so should be controlled. Beyond these factors, researchers may not consider or have access to data on other causal factors. An example is on the study of smoking tobacco on human health. Smoking, drinking alcohol, and diet are lifestyle activities that are related. A risk assessment that looks at the effects of smoking but does not control for alcohol consumption or diet may overestimate the risk of smoking. Smoking and confounding are reviewed in occupational risk assessments such as the safety of coal mining. When there is not a large sample population of non-smokers or non-drinkers in a particular occupation, the risk assessment may be biased towards finding a negative effect on health.

Confounding Variables

A confounding variable is an extraneous variable in a statistical model that correlates (positively or negatively) with both the dependent variable and the independent variable. A perceived relationship between an independent variable and a dependent variable that has been misestimated due to the failure to account for a confounding factor is termed a spurious relationship, and the presence of misestimation for this reason is termed omitted-variable bias.

As an example, suppose that there is a statistical relationship between ice cream consumption and number of drowning deaths for a given period. These two variables have a positive correlation with each other. An individual might attempt to explain this correlation by inferring a causal relationship between the two variables (either that ice cream causes drowning, or that drowning causes ice cream consumption). However, a more likely explanation is that the relationship between ice cream consumption and drowning is spurious and that a third, confounding, variable (the season) influences both variables: during the summer, warmer temperatures lead to increased ice cream consumption as well as more people swimming and, thus, more drowning deaths.

Types of Confounding

Confounding by indication has been described as the most important limitation of observational studies. Confounding by indication occurs when prognostic factors cause bias, such as biased estimates of treatment effects in medical trials. Controlling for known prognostic factors may reduce this problem, but it is always possible that a forgotten or unknown factor was not included or that factors interact complexly. Randomized trials tend to reduce the effects of confounding by indication due to random assignment.

Confounding variables may also be categorised according to their source:

• The choice of measurement instrument (operational confound) – This type of confound occurs when a measure designed to assess a particular construct inadvertently measures something else as well.
• Situational characteristics (procedural confound) – This type of confound occurs when the researcher mistakenly allows another variable to change along with the manipulated independent variable.
• Inter-individual differences (person confound) – This type of confound occurs when two or more groups of units are analyzed together (e.g., workers from different occupations) despite varying according to one or more other (observed or unobserved) characteristics (e.g., gender).

Decreasing the Potential for Confounding

A reduction in the potential for the occurrence and effect of confounding factors can be obtained by increasing the types and numbers of comparisons performed in an analysis. If a relationship holds among different subgroups of analyzed units, confounding may be less likely. That said, if measures or manipulations of core constructs are confounded (i.e., operational or procedural confounds exist), subgroup analysis may not reveal problems in the analysis.

Peer review is a process that can assist in reducing instances of confounding, either before study implementation or after analysis has occurred. Similarly, study replication can test for the robustness of findings from one study under alternative testing conditions or alternative analyses (e.g., controlling for potential confounds not identified in the initial study). Also, confounding effects may be less likely to occur and act similarly at multiple times and locations.

Moreover, depending on the type of study design in place, there are various ways to modify that design to actively exclude or control confounding variables:

1. Case-control studies assign confounders to both groups, cases and controls, equally. In case-control studies, matched variables most often are age and sex.
2. In cohort studies, a degree of matching is also possible, and it is often done by only admitting certain age groups or a certain sex into the study population. this creates a cohort of people who share similar characteristics; thus, all cohorts are comparable in regard to the possible confounding variable.
3. Double blinding conceals the experiment group membership of the participants from the trial population and the observers. By preventing the participants from knowing if they are receiving treatment or not, the placebo effect should be the same for the control and treatment groups. By preventing the observers from knowing of their membership, there should be no bias from researchers treating the groups differently or from interpreting the outcomes differently.
4. A randomized controlled trial is a method where the study population is divided randomly in order to mitigate the chances of self-selection by participants or bias by the study designers. Before the experiment begins, the testers will assign the members of the participant pool to their groups (control, intervention, parallel) using a randomization process such as the use of a random number generator.

2.1.4: Sex Bias in Graduate Admissions

Learning Objective

Illustrate how the phenomenon of confounding can be seen in practice via Simpson’s Paradox.

Key Points

• A study conducted in the aftermath of a law suit filed against the University of California, Berkeley showed that men applying were more likely than women to be admitted.
• Examination of the aggregate data on admissions showed a blatant, if easily misunderstood, pattern of gender discrimination against applicants.
• When examining the individual departments, it appeared that no department was significantly biased against women.
• The study concluded that women tended to apply to competitive departments with low rates of admission even among qualified applicants, whereas men tended to apply to less-competitive departments with high rates of admission among the qualified applicants.
• Simpson’s Paradox is a paradox in which a trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data.

Key Terms

partition

a part of something that had been divided, each of its results

Simpson’s paradox

a paradox in which a trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data

aggregate

a mass, assemblage, or sum of particulars; something consisting of elements but considered as a whole

Women have traditionally had limited access to higher education. Moreover, when women began to be admitted to higher education, they were encouraged to major in less-intellectual subjects. For example, the study of English literature in American and British colleges and universities was instituted as a field considered suitable to women’s “lesser intellects”.

However, since 1991 the proportion of women enrolled in college in the U.S. has exceeded the enrollment rate for men, and that gap has widened over time. As of 2007, women made up the majority — 54 percent — of the 10.8 million college students enrolled in the U.S.

This has not negated the fact that gender bias exists in higher education. Women tend to score lower on graduate admissions exams, such as the Graduate Record Exam (GRE) and the Graduate Management Admissions Test (GMAT). Representatives of the companies that publish these tests have hypothesized that greater number of female applicants taking these tests pull down women’s average scores. However, statistical research proves this theory wrong. Controlling for the number of people taking the test does not account for the scoring gap.

Sex Bias at the University of California, Berkeley

On February 7, 1975, a study was published in the journal Science by P.J. Bickel, E.A. Hammel, and J.W. O’Connell entitled “Sex Bias in Graduate Admissions: Data from Berkeley. ” This study was conducted in the aftermath of a law suit filed against the University, citing admission figures for the fall of 1973, which showed that men applying were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to chance.

Examination of the aggregate data on admissions showed a blatant, if easily misunderstood, pattern of gender discrimination against applicants.

Aggregate Data:

Men: 8,442 applicants – 44% admitted

Women: 4,321 applicants – 35% admitted

When examining the individual departments, it appeared that no department was significantly biased against women. In fact, most departments had a small but statistically significant bias in favor of women. The data from the six largest departments are listed below.

Department A

Men: 825 applicants – 62% admitted

Women: 108 applicants – 82% admitted

Department B

Men: 560 applicants – 63% admitted

Women: 25 applicants – 68% admitted

Department C

Men: 325 applicants – 37% admitted

Women: 593 applicants – 34% admitted

Department D

Men: 417 applicants – 33% admitted

Women: 375 applicants – 35% admitted

Department E

Men: 191 applicants – 28% admitted

Women: 393 applicants – 24% admitted

Department F

Men: 272 applicants – 6% admitted

Women: 341 applicants – 7% admitted

The research paper by Bickel et al. concluded that women tended to apply to competitive departments with low rates of admission even among qualified applicants (such as in the English Department), whereas men tended to apply to less-competitive departments with high rates of admission among the qualified applicants (such as in engineering and chemistry). The study also concluded that the graduate departments that were easier to enter at the University, at the time, tended to be those that required more undergraduate preparation in mathematics. Therefore, the admission bias seemed to stem from courses previously taken.

Confounding Variables and Simpson’s Paradox

The above study is one of the best known real life examples of an experiment suffering from a confounding variable. In this particular case, we can see an occurrence of Simpson’s Paradox . Simpson’s Paradox is a paradox in which a trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data. This result is often encountered in social-science and medical-science statistics, and is particularly confounding when frequency data are unduly given causal interpretations.

The practical significance of Simpson’s paradox surfaces in decision making situations where it poses the following dilemma: Which data should we consult in choosing an action, the aggregated or the partitioned? The answer seems to be that one should sometimes follow the partitioned and sometimes the aggregated data, depending on the story behind the data; with each story dictating its own choice.

As to why and how a story, not data, should dictate choices, the answer is that it is the story which encodes the causal relationships among the variables. Once we extract these relationships we can test algorithmically whether a given partition, representing confounding variables, gives the correct answer.

1.1: Overview

1.1.1: Collecting and Measuring Data

Learning Objective

Distinguish between the nominal, ordinal, interval and ratio methods of data measurement.

Key Points

• Ratio measurements provide the greatest flexibility in statistical methods that can be used for analyzing the data.
• Interval data allows for the degree of difference between items, but not the ratio between them.
• Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values.
• Variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, they are often grouped together as categorical variables.
• Ratio and interval measurements are grouped together as quantitative variables.
• Nominal measurements have no meaningful rank order among values.

Key Terms

sampling

the process or technique of obtaining a representative sample

population

a group of units (persons, objects, or other items) enumerated in a census or from which a sample is drawn

Example

• An example of an observational study is one that explores the correlation between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then performs statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers, perhaps through a case-control study, and then look for the number of cases of lung cancer in each group.

There are four main levels of measurement used in statistics: nominal, ordinal, interval, and ratio. Each of these have different degrees of usefulness in statistical research. Data is collected about a population by random sampling .

Nominal measurements have no meaningful rank order among values. Nominal data differentiates between items or subjects based only on qualitative classifications they belong to. Examples include gender, nationality, ethnicity, language, genre, style, biological species, visual pattern, etc.

Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values. Ordinal data allows for rank order (1st, 2nd, 3rd, etc) by which data can be sorted, but it still does not allow for relative degree of difference between them. Examples of ordinal data include dichotomous values such as “sick” versus “healthy” when measuring health, “guilty” versus “innocent” when making judgments in courts, “false” versus “true”, when measuring truth value. Examples also include non-dichotomous data consisting of a spectrum of values, such as “completely agree”, “mostly agree”, “mostly disagree”, or “completely disagree” when measuring opinion.

Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with longitude and temperature measurements in Celsius or Fahrenheit). Interval data allows for the degree of difference between items, but not the ratio between them. Ratios are not allowed with interval data since 20°C cannot be said to be “twice as hot” as 10°C, nor can multiplication/division be carried out between any two dates directly. However, ratios of differences can be expressed; for example, one difference can be twice another. Interval type variables are sometimes also called “scaled variables”.

Ratio measurements have both a meaningful zero value and the distances between different measurements are defined; they provide the greatest flexibility in statistical methods that can be used for analyzing the data.

Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as categorical variables, whereas ratio and interval measurements are grouped together as quantitative variables, which can be either discrete or continuous, due to their numerical nature.

Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic (bias), but other important types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important.

1.1.2: What Is Statistics?

Learning Objective

Define the field of Statistics in terms of its definition, application and history.

Key Points

• Statistics combines mathematical and non-mathematical procedures into one discipline.
• Statistics is generally broken down into two categories: descriptive statistics and inferential statistics.
• Statistics is an applied science and is used in many fields, including the natural and social sciences, government, and business.
• The use of statistical methods dates back to at least the 5th century BC.

Key Terms

statistics

a mathematical science concerned with data collection, presentation, analysis, and interpretation

empirical

verifiable by means of scientific experimentation

Example

• Say you want to conduct a poll on whether your school should use its funding to build a new athletic complex or a new library. Appropriate questions to ask would include: How many people do you have to poll? How do you ensure that your poll is free of bias? How do you interpret your results?

Statistics Overview

Statistics is the study of the collection, organization, analysis, interpretation, and presentation of data. It deals with all aspects of data, including the planning of its collection in terms of the design of surveys and experiments. Some consider statistics a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation of data, while others consider it a branch of mathematics concerned with collecting and interpreting data. Because of its empirical roots and its focus on applications, statistics is usually considered a distinct mathematical science rather than a branch of mathematics. As one would expect, statistics is largely grounded in mathematics, and the study of statistics has lent itself to many major concepts in mathematics, such as:

• probability,
• distributions ,
• samples and populations,
• estimation, and
• data analysis.

However, much of statistics is also non-mathematical. This includes:

• ensuring that data collection is undertaken in a way that produces valid conclusions,
• coding and archiving data so that information is retained and made useful for international comparisons of official statistics,
• reporting of results and summarized data (tables and graphs) in ways comprehensible to those who must use them, and
• implementing procedures that ensure the privacy of census information.

In short, statistics is the study of data. It includes descriptive statistics (the study of methods and tools for collecting data, and mathematical models to describe and interpret data) and inferential statistics (the systems and techniques for making probability-based decisions and accurate predictions based on incomplete data).

How Do We Use Statistics?

A statistician is someone who is particularly well-versed in the ways of thinking necessary to successfully apply statistical analysis. Such people often gain experience through working in any of a wide number of fields. Statisticians improve data quality by developing specific experimental designs and survey samples. Statistics itself also provides tools for predicting and forecasting the use of data and statistical models. Statistics is applicable to a wide variety of academic disciplines, including natural and social sciences, government, and business. Statistical consultants can help organizations and companies that don’t have in-house expertise relevant to their particular questions.

History of Statistics

Statistical methods date back at least to the 5^th century BC. The earliest known writing on statistics appears in a 9^th century book entitled Manuscript on Deciphering Cryptographic Messages, written by Al-Kindi. In this book, Al-Kindi provides a detailed description of how to use statistics and frequency analysis to decipher encrypted messages. This was the birth of both statistics and cryptanalysis, according to the Saudi engineer Ibrahim Al-Kadi.

The Nuova Cronica, a 14^th century history of Florence by the Florentine banker and official Giovanni Villani, includes much statistical information on population, ordinances, commerce, education, and religious facilities, and has been described as the first introduction of statistics as a positive element in history.

Some scholars pinpoint the origin of statistics to 1663, with the publication of Natural and Political Observations upon the Bills of Mortality by John Graunt. Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its “stat-” etymology. The scope of the discipline of statistics broadened in the early 19^th century to include the collection and analysis of data in general.

1.1.3: The Purpose of Statistics

Learning Objective

Describe how Statistics helps us to make inferences about a population, understand and interpret variation, and make more informed everyday decisions.

Key Points

• Statistics is an extremely powerful tool available for assessing the significance of experimental data and for drawing the right conclusions from it.
• Statistics helps scientists, engineers, and many other professionals draw the right conclusions from experimental data.
• Variation is ubiquitous in nature, and probability and statistics are the fields that allow us to study, understand, model, embrace and interpret this variation.

Key Terms

sample

a subset of a population selected for measurement, observation, or questioning to provide statistical information about the population

population

a group of units (persons, objects, or other items) enumerated in a census or from which a sample is drawn

Example

• A company selling the cat food brand “Cato” (a fictitious name here), may claim quite truthfully in their advertisements that eight out of ten cat owners said that their cats preferred Cato brand cat food to “the other leading brand” cat food. What they may not mention is that the cat owners questioned were those they found in a supermarket buying Cato, which doesn’t represent an unbiased sample of cat owners.

Imagine reading a book for the first few chapters and then being able to get a sense of what the ending will be like. This ability is provided by the field of inferential statistics. With the appropriate tools and solid grounding in the field, one can use a limited sample (e.g., reading the first five chapters of Pride & Prejudice) to make intelligent and accurate statements about the population (e.g., predicting the ending of Pride & Prejudice).

Those proceeding to higher education will learn that statistics is an extremely powerful tool available for assessing the significance of experimental data and for drawing the right conclusions from the vast amounts of data encountered by engineers, scientists, sociologists, and other professionals in most spheres of learning. There is no study with scientific, clinical, social, health, environmental or political goals that does not rely on statistical methodologies. The most essential reason for this fact is that variation is ubiquitous in nature, and probability and statistics are the fields that allow us to study, understand, model, embrace and interpret this variation.

In today’s information-overloaded age, statistics is one of the most useful subjects anyone can learn. Newspapers are filled with statistical data, and anyone who is ignorant of statistics is at risk of being seriously misled about important real-life decisions such as what to eat, who is leading the polls, how dangerous smoking is, et cetera. Statistics are often used by politicians, advertisers, and others to twist the truth for their own gain. Knowing at least a little about the field of statistics will help one to make more informed decisions about these and other important questions.

1.1.4: Inferential Statistics

Learning Objective

Discuss how inferential statistics allows us to draw conclusions about a population from a random sample and corresponding tests of significance.

Key Points

• Inferential statistics is used to describe systems of procedures that can be used to draw conclusions from data sets arising from systems affected by random variation, such as observational errors, random sampling, or random experimentation.
• Samples must be representative of the entire population in order to induce a conclusion about that population.
• Statisticians use tests of significance to determine the probability that the results were found by chance.

Key Term

inferential statistics

A branch of mathematics that involves drawing conclusions about a population based on sample data drawn from it.

In statistics, statistical inference is the process of drawing conclusions from data that is subject to random variation–for example, observational errors or sampling variation. More substantially, the terms statistical inference, statistical induction, and inferential statistics are used to describe systems of procedures that can be used to draw conclusions from data sets arising from systems affected by random variation, such as observational errors, random sampling, or random experimentation. Initial requirements of such a system of procedures for inference and induction are that the system should produce reasonable answers when applied to well-defined situations and that it should be general enough to be applied across a range of situations.

The outcome of statistical inference may be an answer to the question “what should be done next? ” where this might be a decision about making further experiments or surveys, or about drawing a conclusion before implementing some organizational or governmental policy.

Suppose you have been hired by the National Election Commission to examine how the American people feel about the fairness of the voting procedures in the U.S. How will you do it? Who will you ask?

It is not practical to ask every single American how he or she feels about the fairness of the voting procedures. Instead, we query a relatively small number of Americans, and draw inferences about the entire country from their responses. The Americans actually queried constitute our sample of the larger population of all Americans. The mathematical procedures whereby we convert information about the sample into intelligent guesses about the population fall under the rubric of inferential statistics.

In the case of voting attitudes, we would sample a few thousand Americans, drawn from the hundreds of millions that make up the country. In choosing a sample, it is therefore crucial that it be representative. It must not over-represent one kind of citizen at the expense of others. For example, something would be wrong with our sample if it happened to be made up entirely of Florida residents. If the sample held only Floridians, it could not be used to infer the attitudes of other Americans. The same problem would arise if the sample were comprised only of Republicans. Inferential statistics are based on the assumption that sampling is random. We trust a random sample to represent different segments of society in close to the appropriate proportions (provided the sample is large enough).

Furthermore, when generalizing a trend found in a sample to the larger population, statisticians uses tests of significance (such as the Chi-Square test or the T-test). These tests determine the probability that the results found were by chance, and therefore not representative of the entire population.

1.1.5: Types of Data

Learning Objective

Differentiate between primary and secondary data and qualitative and quantitative data.

Key Points

• Primary data is data collected first-hand. Secondary data is data reused from another source.
• Qualitative data is a categorical measurement expressed not in terms of numbers, but rather by means of a natural language description.
• Quantitative data is a numerical measurement expressed not by means of a natural language description, but rather in terms of numbers.

Key Terms

primary data

data that has been compiled for a specific purpose, and has not been collated or merged with others

qualitative data

data centered around descriptions or distinctions based on some quality or characteristic rather than on some quantity or measured value

quantitative

of a measurement based on some quantity or number rather than on some quality

Example

• Qualitative data: race, religion, gender, etc. Quantitative data: height in inches, time in seconds, temperature in degrees, etc.

Primary and Secondary Data

Data can be classified as either primary or secondary. Primary data is original data that has been collected specially for the purpose in mind. This type of data is collected first hand. Those who gather primary data may be an authorized organization, investigator, enumerator or just someone with a clipboard. These people are acting as a witness, so primary data is only considered as reliable as the people who gather it. Research where one gathers this kind of data is referred to as field research. An example of primary data is conducting your own questionnaire.

Secondary data is data that has been collected for another purpose. This type of data is reused, usually in a different context from its first use. You are not the original source of the data–rather, you are collecting it from elsewhere. An example of secondary data is using numbers and information found inside a textbook.

Knowing how the data was collected allows critics of a study to search for bias in how it was conducted. A good study will welcome such scrutiny. Each type has its own weaknesses and strengths. Primary data is gathered by people who can focus directly on the purpose in mind. This helps ensure that questions are meaningful to the purpose, but this can introduce bias in those same questions. Secondary data doesn’t have the privilege of this focus, but is only susceptible to bias introduced in the choice of what data to reuse. Stated another way, those who gather secondary data get to pick the questions. Those who gather primary data get to write the questions. There may be bias either way.

Qualitative and Quantitative Data

Qualitative data is a categorical measurement expressed not in terms of numbers, but rather by means of a natural language description. In statistics, it is often used interchangeably with “categorical” data. Collecting information about a favorite color is an example of collecting qualitative data. Although we may have categories, the categories may have a structure to them. When there is not a natural ordering of the categories, we call these nominal categories. Examples might be gender, race, religion, or sport. When the categories may be ordered, these are called ordinal categories. Categorical data that judge size (small, medium, large, etc. ) are ordinal categories. Attitudes (strongly disagree, disagree, neutral, agree, strongly agree) are also ordinal categories; however, we may not know which value is the best or worst of these issues. Note that the distance between these categories is not something we can measure.

Quantitative data is a numerical measurement expressed not by means of a natural language description, but rather in terms of numbers. Quantitative data always are associated with a scale measure. Probably the most common scale type is the ratio-scale. Observations of this type are on a scale that has a meaningful zero value but also have an equidistant measure (i.e. the difference between 10 and 20 is the same as the difference between 100 and 110). For example, a 10 year-old girl is twice as old as a 5 year-old girl. Since you can measure zero years, time is a ratio-scale variable. Money is another common ratio-scale quantitative measure. Observations that you count are usually ratio-scale (e.g. number of widgets). A more general quantitative measure is the interval scale. Interval scales also have an equidistant measure. However, the doubling principle breaks down in this scale. A temperature of 50 degrees Celsius is not “half as hot” as a temperature of 100, but a difference of 10 degrees indicates the same difference in temperature anywhere along the scale.

1.1.6: Applications of Statistics

Learning Objective

Describe how statistics is applied to scientific, industrial, and societal problems.

Key Points

• Statistics can be used to improve data quality by developing specific experimental designs and survey samples.
• Statistics includes the planning of data collection in terms of the design of surveys and experiments.
• Statistics provides tools for prediction and forecasting and is applicable to a wide variety of academic disciplines, including natural and social sciences, as well as government, and business.

Key Terms

population

a group of units (persons, objects, or other items) enumerated in a census or from which a sample is drawn

statistics

The study of the collection, organization, analysis, interpretation, and presentation of data.

sample

a subset of a population selected for measurement, observation, or questioning to provide statistical information about the population

Example

• In calculating the arithmetic mean of a sample, for example, the algorithm works by summing all the data values observed in the sample and then dividing this sum by the number of data items. This single measure, the mean of the sample, is called a statistic; its value is frequently used as an estimate of the mean value of all items comprising the population from which the sample is drawn. The population mean is also a single measure; however, it is not called a statistic; instead it is called a population parameter.

Statistics deals with all aspects of the collection, organization, analysis, interpretation, and presentation of data. It includes the planning of data collection in terms of the design of surveys and experiments.

Statistics can be used to improve data quality by developing specific experimental designs and survey samples. Statistics also provides tools for prediction and forecasting. Statistics is applicable to a wide variety of academic disciplines, including natural and social sciences as well as government and business. Statistical consultants can help organizations and companies that don’t have in-house expertise relevant to their particular questions.

Descriptive and Inferential Statistics

Statistical methods can summarize or describe a collection of data. This is called descriptive statistics . This is particularly useful in communicating the results of experiments and research. Statistical models can also be used to draw statistical inferences about the process or population under study—a practice called inferential statistics. Inference is a vital element of scientific advancement, since it provides a way to draw conclusions from data that are subject to random variation. Conclusions are tested in order to prove the propositions being investigated further, as part of the scientific method. Descriptive statistics and analysis of the new data tend to provide more information as to the truth of the proposition.

The Statistical Process

When applying statistics to a scientific, industrial, or societal problems, it is necessary to begin with a population or process to be studied. Populations can be diverse topics such as “all persons living in a country” or “every atom composing a crystal”. A population can also be composed of observations of a process at various times, with the data from each observation serving as a different member of the overall group. Data collected about this kind of “population” constitutes what is called a time series. For practical reasons, a chosen subset of the population called a sample is studied—as opposed to compiling data about the entire group (an operation called census). Once a sample that is representative of the population is determined, data is collected for the sample members in an observational or experimental setting. This data can then be subjected to statistical analysis, serving two related purposes: description and inference.

Descriptive statistics summarize the population data by describing what was observed in the sample numerically or graphically. Numerical descriptors include mean and standard deviation for continuous data types (like heights or weights), while frequency and percentage are more useful in terms of describing categorical data (like race). Inferential statistics uses patterns in the sample data to draw inferences about the population represented, accounting for randomness. These inferences may take the form of: answering yes/no questions about the data (hypothesis testing), estimating numerical characteristics of the data (estimation), describing associations within the data (correlation) and modeling relationships within the data (for example, using regression analysis). Inference can extend to forecasting, prediction and estimation of unobserved values either in or associated with the population being studied. It can include extrapolation and interpolation of time series or spatial data and can also include data mining.

Statistical Analysis

Statistical analysis of a data set often reveals that two variables of the population under consideration tend to vary together, as if they were connected. For example, a study of annual income that also looks at age of death might find that poor people tend to have shorter lives than affluent people. The two variables are said to be correlated; however, they may or may not be the cause of one another. The correlation could be caused by a third, previously unconsidered phenomenon, called a confounding variable. For this reason, there is no way to immediately infer the existence of a causal relationship between the two variables.

To use a sample as a guide to an entire population, it is important that it truly represent the overall population. Representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. A major problem lies in determining the extent that the sample chosen is actually representative. Statistics offers methods to estimate and correct for any random trending within the sample and data collection procedures. There are also methods of experimental design for experiments that can lessen these issues at the outset of a study, strengthening its capability to discern truths about the population. Randomness is studied using the mathematical discipline of probability theory. Probability is used in “mathematical statistics” (alternatively, “statistical theory”) to study the sampling distributions of sample statistics and, more generally, the properties of statistical procedures. The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method.

1.1.7: Fundamentals of Statistics

Learning Objective

Recall that the field of Statistics involves using samples to make inferences about populations and describing how variables relate to each other.

Key Points

• For practical reasons, a chosen subset of the population called a sample is studied—as opposed to compiling data about the entire group (an operation called census).
• Descriptive statistics summarizes the population data by describing what was observed in the sample numerically or graphically.
• Inferential statistics uses patterns in the sample data to draw inferences about the population represented, accounting for randomness.
• Statistical analysis of a data set often reveals that two variables (properties) of the population under consideration tend to vary together, as if they were connected.
• To use a sample as a guide to an entire population, it is important that it truly represent the overall population.

Key Terms

sample

a subset of a population selected for measurement, observation, or questioning to provide statistical information about the population

variable

a quantity that may assume any one of a set of values

population

a group of units (persons, objects, or other items) enumerated in a census or from which a sample is drawn

Example

• A population can be composed of observations of a process at various times, with the data from each observation serving as a different member of the overall group. Data collected about this kind of “population” constitutes what is called a time series.

In applying statistics to a scientific, industrial, or societal problem, it is necessary to begin with a population or process to be studied. Populations can be diverse topics such as “all persons living in a country” or “every atom composing a crystal.”. A population can also be composed of observations of a process at various times, with the data from each observation serving as a different member of the overall group. Data collected about this kind of “population” constitutes what is called a time series.

For practical reasons, a chosen subset of the population called a sample is studied—as opposed to compiling data about the entire group (an operation called census). Once a sample that is representative of the population is determined, data is collected for the sample members in an observational or experimental setting. This data can then be subjected to statistical analysis, serving two related purposes: description and inference.

• Descriptive statistics summarizes the population data by describing what was observed in the sample numerically or graphically. Numerical descriptors include mean and standard deviation for continuous data types (like heights or weights), while frequency and percentages are more useful in terms of describing categorical data (like race).
• Inferential statistics uses patterns in the sample data to draw inferences about the population represented, accounting for randomness. These inferences may take the form of: answering yes/no questions about the data (hypothesis testing), estimating numerical characteristics of the data (estimation), describing associations within the data (correlation ) and modeling relationships within the data (for example, using regression analysis). Inference can extend to forecasting, prediction and estimation of unobserved values either in or associated with the population being studied. It can include extrapolation and interpolation of time series or spatial data, and can also include data mining.

The concept of correlation is particularly noteworthy for the potential confusion it can cause. Statistical analysis of a data set often reveals that two variables (properties) of the population under consideration tend to vary together, as if they were connected. For example, a study of annual income that also looks at age of death might find that poor people tend to have shorter lives than affluent people. The two variables are said to be correlated; however, they may or may not be the cause of one another. The correlation phenomena could be caused by a third, previously unconsidered phenomenon, called a confounding variable. For this reason, there is no way to immediately infer the existence of a causal relationship between the two variables.

To use a sample as a guide to an entire population, it is important that it truly represent the overall population. Representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. A major problem lies in determining the extent that the sample chosen is actually representative. Statistics offers methods to estimate and correct for any random trending within the sample and data collection procedures. There are also methods of experimental design for experiments that can lessen these issues at the outset of a study, strengthening its capability to discern truths about the population.

Randomness is studied using the mathematical discipline of probability theory. Probability is used in “mathematical statistics” (alternatively, “statistical theory”) to study the sampling distributions of sample statistics and, more generally, the properties of statistical procedures. The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method.

1.1.8: Critical Thinking

Learning Objective

Interpret the role that the process of critical thinking plays in statistical literacy.

Key Points

• Statistics can be made to produce misrepresentations of data that may seem valid.
• Statistical literacy is necessary to understand what makes a poll trustworthy and to properly weigh the value of poll results and conclusions.
• Critical thinking is a way of deciding whether a claim is always true, sometimes true, partly true, or false.
• The list of core critical thinking skills includes observation, interpretation, analysis, inference, evaluation, explanation, and meta-cognition.

Key Terms

statistical literacy

the ability to understand statistics, necessary for citizens to understand mateiral presented in publications such as newspapers, television, and the Internet

critical thinking

the application of logical principles, rigorous standards of evidence, and careful reasoning to the analysis and discussion of claims, beliefs, and issues

Each day people are inundated with statistical information from advertisements (“4 out of 5 dentists recommend”), news reports (“opinion polls show the incumbent leading by four points”), and even general conversation (“half the time I don’t know what you’re talking about”). Experts and advocates often use numerical claims to bolster their arguments, and statistical literacy is a necessary skill to help one decide what experts mean and which advocates to believe. This is important because statistics can be made to produce misrepresentations of data that may seem valid. The aim of statistical literacy is to improve the public understanding of numbers and figures.

For example, results of opinion polling are often cited by news organizations, but the quality of such polls varies considerably. Some understanding of the statistical technique of sampling is necessary in order to be able to correctly interpret polling results. Sample sizes may be too small to draw meaningful conclusions, and samples may be biased. The wording of a poll question may introduce a bias, and thus can even be used intentionally to produce a biased result. Good polls use unbiased techniques, with much time and effort being spent in the design of the questions and polling strategy. Statistical literacy is necessary to understand what makes a poll trustworthy and to properly weigh the value of poll results and conclusions.

Critical Thinking

The essential skill of critical thinking will go a long way in helping one to develop statistical literacy. Critical thinking is a way of deciding whether a claim is always true, sometimes true, partly true, or false. The list of core critical thinking skills includes observation, interpretation, analysis, inference, evaluation, explanation, and meta-cognition. There is a reasonable level of consensus that an individual or group engaged in strong critical thinking gives due consideration to establish:

• Evidence through observation,
• Context skills,
• Relevant criteria for making the judgment well,
• Applicable methods or techniques for forming the judgment, and
• Applicable theoretical constructs for understanding the problem and the question at hand.

Critical thinking calls for the ability to:

• Recognize problems, to find workable means for meeting those problems,
• Understand the importance of prioritization and order of precedence in problem solving,
• Gather and marshal pertinent (relevant) information,
• Recognize unstated assumptions and values,
• Comprehend and use language with accuracy, clarity, and discernment,
• Interpret data, to appraise evidence and evaluate arguments,
• Recognize the existence (or non-existence) of logical relationships between propositions,
• Draw warranted conclusions and generalizations,
• Put to test the conclusions and generalizations at which one arrives,
• Reconstruct one’s patterns of beliefs on the basis of wider experience, and
• Render accurate judgments about specific things and qualities in everyday life.

1.1.9: Experimental Design

Learning Objective

Outline the methodology for designing experiments in terms of comparison, randomization, replication, blocking, orthogonality, and factorial experiments

Key Points

• The experimenter is often interested in the effect of some process or intervention (the “treatment”) on some objects (the “experimental units”), which may be people, parts of people, groups of people, plants, animals, etc.
• A methodology for designing experiments involves comparison, randomization, replication, blocking, orthogonality, and factorial considerations.
• It is best that a process be in reasonable statistical control prior to conducting designed experiments.
• One of the most important requirements of experimental research designs is the necessity of eliminating the effects of spurious, intervening, and antecedent variables.

Key Terms

dependent variable

in an equation, the variable whose value depends on one or more variables in the equation

independent variable

in an equation, any variable whose value is not dependent on any other in the equation

experiment

A test under controlled conditions made to either demonstrate a known truth, examine the validity of a hypothesis, or determine the efficacy of something previously untried.

Example

• For example, if a researcher feeds an experimental artificial sweetener to sixty laboratory rats and observes that ten of them subsequently become sick, the underlying cause could be the sweetener itself or something unrelated. Other variables, which may not be readily obvious, may interfere with the experimental design. For instance, perhaps the rats were simply not supplied with enough food or water, or the water was contaminated and undrinkable, or the rats were under some psychological or physiological stress, etc. Eliminating each of these possible explanations individually would be time-consuming and difficult. However, if a control group is used that does not receive the sweetener but is otherwise treated identically, any difference between the two groups can be ascribed to the sweetener itself with much greater confidence.

In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. Formal planned experimentation is often used in evaluating physical objects, chemical formulations, structures, components, and materials. In the design of experiments, the experimenter is often interested in the effect of some process or intervention (the “treatment”) on some objects (the “experimental units”), which may be people, parts of people, groups of people, plants, animals, etc. Design of experiments is thus a discipline that has very broad application across all the natural and social sciences and engineering.

A methodology for designing experiments was proposed by Ronald A. Fisher in his innovative books The Arrangement of Field Experiments (1926) and The Design of Experiments (1935). These methods have been broadly adapted in the physical and social sciences.

• Comparison: In some fields of study it is not possible to have independent measurements to a traceable standard. Comparisons between treatments are much more valuable and are usually preferable. Often one compares against a scientific control or traditional treatment that acts as baseline.
• Randomization: Random assignment is the process of assigning individuals at random to groups or to different groups in an experiment. The random assignment of individuals to groups (or conditions within a group) distinguishes a rigorous, “true” experiment from an adequate, but less-than-rigorous, “quasi-experiment”. Random does not mean haphazard, and great care must be taken that appropriate random methods are used.
• Replication: Measurements are usually subject to variation and uncertainty. Measurements are repeated and full experiments are replicated to help identify the sources of variation, to better estimate the true effects of treatments, to further strengthen the experiment’s reliability and validity, and to add to the existing knowledge of the topic.
• Blocking: Blocking is the arrangement of experimental units into groups (blocks) consisting of units that are similar to one another. Blocking reduces known but irrelevant sources of variation between units and thus allows greater precision in the estimation of the source of variation under study.
• Orthogonality: Orthogonality concerns the forms of comparison (contrasts) that can be legitimately and efficiently carried out. Contrasts can be represented by vectors and sets of orthogonal contrasts are uncorrelated and independently distributed if the data are normal. Because of this independence, each orthogonal treatment provides different information to the others. If there are
  treatments and
  orthogonal contrasts, all the information that can be captured from the experiment is obtainable from the set of contrasts.
• Factorial experiments: Use of factorial experiments instead of the one-factor-at-a-time method. These are efficient at evaluating the effects and possible interactions of several factors (independent variables). Analysis of experiment design is built on the foundation of the analysis of variance, a collection of models that partition the observed variance into components, according to what factors the experiment must estimate or test.

It is best that a process be in reasonable statistical control prior to conducting designed experiments. When this is not possible, proper blocking, replication, and randomization allow for the careful conduct of designed experiments. To control for nuisance variables, researchers institute control checks as additional measures. Investigators should ensure that uncontrolled influences (e.g., source credibility perception) are measured do not skew the findings of the study.

One of the most important requirements of experimental research designs is the necessity of eliminating the effects of spurious, intervening, and antecedent variables. In the most basic model, cause (
) leads to effect (
). But there could be a third variable (
) that influences (
), and
might not be the true cause at all.
is said to be a spurious variable and must be controlled for. The same is true for intervening variables (a variable in between the supposed cause (
) and the effect (
)), and anteceding variables (a variable prior to the supposed cause (
) that is the true cause). In most designs, only one of these causes is manipulated at a time.

1.1.10: Random Samples

Learning Objective

Explain how simple random sampling leads to every object having the same possibility of being chosen.

Key Points

• Simple random sampling merely allows one to draw externally valid conclusions about the entire population based on the sample.
• Advantages of random sampling are that it is free of classification error, and it requires minimum advance knowledge of the population other than the frame.
• Simple random sampling best suits situations where not much information is available about the population and data collection can be efficiently conducted on randomly distributed items, or where the cost of sampling is small enough to make efficiency less important than simplicity.

Key Terms

population

a group of units (persons, objects, or other items) enumerated in a census or from which a sample is drawn

random sample

a sample randomly taken from an investigated population

Sampling is concerned with the selection of a subset of individuals from within a statistical population to estimate characteristics of the whole population . Two advantages of sampling are that the cost is lower and data collection is faster than measuring the entire population.

Each observation measures one or more properties (such as weight, location, color) of observable bodies distinguished as independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly stratified sampling (blocking). Results from probability theory and statistical theory are employed to guide practice. In business and medical research, sampling is widely used for gathering information about a population.

A simple random sample is a subset of individuals chosen from a larger set (a population). Each individual is chosen randomly and entirely by chance, such that each individual has the same probability of being chosen at any stage during the sampling process and each subset of
individuals has the same probability of being chosen for the sample as any other subset of
 individuals. A simple random sample is an unbiased surveying technique.

Simple random sampling is a basic type of sampling, since it can be a component of other more complex sampling methods. The principle of simple random sampling is that every object has the same possibility to be chosen. For example,
college students want to get a ticket for a basketball game, but there are not enough tickets (
) for them, so they decide to have a fair way to see who gets to go. Then, everybody is given a number (0 to
), and random numbers are generated. The first
numbers would be the lucky ticket winners.

In small populations and often in large ones, such sampling is typically done “without replacement” (i.e., one deliberately avoids choosing any member of the population more than once). Although simple random sampling can be conducted with replacement instead, this is less common and would normally be described more fully as simple random sampling with replacement. Sampling done without replacement is no longer independent, but still satisfies exchangeability. Hence, many results still hold. Further, for a small sample from a large population, sampling without replacement is approximately the same as sampling with replacement, since the odds of choosing the same individual twice is low.

An unbiased random selection of individuals is important so that, in the long run, the sample represents the population. However, this does not guarantee that a particular sample is a perfect representation of the population. Simple random sampling merely allows one to draw externally valid conclusions about the entire population based on the sample.

Conceptually, simple random sampling is the simplest of the probability sampling techniques. It requires a complete sampling frame, which may not be available or feasible to construct for large populations. Even if a complete frame is available, more efficient approaches may be possible if other useful information is available about the units in the population.

Advantages are that it is free of classification error, and it requires minimum advance knowledge of the population other than the frame. Its simplicity also makes it relatively easy to interpret data collected via SRS. For these reasons, simple random sampling best suits situations where not much information is available about the population and data collection can be efficiently conducted on randomly distributed items, or where the cost of sampling is small enough to make efficiency less important than simplicity. If these conditions are not true, stratified sampling or cluster sampling may be a better choice.