A logical place to begin analysis of data is simply to describe the distribution of one variable at a time.  (For that reason, such analysis is often referred to as “univariate” analysis.)

Central Tendency

“The “central tendency” of a variable is nothing more than its average value.  There are, however, different kinds of averages: the mean, the median, and the mode.

The Mean

Also known as the “arithmetic mean,” this is probably what most of us think of when we use the term average.  The mean value of a variable is calculated simply by adding up all the values and then dividing by the number of cases.  The mean requires that data be at least interval level.  Take, for example, a variable with three cases having the values 2, 3, and 4 respectively.  The mean for this variable would obviously be 3 (2+3+4=9; 9/3=3), but this makes no sense unless the difference between 2 and 3 is the same as the difference between 3 and 4.

In mathematical notation, the mean for population data is represented by the symbol μ (the lower case Greek letter mu).  If we are using sample data, and the mean (of a variable named X) is represented by the symbol  (pronounced “X bar”).

The Median

The median is calculated by ranking cases from high to low (or vice-versa) and then finding the value of the case that is in the middle (also called the 50^th percentile) of the distribution.  By definition, half of all cases are at or above the median, and half below[1].  In a distribution of 21 cases, for example, the median value is the value of the 11^th highest case, since there are 10 cases with higher values and 10 cases with lower values.  If there is an even number of cases, the median value is the value half way between the values of the two cases closest to the middle.  For example, in a distribution of 20 cases, the median value is half way between the values of the 10^th and 11^th highest cases.

The notion of a “middle” case makes sense only if cases can be rank-ordered. Calculation of a median, therefore, requires at least ordinal level data.  Sometimes, it makes sense to calculate a median instead of, or in addition to, a mean even with interval or ratio data.  If the distribution of the values of a variable is heavily “skewed” by a few very high or very low scores, the mean of the distribution will be misleading.  Suppose, for example, that there are 100 households in your neighborhood, and that both their mean and the median incomes are about $40,000 per year[2].  Now suppose that Bill Gates and his family move in next door.  The median household income will not change much (it will be the income of the family ranked 51^st), but the mean household income will be in the hundreds of millions of dollars.  Which figures better describes the “average” family in the neighborhood?

The Mode

The mode of a variable is the value that occurs most frequently.  In the United States, the modal religion is Protestant and the modal gender is female.  In an election, the modal candidate is the one who receives more votes than anyone else.  In politics, a mode is often referred to as a plurality.  It can be used with any level of measurement.

Sometimes the question of which measure of central tendency is used can be a hot political topic.   Many government contracts require contractors to pay their employees the “prevailing wage.”  In federal law and in most states, the prevailing wage is defined as the mean.  In a couple of states, including California, the prevailing wage is defined as the mode.   Since the most common wage is usually that called for by union contracts, and since unionized workers usually receive higher pay than non-unionized workers, the mean is obviously preferred by business, while the mode is favored by labor[3].

Dispersion

In addition to wanting to know the average value of a variable, we would probably also want some information about how spread out the values are.  One measure of this is the range (the difference between the maximum and minimum values), but this provides us with only very limited information.  There are some other, more useful, measures.

The Variance and the Standard Deviation

The variance and the standard deviation are related measures of how far away, on average, the values of a variable are from the mean.  Since the mean requires at least interval level measurement, so do the variance and the standard deviation. 

Consider the two sets of numbers shown below.  Both have the same mean (10), but the numbers on the right are clearly more spread out than those on the left.

Set 1	Set 2
12	14
11	12
10	10
9	8
8	6

Table 1 shows how the variance in the group of numbers on the left is calculated.  In the first column, the individual values of the variable (which we will represent with the symbol “X_i”) are listed.  In the second column, the “deviation” from the mean value (µ) of 10 is subtracted from each value.  If we simply took an average of the deviations, the result would always be zero.  Instead, in the third column we square the deviations from the mean.  Finally we sum (Σ, the upper-case Greek letter sigma) these individual numbers from the first through the last, or n^th (  ), and divide by the number of cases (5).  The result is the “mean squared deviation from the mean,” or the variance.  For population data, the variance[4] is represented by the symbol ² (the square of the lower-case Greek letter sigma) for population data, and s² for sample data.

Table1: Calculating Variance
Xi	Xi – µ	(Xi - µ)2
12	2	4
11	1	1
10	0	0
9	-1	1
8	-2	4
(Xi - µ)2/N = 10/5 = 2

The standard deviation ( for population data, s for sample data), like the variance, is a measure of dispersion, and is the one usually reported.  It is simply the positive square root of the variance.  In the above example, 

 

The variance and the standard deviation are usually not of great interest in and of themselves.  They are, however, central to a wide variety other statistical methods.  Occasionally, they do have direct application.  Beck, for example, demonstrates the nationalization of American politics by showing that the standard deviation in presidential vote by state declined fairly steadily between 1896 and 1992[5].

 

 

Boxplots

 

A boxplot (also known as a “box and whiskers” plot) is another way of examining the distribution of a continuous variable.   This figure (using data from the CIA World Factbook – see the countries file) shows the boxplot for unemployment rates by country.
 

The “box” in the figure shows the “interquartile range.”  That is, the line at the top of the box represents the value of the 75^th percentile, while the line at the bottom of the box represents the value of the 25^th percentile.  In other words, the middle half of all counties are within the box.  The value of the 50^th percentile (that is, of the median value) is represented by the horizontal line within the box.  The lines extending from the box are the “whiskers,” and the horizontal lines at the end of the whiskers represent the highest and lowest values that are outside the box but within 1.5 times the inter-quartile range (1.5*IQR).    The circles beyond the whiskers represent “outliers,” that is, cases outside the box by more than 1.5*IQR, while asterisks represent “extreme” values, that is, those outside the box by more than 3*IQR.  The figure shows that the distribution of unemployment is positively “skewed”:  there is a bigger difference between the 75^th and 50^th percentile than between the 50^th and 25^th percentiles, the upper whisker is longer than the lower one, and all outliers and extreme values are at the top of the plot, indicating countires with very high rates of unemployment. 

 

 

 

Exercises

Note:  There are several ways to produce the statistics and graphs described in this topic.  The frequencies procedure can produce all of the statistics described; the descriptives procedure can produce all but the median and the mode; the explore procedure can produce all but the mode, and also produces boxplots.  There is also a separate and more powerful procedure specifically for generating boxplots.

1.         Calculate variance and standard deviation for the second set of numbers listed above.

 2.        A number of years ago, Scammon and Wattenberg famously described the average American voter as “A 47 year-old housewife from the outskirts of Dayton, Ohio, whose husband is a machinist.”[6]   For each italicized term in this description, try to figure out if, by average, Scammon and Wattenberg are referring to a mean, a median, or a mode.

3.         Using Webstats,  pick one of the datasets and locating the interval and ratio level variables in this dataset, use frequencies to calculate and compare the means and the medians.  For which variables are they markedly different?  Speculate as to why.

4.         Using Webstats open the Canadian National Election Study.  You will find “feeling thermometers” in which respondents were asked to indicate how “warmly” they felt about various political figures.  On these thermometers, which range from 0 to 100, the higher the score, the more “favorable and warm” the respondent reported feeling about the person in question.  Toward which political figures did respondents have the warmest feelings?  Where there are both pre and post election thermometers for the same person, were these people perceived more favorably after the election than they had been before? 

Some political leaders are very polarizing figures, toward whom many people have either very warm or very cool feelings.  Others provoke more lukewarm responses.  Presumably, the more polarizing figures would have higher standard deviations in their thermometers.  Compare the different leaders in this regard.  Where available, compare their thermometers in the pre-election survey versus the post-election survey to test the notion that the campaign may have caused peoples reactions to become more divided. 

 

 

For Further Study

 

Lane, David M., “Chapter 2: Describing Univariate Data,” Hyperstat Online. http://davidmlane.com/hyperstat/

 

Lowry, Richard, “Chapter 2: Distributions,” Concepts and Applications of Inferential Statistics. http://faculty.vassar.edu/lowry/webtext.html.