University Maths Notes: Probability and Statistics – Normal Probability Plots – Testing for Normality
Suppose we have n observations
–
which we assume are in ascending order - and we want to know whether
they could have arisen from a normal distribution. We plot the data
points against the corresponding standard normal quantiles
given
by
for
If
the data is from a normal population then the points
should
lie approximately on a straight line. This is called anormal
probability plot and is especially suited to small data sets.
Example:
We have some silver coins from ancient Byzantine. We want to test whether the % silver content follows a normal distribution.
|
5.9 |
6.8 |
6.4 |
7 |
6.6 |
7.7 |
7.2 |
6.9 |
6.2 |
First arrange the data in order.
|
5.9 |
6.2 |
6.4 |
6.6 |
6.8 |
6.9 |
7 |
7.2 |
7.7 |
Now find the quantiles:
etc.
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
|
5.9 |
6.2 |
6.4 |
6.6 |
6.8 |
6.9 |
7 |
7.2 |
7.7 |
|
|
0.1 |
0.2 |
0.3 |
0.4 |
0.5 |
0.6 |
0.7 |
0.8 |
0.9 |
|
|
-1.282 |
-0.842 |
-0.524 |
-0.253 |
0.000 |
0.253 |
0.524 |
0.842 |
1.282 |
Plot the points
The correlation coefficient (
)
indicates a very good fit to a straight line, hence we may take the
silver content as normally distributed.
