Quantile Plot
Quantile plots can be used to determine if
a data set is normally distributed.
Construct a quantile plot of the data. This can be done by plotting
the data and its probability value on special log normal plotting paper
or the data can be transformed into lognormal values and plotted. The value
of the standard normal variable, z, is used as the horizontal axis
to linearize the plot; this is equivalent to using normal probability plotting
paper. (from Applied Hydrology, Chow/Maidment/Mays, Chapter 12)
The following is the procedure for transforming the data.
i) Arrange the data in descending order
ii) Give each value a rank number, i, .
iii) Calculate the probability value p (Blom, 1958) for each
data point: 
iv) An intermediate variable w is calculated: 
v) The frequency factor for normal distribution, z, is calculated
using the following formula: 
vi) Plot the data versus the corresponding value of z.
vii) Plot the normal distribution values which correspond to the data
set versus the corresponding value of z using the following formula for
the predicted value: 
where:
is the mean of the data set,
m is the standard deviation of the data set
From the plot a visual inspection of how close to two plots correspond
can give an indication of whether the data is normally distributed and
the effects of extreme values. A more accurate method is to calculate the
correlation coefficient of the two lines. This value can be compared to
the values in Table 1 (from Handbook of Hydrology, Maidment, Table
18.3.3), which is specifically for pi = (i-0.375)/(n+0.25),
to determine if the data is normally distributed.
Table 1
Just one or a few outlier data points can cause the correlation coefficient
to fall beyond the limits specified in the table. If this is the case a
judgement should be made as to whether these values should be excluded
from the calculation. Figure 1 is shows the spreadsheet used to construct
the quantile plot shown in Figure 2.
Figure 1
In this example the two extreme data values (16.15 and 13.4 mg/l) caused
the correlation factor be lower than the lower limit (0.927 vs 0.971) for
normal distribution. By eliminating these values the correlation coefficient
value is 0.997.
Figure 2
These results indicate that the data set has a normal distribution.