Variability is a numerical way of describing how much spread there is in a distribution.

MASTERING THE CONCEPT

4.3: Variability is the second most common concept (after central tendency) to help us understand the shape of a distribution. Common indicators of variability are range, variance, and standard deviation.

The range is a measure of variability calculated by subtracting the lowest score (the minimum) from the highest score (the maximum).

MASTERING THE FORMULA

4-2: The formula for the range is: range = X_highest − X_lowest. We simply subtract the lowest score from the highest score to calculate the range.

EXAMPLE 4.8

We can determine the highest and lowest scores either by reading through the data or, more easily, by glancing at the frequency table for these data.

STEP 1: Determine the highest score.

In this case, the highest score is 10.

STEP 2: Determine the lowest score.

In this case, the lowest score is 1.

STEP 3: Calculate the range by subtracting the lowest score from the highest score:

The range can be a useful first indicator of variability, but it is influenced only by the highest and lowest scores. All the other scores in between could be clustered near the highest score, huddled near the center, spread out evenly, or have some other unexpected pattern. We can’t know based only on the range.

Variance is the average of the squared deviations from the mean.

EXAMPLE 4.9

Students who seek therapy at university counseling centers often do not attend many sessions. For example, in one study, the median number of therapy sessions was 3 and the mean was 4.6 (Hatchett, 2003). Let’s examine the spread of fictional scores for a sample of five students: 1, 2, 4, 4, and 10 therapy sessions, with a mean of 4.2. We find out how far each score deviates from the mean by subtracting the mean from every score. First, we label the column that lists the scores with an X. Here, the second column includes the results we get when we subtract the mean from each score, or X − M. We call each of these a deviation from the mean (or just a deviation)—the amount that a score in a sample differs from the mean of the sample.

But we can’t just take the mean of the deviations. If we do (and if you try this, don’t forget the signs—negative and positive), we get 0—every time. Are you surprised? Remember, the mean is the point at which all scores are perfectly balanced. Mathematically, the scores have to balance out. Yet we know that there is variability among these scores. The number representing the amount of variability is certainly not 0!

When we ask students for ways to eliminate the negative signs, two suggestions typically come up: (1) Take the absolute value of the deviations, thus making them all positive, or (2) square all the scores, again making them all positive. It turns out that the latter, squaring all the deviations, is how statisticians solve this problem. Once we square the deviations, we can take their average and get a measure of variability. Later (using a beautifully descriptive term created by our students), we will “unsquare” those deviations in order to calculate the standard deviation.

To recap:

STEP 1: Subtract the mean from every score.

We call these deviations from the mean.

STEP 2: Square every deviation from the mean.

We call these squared deviations.

STEP 3: Sum all of the squared deviations.

This is often called the sum of squared deviations, or the sum of squares, for short.

STEP 4: Divide the sum of squares by the total number in the sample (N).

This number represents the mathematical definition of variance—the average of the squared deviations from the mean.

To calculate the variance for the therapy session data, we add a third column to contain the squares of each of the deviations. Then we add all of these numbers up to compute the sum of squares (symbolized as SS), the sum of each score’s squared deviation from the mean. In this case, the sum of the squared deviations is 48.80, so the average squared deviation is 48.80/5 = 9.76. Thus, the variance equals 9.76.

Language Alert! We need a few more symbols to use symbolic notation to represent the idea of variance. Each of these symbols represents the same idea (variance) applied to slightly different situations. The symbols that represent the variance of a sample include SD², s², and MS. The first two symbols, SD² and s², both represent the words standard deviation squared. The symbolic notation MS comes from the words mean square (referring to the average of the squared deviations). We’ll use SD² at this point, but we will alert you when we switch to other symbols for variance later. The variance of the sample uses all three symbolic notations; however, the variance of a population uses just one symbol: σ² (pronounced “sigma squared”). Table 4-2 summarizes the symbols and language used to describe different versions of the mean and variance, but we will keep reminding you as we go along.

We already know all the other symbols needed to calculate variance: X to indicate the individual scores, M to indicate the mean, and N to indicate the sample size.

As you can see, variance is really just a mean—the mean of squared deviations.

EXAMPLE 4.10

Language Alert! Variance and standard deviation refer to the same core idea. The standard deviation is more useful because it is the typical amount that each score varies from the mean. Mathematically, the standard deviation is the square root of the average of the squared deviations from the mean, or, more simply, the square root of the variance. The beauty of the standard deviation—compared to the variance—is that we can understand it at a glance.

For example, the numbers of therapy sessions for the five students were 1, 2, 4, 4, and 10, with a mean of 4.2. The typical score does not vary from the mean by 9.76. The variance is based on squared deviations, not deviations, so it is too large. When we ask our students how to solve this problem, they invariably say, “Unsquare it,” and that’s just what we do. We take the square root of variance to come up with a much more useful number, the standard deviation. The square root of 9.76 is 3.12. Now we have a number that “makes sense” to us. We can now say that the typical number of therapy sessions for students in this sample is 4.2 and the typical amount a student varies from that is 3.12.

As you read journal articles, you often will see the mean and standard deviation reported as: (M = 4.2, SD = 3.12). A glance at the original data (1, 2, 4, 4, 10) tells us that these numbers make sense: 4.2 does seem to be approximately in the center, and scores do seem to vary from 4.2 by roughly 3.12. The score of 10 is a bit of an outlier—but not so much of one; the mean and the standard deviation are still somewhat representative of the typical score and typical deviation.

We didn’t actually need a formula to get the standard deviation. We just took the square root of the variance. Perhaps you guessed the symbols for standard deviation by just taking the square root of those for variance. With a sample, standard deviation is either SD or s. With a population, standard deviation is σ. Table 4-2 presents this information concisely. We can write the formula showing how standard deviation is calculated from variance:

We can also write the formula showing how standard deviation is calculated from the original X’s, M, and N:

Next Steps

The Interquartile Range

As we noted earlier in the chapter, the range has a major limitation: it is completely dependent on the maximum and minimum scores. For example, the $17 million home at the high end or the shack at the low end of the distribution skew a distribution of home prices. Whenever we have outliers, the range will be an exaggerated measure of the variability. Fortunately, we have an alternative to the range: the interquartile range.

The interquartile range is a measure of the distance between the first and third quartiles. As we learned earlier, the median marks the 50th percentile of a data set. Similarly, the first quartile marks the 25th percentile of a data set, and the third quartile marks the 75th percentile of a data set. Essentially, the first and third quartiles are the medians of the two halves of the data—the half below the median and the half above the median. We calculate the first quartile and the third quartile in a similar manner to how we calculate the median.

Here are the steps for finding the interquartile range:

Step 1: Calculate the median.

Step 2: Look at all of the scores below the median. The median of these scores, the lower half of the scores, is the first quartile, often called Q1 for short.

Step 3: Look at all of the scores above the median. The median of these scores, the upper half of the scores, is the third quartile, often called Q3 for short.

Step 4: Subtract Q1 from Q3. The interquartile range, often abbreviated as IQR, is the difference between the first and third quartiles: IQR = Q3 − Q1.

Because the interquartile range is the distance between the 25th and 75th percentile of the data, it can be thought of as the range of the middle 50% of the data.

The interquartile range has an important advantage over the range. Because it is not based on the minimum and maximum—the most extreme scores—it is less susceptible to outliers. Let’s look at an example.

CHECK YOUR LEARNING

Clarifying the Concepts

• 4-6 In your own words, what is variability?
• 4-7 Distinguish the range from the standard deviation. What does each tell us about the distribution?

Calculating the Statistics

• 4-8 Calculate the range, variance, and standard deviation for the following data sets (the same ones from the section on central tendency).
  1. 10 8 22 5 6 1 19 8 13 12 8
  2. 122.5 123.8 121.2 125.8 120.2 123.8 120.5 119.8 126.3 123.6
  3. 0.100 0.866 0.781 0.555 0.222 0.245 0.234

Applying the Concepts

• 4-9 Final exam week is approaching, and students are not eating as well as usual. Four students were asked how many calories of junk food they had consumed between noon and 10:00 p.m. on the day before an exam. The estimated numbers of nutritionless calories, calculated with the help of a nutritional software program, were 450, 670, 1130, and 1460.
  1. Using the formula, calculate the range for these scores.
  2. What information can’t you glean from the range?
  3. Using the formula, calculate variance for these scores.
  4. Using the formula, calculate standard deviation for these scores.
  5. If a researcher were interested only in these four students, what symbols would he use for variance and standard deviation, respectively?
  6. If another researcher hoped to generalize from these four students to all students at the university, what symbols would she use for variance and standard deviation?