Statistics in Psychology A-9This procedure may look very complicated. Let us assure you that computers
and inexpensive calculators can figure out the standard deviation simply by enter-
ing the numbers and pressing a button. No one does a standard deviation by hand
anymore.
How does the standard deviation relate to the normal curve? Let’s look at the
classic distribution of IQ scores. It has a mean of 100 and a standard deviation of 15
as set up by the test designers. It is a bell curve. With a true normal curve, researchers
know exactly what percentage of the population lies under the curve between each
standard deviation from the mean. For example, notice that in the percentages in Fig-
ure A.8, one standard deviation above the mean has 34.13 percent of the population
represented by the graph under that section. These are the scores between the IQs of
100 and 115. One standard deviation below the mean (−1) has exactly the same per-
cent, 34.13, under that section—the scores between 85 and 100. This means that 68.26
percent of the population falls within one standard deviation from the mean, or one
average “spread” from the center of the distribution. For example, “giftedness” is nor-
mally defined as having an IQ score that is two standard deviations above the mean.
On the Wechsler Intelligence Scales, this means having an IQ of 130 or greater because
the Wechsler ’s standard deviation is 15. But if the test a person took to determine
giftedness was the Stanford-Binet Fourth Edition (the previous version of the test),
the IQ score must have been 132 or greater because the standard deviation of that test
was 16, not 15. The current version, the Stanford-Binet Fifth Edition, was published
in 2003, and it now has a mean of 100 and a standard deviation of 15 for composite
scores.
Although the “tails” of this normal curve seem to touch the bottom of the graph,
in theory they go on indefinitely, never touching the base of the graph. In reality,
though, any statistical measurement that forms a normal curve will have 99.72 percent
of the population it measures falling within three standard deviations either above or
below the mean. Because this relationship between the standard deviation and the nor-
mal curve does not change, it is always possible to compare different test scores or sets
of data that come close to a normal curve distribution. This is done by computing a z
score, which indicates how many standard deviations you are away from the mean.
It is calculated by subtracting the mean from your score and dividing by the standard
deviation. For example, if you had an IQ of 115, your z score would be 1.0. If you had an
IQ of 70, your z score would be −2.0. So on any exam, if you had a positive z score, you
did relatively well. A negative z score means you didn’t do as well. The formula for a z
score is:
Z = (X − M)#SDFigure A.8 IQ Normal Curve
Scores on intelligence tests are typically
represented by the normal curve. The
dotted vertical lines each represent one
standard deviation from the mean, which
is always set at 100. For example, an IQ of
116 on the Stanford-Binet Fourth Edition
(Stanford-Binet 4) represents one standard
deviation above the mean, and the area
under the curve indicates that 34.13 percent
of the population falls between 100 and
116 on that test. The Stanford-Binet Fifth
Edition was published in 2003 and it now
has a mean of 100 and a standard deviation
of 15 for composite scores.23
55
52
0.13524
40
36
0.00322
70
68
2.27521
85
84
15.8560
100
100
50.001
115
116
84.1342
130
132
97.7253
145
148
99.8654
160
164
99.997Standard Deviations
Wechsler IQ
Stanford-Binet 4 IQ
Cumulative %34.13%z score
a statistical measure that indicates
how far away from the mean a
particular score is in terms of the
number of standard deviations that
exist between the mean and that
score.Z01_CICC7961_05_SE_APPA.indd 9 9/2/16 11:57 PM