## 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 quantilesgiven byforIf the data is from a normal population then the pointsshould lie approximately on a straight line. This is called a normal 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.