6.2: Describing Variables

Learning Objectives

1. Use frequency tables and histograms to display and interpret the distribution of a variable.
2. Compute and interpret the mean, median, and mode of a distribution and identify situations in which the mean, median, or mode is the most appropriate measure of central tendency.
3. Compute and interpret the range and standard deviation of a distribution.

Introduction to Analysis

In the chapter on reading a research article, there was a section that focused on interpreting statistics in a research article. Now, the sections in this chapter will go into more detail about organizing and describing quantitative data, null hypothesis significance testing to compare means or test (linear) relationships, and introduce you to qualitative research and analyses. So why is there another whole chapter about analyses? There are a couple reasons. First, the additional details in this chapter can help you better interpret many analyses that you may across in the articles that you're starting to read. The section in the previous chapter provided a quick refresher, but was not designed to provide much detail or context. A second reason for this chapter is a review of statistical analyses for when you are asked to analyze data for any class projects. It is expected that you previously passed a statistics class, but it is also likely that you forgot much of what you learned. This chapter will provide a refresher. This is particularly important for those of you who took a statistics class through your math department instead of a behavioral statistics course through your psychology or sociology departments; the statistical analyses are the same, but the emphasis differs between mathematicians and social scientists. The summary in this section will orient you to how social scientists approach statistical analyses.

Descriptive Statistics

The first step to analyzing quantitative data is to organize and summarize the data. Descriptive statistics refers to a set of techniques for summarizing and displaying data. Although in most cases the primary research question will be about one or more statistical relationships between variables, it is also important to describe each variable individually. For this reason, we begin by looking at some of the most common techniques for describing single variables. This should be a reminder of what you learned in many math classes throughout your educational journey.

The Distribution of a Variable

Every variable has a distribution, which is the way the scores are distributed across the levels of that variable. For example, in a sample of 100 university students, the distribution of the variable “number of siblings” might be such that 10 of them have no siblings, 30 have one sibling, 40 have two siblings, and so on. In the same sample, the distribution of the variable “sex” might be such that 44 have a score of “male” and 56 have a score of “female.”

Frequency Tables

One way to display the distribution of a variable is in a frequency table. Table \(\PageIndex{1}\), for example, is a frequency table showing a hypothetical distribution of scores on the Rosenberg Self-Esteem Scale for a sample of 40 college students. The first column lists the values of the variable—the possible scores on the Rosenberg scale—and the second column lists the frequency of each score. This table shows that there were three students who had self-esteem scores of 24, five who had self-esteem scores of 23, and so on. From a frequency table like this, one can quickly see several important aspects of a distribution, including the range of scores (from 15 to 24), the most and least common scores (22 and 17, respectively), and any extreme scores that stand out from the rest.

Table \(\PageIndex{1}\): Frequency Table Showing a Hypothetical Distribution of Scores on the Rosenberg Self-Esteem Scale

Self-esteem	Frequency
24	3
23	5
22	10
21	8
20	5
19	3
18	3
17	0
16	2
15	1

Frequency Charts

A frequency polygon, often referred to as a frequency line graph, is a graphical display of a distribution of quantitative variables. It presents the same information as a frequency table but in a way that is even quicker and easier to grasp. The line graph in Figure \(\PageIndex{1}\) presents the distribution of self-esteem scores in Table \(\PageIndex{1}\). The x-axis of the histogram represents the variable and the y-axis represents frequency. Similar information can be displayed in a histogram, but there are more rules to how to create a histogram. Additionally, a histogram uses bars to represent the frequency of categories of scores. There are no gaps or spaces between the bars; this is what makes a histogram visually different from a bar chart. What makes a histogram conceptually different from a bar chart is that the frequency of quantitative variables are displayed on histogram, while the frequency of categorical variables are displayed on bar charts.

Figure \(\PageIndex{1}\): Frequency Polygon Showing the Distribution of Self-Esteem Scores Presented in Table \(\PageIndex{1}\)

Distribution Shapes

When the distribution of a quantitative variable is displayed in a frequency polygon (frequency line graph) or a histogram, it has a shape. The shape of the distribution of self-esteem scores in Figure \(\PageIndex{1}\) is a little negatively skewed since there are more people who score higher, and few who score lower; in other words, there is a peak somewhere towards the right of the distribution and “tails” that taper in to the left. Figure \(\PageIndex{2}\) shows the two kinds of skew (negative or positive), plus what a symmetrical histogram might look like. The distribution on the left is negatively skewed, with its peak shifted toward the upper end of its range and a relatively long negative tail. The distribution on the right is positively skewed, with its peak toward the lower end of its range and a relatively long positive tail.

Figure \(\PageIndex{2}\): Histograms Showing Negatively Skewed, Symmetrical, and Positively Skewed Distributions

An outlier is an extreme score that is much higher or lower than the rest of the scores in the distribution. Sometimes outliers represent truly extreme scores on the variable of interest. For example, on the Beck Depression Inventory, a single clinically depressed person might be an outlier in a sample of otherwise happy and high-functioning peers. However, outliers can also represent errors or misunderstandings on the part of the researcher or participant, equipment malfunctions, or similar problems.

Another thing to look at in a distribution is whether there is only one, or more, frequently occurring scores. The distribution of Figure \(\PageIndex{1}\) is unimodal, meaning it has one distinct peak, but distributions can also be bimodal, meaning they have two distinct peaks. Figure \(\PageIndex{3}\), for example, shows a hypothetical bimodal distribution of scores on the Beck Depression Inventory. Distributions can also have more than two distinct peaks, but these are relatively rare when your sample size is big enough.

Figure \(\PageIndex{3}\): Histogram Showing a Hypothetical Bimodal Distribution of Scores on the Beck Depression Inventory

Measures of Central Tendency and Variability

It is also useful to be able to describe the characteristics of a distribution more precisely. Here we look at how to do this in terms of two important characteristics: their central tendency and their variability.

Central Tendency

The central tendency of a distribution is its middle—the point around which the scores in the distribution tend to cluster. Looking back at Figure \(\PageIndex{1}\), for example, we can see that the self-esteem scores tend to cluster around the values of 20 to 22. Here we will consider the three most common measures of central tendency: the mean, the median, and the mode.

The mean of a distribution (symbolized M) is the sum of the scores divided by the number of scores. It is an average. As a formula, it looks like this:

\[M=\dfrac{\displaystyle \sum X}{N}\]

In this formula, the symbol Σ (the Greek letter sigma) is the summation sign and means to sum across the values of the variable X. N represents the number of scores. The mean is by far the most common measure of central tendency, and there are some good reasons for this. It usually provides a good indication of the central tendency of a distribution, and it is easily understood by most people. In addition, the mean has statistical properties that make it especially useful in doing inferential statistics.

An alternative to the mean is the median. The median is the middle score in the sense that half the scores in the distribution are less than it and half are greater than it. The simplest way to find the median is to organize the scores from lowest to highest and locate the score in the middle. Consider, for example, the following set of seven scores:

8 4 12 14 3 2 3

To find the median, simply rearrange the scores from lowest to highest and locate the one in the middle.

2 3 3 4 8 12 14

In this case, the median is 4 because there are three scores lower than 4 and three scores higher than 4. When there is an even number of scores, there are two scores in the middle of the distribution, in which case the median is the value halfway between them. For example, if we were to add a score of 15 to the preceding data set, there would be two scores (both 4 and 8) in the middle of the distribution, and the median would be halfway between them (6).

One final measure of central tendency is the mode. The mode is the most frequent score in a distribution. In the self-esteem distribution presented in Table \(\PageIndex{1}\) and Figure \(\PageIndex{1}\), for example, the mode is 22. More students had that score than any other. The mode is the only measure of central tendency that can also be used for categorical variables.

In a distribution that is both unimodal and symmetrical, the mean, median, and mode will be very close to each other at the peak of the distribution. In a bimodal or asymmetrical distribution, the mean, median, and mode can be quite different. In a bimodal distribution, the mean and median will tend to be between the peaks, while the mode will be at the tallest peak. In a skewed distribution, the mean will differ from the median in the direction of the skew (i.e., the direction of the longer tail). For highly skewed distributions, the mean can be pulled so far in the direction of the skew that it is no longer a good measure of the central tendency of that distribution. Imagine, for example, a set of four simple reaction times of 200, 250, 280, and 250 milliseconds (ms). The mean is 245 ms. But the addition of one more score of 5,000 ms—perhaps because the participant was not paying attention—would raise the mean to 1,445 ms. Not only is this measure of central tendency greater than 80% of the scores in the distribution, but it also does not seem to represent the behavior of anyone in the distribution very well. This is why researchers often prefer the median for highly skewed distributions (such as distributions of reaction times).

Keep in mind, though, that you are not required to choose a single measure of central tendency in analyzing your data. Each one provides slightly different information, and all of them can be useful.

Measures of Variability

The variability of a distribution is the extent to which the scores vary around their central tendency. Consider the two distributions in Figure \(\PageIndex{4}\), both of which have the same central tendency. The mean, median, and mode of each distribution are 10. Notice, however, that the two distributions differ in terms of their variability. The top one has relatively low variability, with all the scores relatively close to the center. The bottom one has relatively high variability, with the scores are spread across a much greater range.

Figure \(\PageIndex{4}\): Histograms Showing Hypothetical Distributions With the Same Mean, Median, and Mode (10) but With Low Variability (Top) and High Variability (Bottom)

One simple measure of variability is the range, which is simply the difference between the highest and lowest scores in the distribution. The range of the self-esteem scores in Table \(\PageIndex{1}\), for example, is the difference between the highest score (24) and the lowest score (15). That is, the range is 24 − 15 = 9. Although the range is easy to compute and understand, it can be misleading when there are outliers. Imagine, for example, an exam on which all the students scored between 90 and 100. It has a range of 10. But if there was a single student who scored 20, the range would increase to 80—giving the impression that the scores were quite variable when in fact only one student differed substantially from the rest.

By far the most common measure of variability is the standard deviation. The standard deviation of a distribution is the average distance between the scores and the mean. For example, the standard deviations of the distributions in Figure \(\PageIndex{4}\) are 1.69 for the top distribution and 4.30 for the bottom one. That is, while the scores in the top distribution differ from the mean by about 1.69 units on average, the scores in the bottom distribution differ from the mean by about 4.30 units on average.

Computing the standard deviation involves a slight complication. Specifically, it involves finding the difference between each score and the mean, squaring each difference, finding the mean of these squared differences, and finally finding the square root of that mean. The formula for the standard deviation for a sample looks like this:

s_sample= \[\sqrt{\frac{\displaystyle \sum (X-M)^{2}}{N-1}}\]

The computations for the standard deviation are illustrated for a small set of data in Table \(\PageIndex{2}\). The first column is a set of eight scores that has a mean of 5. The second column is the difference between each score and the mean. The third column is the square of each of these differences. Notice that although the differences can be negative, the squared differences are always positive—meaning that the standard deviation is always positive. At the bottom of the third column is the variance (symbolized \(s\)²), which is the sum of the squares divided by the total number of scores minus 1 (N-1). Although the variance is itself a measure of variability, it generally plays a larger role in inferential statistics than in descriptive statistics. Finally, below the variance is the square root of the variance, which is the standard deviation.

Table \(\PageIndex{3}\): Computations for the Standard Deviation

\(X\)	\(X_M\)	\((X − M)^2\)
3	−2	4
5	0	0
4	−1	1
2	−3	9
7	2	4
6	1	1
5	0	0
8	3	9
M = 5		s^2=28/(8-1)=4.00
		s=√4.00=2.00

If you have already taken a statistics course, you may have learned to divide the sum of the squared differences by N − 1 rather than by N when you compute the variance and standard deviation. Why is this? This is because the standard deviation of a sample tends to be a bit lower than the standard deviation of the population the sample was selected from. Dividing the sum of squares by N − 1 corrects for this tendency and results in a better estimate of the population standard deviation. Because researchers generally think of their data as representing a sample selected from a larger population—and because they are generally interested in drawing conclusions about the population—it makes sense to routinely apply this correction.

Describe to Infer

Now that you've had a refresher on descriptive the data that you collected (or understand the description in an article that you're reading), we will move to another reminder of null hypothesis significance testing, then a brief refresh about the statistical analyses to test research hypotheses. As always, you can find a selection of openly-licensed textbooks on statistics (https://stats.libretexts.org/) on LibreTexts, including textbooks for different social sciences (https://stats.libretexts.org/Bookshe...ied_Statistics).