From Appendixzwe know that the area from the mean to z 52 2.0 is 0.4772 and from
the mean to z 52 1.0 is 0.3413. The difference is these two areas must represent the area
between z 52 2.0 and z 52 1.0. This area is 0.4772 2 0.3413 5 0.1359. Thus, the proba-
bility that Behavior Problem scores drawn at random from a normally distributed popula-
tion will be between 30 and 40 is .1359.
Discussing areas under the normal distribution as we have done in the last two para-
graphs is the traditional way of presenting the normal distribution. However, you might le-
gitimately ask why I would ever want to know the probability that someone would have a
Total Behavior Problem score between 50 and 60. The simple answer is that you probably
don’t care. But, suppose that you took your child in for an evaluation because you were
worried about his behavior. And suppose that your child had a score of 75. A little arith-
metic will show that z 5 (75 – 50)/10 5 2.5, and from Appendix zwe can see that only
0.62% of normal children score that high. If I were you, I’d start worrying. Seventy five
really is a high score.
3.4 Setting Probable Limits on an Observation
For a final example, consider the situation in which we want to identify limits within
which we have some specified degree of confidence that a child sampled at random will
fall. In other words we want to make a statement of the form, “If I draw a child at ran-
dom from this population, 95% of the time her score will lie between and
.” From Figure 3.9 you can see the limits we want—the limits that include 95%
of the scores in the population.
If we are looking for the limits within which 95% of the scores fall, we also are look-
ing for the limits beyond which the remaining 5% of the scores fall. To rule out this remain-
ing 5%, we want to find that value of zthat cuts off 2.5% at each end, or “tail,” of the
distribution. (We do not need to use symmetric limits, but we typically do because they
usually make the most sense and produce the shortest interval.) From Appendixzwe see
that these values are z 56 1.96. Thus, we can say that 95% of the time a child’s score sam-
pled at random will fall between 1.96 standard deviations above the mean and 1.96 stan-
dard deviations below the mean.
Because we generally want to express our answers in terms of raw Behavior Problem
scores, rather than zscores, we must do a little more work. To obtain the raw score limits, we
simply work the formula for zbackward, solving for Xinstead of z. Thus, if we want to state
Section 3.4 Setting Probable Limits on an Observation 75
–3.0
z
f(
X
)
0.40
0.30
0.20
0.10
0
–2.0 –1.0 0 1.0 2.0 3.0
Figure 3.8 Area between 1.0 and 2.0 standard deviations below the mean
