Basic Statistics

AREAS UNDER THE NORMAL CURVE 65

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68 inches

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60 inches

Height (in.)

Figure 6.1 Normal curves with different means.

Height (in.)

Figure 6.2 Normal curves with different standard deviations.

dard deviation, we know that the data are normally distributed, the entire frequency

distribution can be drawn. For a population with a normal distribution, the mean and

standard deviation tell the whole story.

6.2 AREAS UNDERTHE NORMAL CURVE

As we discussed in Section 4.2 on constructing histograms, the proportion of obser-
vations that fall between any two numbers, say XI and X2, can be determined by
measuring the area under the histogram between XI and Xz. Similarly, if a popula-
tion is normally distributed, the proportion of X’s that lie between XI and Xz is equal
to the area above the horizontal axis and under the normal curve and lying between
XI and X2.
It becomes important to measure these areas, and tables have been prepared that
do so. Since it would be impossible to make a table for every normal curve, a
transformation (a change in the mean and standard deviation) is made on the data so
that a table for just one particular normal curve, called the standard normal curve,
will suffice.