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Measures of Variability

How much do students differ from one another in their perceptions of their instructor? Some students will view their instructor as a high performer, and others may view the same instructor as not so great. In other words, there will be variability in how students view the instructor’s performance.

Alternatively, consider a manufacturing process with two machines producing the same product. During a given day, one of the machines produces parts that vary in length by .01 inches, and the other machine produces parts that vary by .001 inches. The production manager will be interested in the variability of the lengths and may have a problem with the first machine if the standard variability is supposed to be less than .01 inches.

Variability in the real world is everywhere, and researchers seek to understand and explain it. Researchers often ask, “How much variance is explained?” A first step in explaining variability is measuring it.

Variance and Standard Deviation

There are a number of measures of variability, but variance and standard deviation are the two measures most often used for continuous variables (i.e., those involving interval and ratio levels of measurement). As you’ll see below, these two measures are related to one another. Standard deviation, however, is the measure of variability that researchers typically report because it conveys the amount of variability using the same units as that of the mean (and also the same units as that of the variable itself). This allows readers of the research to easily understand the typical number of units that individuals in the sample vary from the mean.

Fortunately, you usually do not need to perform the calculations for variance or standard deviation by hand; instead, you can rely on programs like SPSS to do the work for you. It’s important, though, to understand, to a degree, how these measures of variability are calculated. For example, note the subtraction step in calculating the variance. With ratio and interval scales of measurement, differences between items being measured in the real-world can be quantified, and hence, the subtraction process is meaningful for interval and ratio data. Because nominal data do not quantify differences between elements in the real world in a meaningful way, the standard deviation and variance are not used with this scale of measurement. Although the same could be argued for ordinal data, many researchers report the standard deviation for this type of data. If a researcher reports a standard deviation or mean, there has been an implicit willingness to accept the data as being at an interval or ratio level of measurement

Understanding and Reporting Variability Measures

Descriptive Statistics

Consider the following scenarios.  

Scenario 1

The data presented in the table above show summary information about performance ratings for two different instructors.  One had a reputation for being predictable and the other controversial. Each instructor’s performance was rated by 120 students. 

Table: Performance Rating for Two Different Instructors

Teacher	N	Mean	Standard Deviation	Variance
Predictable Teacher	120	6.8740	1.01636	1.033
Controversial Teacher	120	7.2828	2.95288	8.719

Although the means (simple averages) for the two teachers differ by less than one-half unit, note that the standard deviation for the controversial teacher is almost three times larger than the standard deviation for the predictable teacher. 

Because you can think of the standard deviation as being an approximate average difference between the observations and the mean, for the first instructor, the average difference between a student’s rating and the mean of the 120 students’ rating is about 1.0.  For the second instructor, the average difference is almost 3.0. Remember, also, that the variance is the standard deviation squared. In the example, the ratio of the two variances is almost 9 to 1. Often times, knowing about variability in data will be as important as knowing about differences in means.

Researchers typically report the standard deviation along with the mean. For example, the results in the table above could be reported as, “The mean rating for the predictable teacher was 6.87 (SD = 1.02), and the mean for the controversial teacher was 7.28 (SD = 2.95).” You will want to use a similar format when you report your own research results.

At the end of a statistics course, the 27 students in the class were asked to rate the instructor on a number scale of 1 to 9 (1 being "very poor," and 9 being "best instructor I've ever had"). The following table provides three sets of hypothetical data:

Table: Student Rating for the Instructor

Rating	1	2	3	4	5	6	7	8	9
Class I	1	0	0	0	22	0	0	0	1
Class II	12	0	0	0	1	0	0	0	12
Class III	2	2	2	2	2	2	2	2	2

Note: The average rating in each of the three classes is 5 (which should be visually clear from the histograms), and recall that the interpretation of the SD is the “typical” or “average” distance between the data points and their mean.