1
Normal Probability Plots
The following example shows how to construct a normal probability plot (also called a normal
quantile-quantile plot) to determine whether it is reasonable to believe that a data set was sampled
from a normal distribution.
A soft drink bottler is studying the internal pressure strength of 1-liter glass bottles. A random
sample of 16 bottles is tested, and the pressure strengths are obtained. The r.v. X in this case is the
internal pressure strength of a randomly selected bottle. We want to decide whether it is
reasonable to conclude that X is normally distributed. The data values are listed in the fourth
column of the table below, which gives the Excel development of the normal quantile-quantile plot.
The values in the second column are the cumulative relative frequencies, found by calculating
i  0.5
 i  0.5

, 0,1 .
. The third column lists the normal quantiles, found using zi   NORMINV 
n
 n

Each of these numbers is the cut-off point on the standard normal scale corresponding to a lefthand area of (i – 0.5)/n. The NORMINV function of Excel has three arguments – the left-hand tail
probability (here given in column 2), the mean of the particular normal distribution (here taken to
be 0), and the standard deviation of the normal distribution (here taken to be 1). The fourth column
of the table below lists the ordered values of the data. The values in the last column are the
standardized ordered values of the data, found by subtracting the sample mean from each data
value, then dividing the result by the sample standard deviation. These standardized scores will be
plotted against the expected standard normal quantiles in the third column.
I
i  0.5
n
 i  0.5

zi   NORMINV 
, 0,1
 n

x i 
x i   x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.03125
0.09375
0.15625
0.21875
0.28125
0.34375
0.40625
0.46875
0.53125
0.59375
0.65625
0.71875
0.78125
0.84375
0.90625
0.96875
-1.862731
-1.318011
-1.009990
-0.776422
-0.579132
-0.402250
-0.237202
-0.078412
0.078412
0.237202
0.402250
0.579132
0.776422
1.009990
1.318011
1.862731
188.12
193.71
193.73
195.45
200.81
201.63
202.20
202.21
203.62
204.55
208.15
211.14
219.54
221.31
224.39
226.16
s
-1.54904
-1.06597
-1.06424
-0.9156
-0.4524
-0.38154
-0.33228
-0.33141
-0.20956
-0.12919
0.18191
0.440299
1.16621
1.31917
1.585337
1.738297
What we want to do in order to construct the graph is the following:
1) Insert two columns before column 5. Copy column 3 into each of these two columns.
2) Highlight the last three columns in the table (the two columns just inserted, plus the column of
standardized order statistics).
2
3) Go to Insert, Chart, and choose Scatterplot. Follow the prompts in the dialog boxes. You will
need to create a title for the graph (e.g., Normal Q-Q Plot for Bottle Strength Data). You will
also need to label the axes. The vertical axis should be labeled, “Standardized Order Statistics
of the Data.” The horizontal axis should be labeled, “Standard Normal Quantiles.”
4) The resulting graph is shown below. It appears that the data points do not differ substantially
from the straight line, so it is reasonable to conclude that the data were sampled from a normal
distribution.
Normal Probability Plot, Ch. 3, Exercise 3-63
Standardized Order Statistics
2.5
2
1.5
1
0.5
0
-3
-2
-1
-0.5 0
1
2
3
-1
-1.5
-2
-2.5
Standard Normal Quantiles
Note: The points indicated by squares are obtained from the data by plotting the standardized data
values against the calculated standard normal quantiles. The points indicated by diamonds are
obtained by plotting the standard normal quantiles against themselves. The closer the squares are
to the diamonds, overall, the more plausible it is that the data were sampled from a normally
distributed population.