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

Typically raw data that may possibly be fitted by aPoisson distribution comes summarised in a frequency table. The tablebelow contains data that shows the numbers of errors per page made bya 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 hasonly one variable – the mean. For the Poisson distribution, thevariance is equal to the mean. We can find the variance and mean ofthe above data, and if they are approximately equal, then a Poissondistribution may be possible.

The mean of the above table is

and the variance is

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