Page 161 - Statistics for Dummies

Every normal distribution has certain properties. You use these properties to
determine the relative standing of any particular result on the distribution, and
to find probabilities. The properties of any normal distribution are as follows:
✓ Its shape is symmetric (that is, when you cut it in half the two pieces are
mirror images of each other).
✓ Its distribution has a bump in the middle, with tails going down and out
to the left and right.
✓ The mean and the median are the same and lie directly in the middle of
the distribution (due to symmetry).
✓ Its standard deviation measures the distance on the distribution from
the mean to the inflection point (the place where the curve changes from
an “upside-down-bowl” shape to a “right-side-up-bowl” shape).
✓ Because of its unique bell shape, probabilities for the normal distribu-
tion follow the Empirical Rule (full details in Chapter 5), which says the
following: Chapter 9: The Normal Distribution 145
• About 68 percent of its values lie within one standard deviation
of the mean. To find this range, take the value of the standard devi-
ation, then find the mean plus this amount, and the mean minus
this amount.
• About 95 percent of its values lie within two standard deviations of
the mean. (Here you take 2 times the standard deviation, then add
it to and subtract it from the mean.)
• Almost all of its values (about 99.7 percent of them) lie within
three standard deviations of the mean. (Take 3 times the standard
deviation and add it to and subtract it from the mean.)
✓ Precise probabilities for all possible intervals of values on the normal
distribution (not just for those within 1, 2, or 3 standard deviations from
the mean) are found using a table with minimal (if any) calculations.
(The next section gives you all the info on this table.)
Take a look again at Figure 9-1. To compare and contrast the distributions
shown in Figure 9-1a, b, and c, you first see they are all symmetric with the
signature bell shape. The examples in Figure 9-1a and Figure 9-1b have the
same standard deviation, but their means are different; Figure 9-1b is located
30 units to the right of Figure 9-1a because its mean is 120 compared to 90.
Figures 9-1a and c have the same mean (90), but Figure 9-1a has more variabil-
ity than Figure 9-1c due to its higher standard deviation (30 compared to 10).
Because of the increased variability, the values in Figure 9-1a stretch from 0 to
180 (approximately), while the values in Figure 9-1c only go from 60 to 120.
Finally, Figures 9-1b and c have different means and different standard devia-
tions entirely; Figure 9-1b has a higher mean which shifts it to the right, and
Figure 9-1c has a smaller standard deviation; its values are the most concen-
trated around the mean.

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