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p
0.5
p
p 0.95 p
0.9 = 10.23
0.99
0 2 4 6 8 10
y
p
0.5

p
0.9
p
0.05
p
0.99
-4 -3 -2 -1 0 1 2 3 4
x = In(y)

FIGURE 8.1 Correspondence of percentiles on the lognormal and normal distributions. The transformation x = ln(y) converts
the lognormal distribution to a normal distribution. The percentiles also transform.

The normal distribution is completely specified by the mean η and standard deviation σ of the
distribution, respectively. The true pth quantile of the normal distribution is y p = η + z p σ, where z p is
the pth quantile of the standard normal distribution. Generally, the parameters η and σ are unknown and
we must estimate them by the sample average, , and the sample standard deviation, s. The quantile, y p ,y
of a normal distribution is estimated using:

y ˆ p = y + z p s

The appropriate value of z p can be found in a table of the normal distribution.

Example 8.1

Suppose that a set of data is normally distributed with estimated mean and standard deviation
of 10.0 and 1.2. To estimate the 99th quantile, look up z 0.99 = 2.326 and compute:

(
y ˆ p = 10 + 2.326 1.2) = 12.8
This method can be used even when a set of data indicates that the population distribution is not
normally distributed if a transformation will make the distribution normal. For example, if a set of
observations y appears to be from a lognormal distribution, the transformed values x = log(y) will be
normally distributed. The pth quantile of y on the original measurement scale corresponds to the pth
quantile of x on the log scale. Thus, x p = log(y p ) and y p = antilog(x p ).

Example 8.2

A sample of observations, y, appears to be from a lognormal distribution. A logarithmic trans-
formation, x = ln(y), produces values that are normally distributed. The log-transformed values
have an average value of 1.5 and a standard deviation of 1.0. The 99th quantile on the log scale
is located at z 0.99 = 2.326, which corresponds to:

(
x ˆ 0.99 = 1.5 + 2.326 1.0) = 3.826.
© 2002 By CRC Press LLC