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Part III: Distributions and the Central Limit Theorem
In Problem 3, you find p(0 < Z < 2.00); this requires Step 5c. First find p(Z <
2.00), which is 0.9772 from the Z-table. Then find p(Z < 0), which is 0.5000
from the Z-table. Subtract them to get 0.9772 – 0.5000 = 0.4772. The chance
of a fish being between 16 and 24 inches is 0.4772.
The Z-table does not list every possible value of Z; it just carries them out to two
digits after the decimal point. Use the one closest to the one you need. And just
like in an airplane where the closest exit may be behind you, the closest z-value
may be the one that is lower than the one you need.
Finding X When You Know the Percent
Another popular normal distribution problem involves finding percentiles
for X (see Chapter 5 for a detailed rundown on percentiles). That is, you are
given the percentage or probability of being at or below a certain x-value,
and you have to find the x-value that corresponds to it. For example, if you
know that the people whose golf scores were in the lowest 10% got to go to
the tournament, you may wonder what the cutoff score was; that score would
represent the 10th percentile.
A percentile isn’t a percent. A percent is a number between 0 and 100; a
percentile is a value of X (a height, an IQ, a test score, and so on).
Figuring out a percentile
for a normal distribution
Certain percentiles are so popular that they have their own names and their
own notation. The three “named” percentiles are Q — the first quartile,
1
or the 25th percentile; Q — the 2nd quartile (also known as the median or
2
the 50th percentile); and Q — the 3rd quartile or the 75th percentile. (See
3
Chapter 5 for more information on quartiles.)
Here are the steps for finding any percentile for a normal distribution X:
1a. If you’re given the probability (percent) less than x and you need to
find x, you translate this as: Find a where p(X < a) = p (and p is the

given probability). That is, find the pth percentile for X. Go to Step 2.
1b. If you’re given the probability (percent) greater than x and you need
to find x, you translate this as: Find b where p(X > b) = p (and p is
given). Rewrite this as a percentile (less-than) problem: Find b where
p(X < b) = 1 – p. This means find the (1 – p)th percentile for X.

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