## Testing for Goodness of Fit to a Poisson Distribution by Comparing Variance and Mean

Typically raw data that may possibly be fitted by a Poisson distribution comes summarised in a frequency table. The table below contains data that shows the numbers of errors per page made by a secretary, and the associated frequencies.

No. of errors | 0 | 1 | 2 | 3 | 4 | 5 |

No. of Pages | 37 | 65 | 60 | 49 | 27 | 12 |

It is a feature of the Poisson distribution that it has only one variable – the mean. For the Poisson distribution, the variance is equal to the mean. We can find the variance and mean of the above data, and if they are approximately equal, then a Poisson distribution may be possible.

The mean of the above table is

and the variance is

The sample variance is very close to the mean, so a Poisson distribution is possible. To be conclusive we would need to conduct a hypothesis test.