Understanding Psychology (10th Ed)

variability
The spread, or dispersion, 
of scores in a distribution.
Key Concept 
How can we assess the vari-
ability of a set of data?
M O D U L E 5 7 
Measures of Variability 
FIGURE 1 
Although the mean is iden-
tical in these two distributions, the variability, or spread of scores, is very diff erent. Specifi cally, 
the variability is considerably greater in (a) than in (b).
Mean=50 
Mean=50 
(a)
(b)
The Range: 
Highest Minus Lowest
The simplest measure of variability is the range. The range is the difference between 
the highest score in a distribution and the lowest score. In the following set of scores 
the distribution has a range of 47 (96 
2
49 
5
47).
96 92 90 90 89 86 
86 85 85 83 83 80 
78 78 78 73 72 72 
69 69 59 58 49 
range
The difference between the 
highest score and the lowest score in a 
distribution.
feL82795_app_A2-A19.indd Page A-10 8/6/10 8:14 PM user-f465
feL82795_app_A2-A19.indd Page A-10 8/6/10 8:14 PM user-f465
/Users/user-f465/Desktop
/Users/user-f465/Desktop

---

 
Module 57 
Measures of Variability 
A-11
The fact that a range is simple to calculate is about its only virtue. The problem 
with this particular measure of variability is that it is based entirely on extreme 
scores, and a single score that is very different from the others in a distribution can 
distort the picture of the distribution as a whole. For example, the addition of a score 
of 20 to the test score distribution we are considering would almost double the range 
even though the variability of the remaining scores in the distribution would not 
have changed at all.
The Standard Deviation: 
Diff erences from the Mean
The most frequently used method of characterizing the variability of a distribution 
of scores is the standard deviation. The standard deviation bears a conceptual rela-
tionship to a mean. Recall that the mean is the average score in a distribution of 
scores. A standard deviation is an index of the average deviation of a set of scores 
from the center of the distribution.
Consider, for instance, the distributions in Figure 1. The distribution on the left 
is widely dispersed; on the average an individual score in the distribution can be 
thought of as deviating quite a bit from the center of the distribution. Certainly the 
scores in the distribution on the left are going to deviate more from the center of the 
distribution than those in the distribution on the right. 
In contrast, in the distribution on the right, the scores are closely packed together 
and there is little deviation of a typical score from the center of the distribution. On 
the basis of this analysis, then, it would be expected that a good measure of vari-
ability would yield a larger value for the distribution on the left than it would for 
the one on the right—and, in fact, a standard deviation would do exactly this by 
indicating how far away a typical score lies from the center of the distribution. 
In a normal distribution, 68% of the scores fall within one standard deviation of 
the mean (34% on either side of it), 95% of the scores fall within two standard devi-
ations, and 99.7% fall within three standard deviations. In the general population, IQ 
scores of intelligence fall into a normal distribution, and they have a mean of 100 
and a standard deviation of 15. Consequently, an IQ score of 100 does not deviate 
from the mean, whereas an IQ score that is three standard deviations above the mean 
(or 145) is very unusual (higher than 99% of all IQ scores). 
The calculation of the standard deviation follows the logic of calculating the dif-
ference of individual scores from the mean of the distribution (see Figure 2). Not 
only does the standard deviation provide an excellent indicator of the variability of 
a set of scores, it provides a means for converting initial scores on standardized tests 
such as the SAT (the college admissions exam) into the scales used to report results. 
In this way, it is possible to make a score of 585 on the verbal section of the SAT 
exam, for example, equivalent from one year to the next even though the specifi c test 
items differ from year to year.