Mean, median and mode for a discrete distribution

λ (mean):

When the median, mode and mean of a distribution are different, then the median is usually between the mode and the mean, i.e. either


However, this is not always true.

The graph shows the number of accidents per week on a stretch of road, averaged over many weeks.

This follows a Poisson distribution. This can only take whole-number values: there is no such thing as half a road accident.

The Poisson distribution is described completely by one parameter called λ (lambda), which you can think of as measuring the "dangerousness" of the road.

Note the skewness of this distribution. There is a short tail to the left, because you can't have fewer than zero accidents per week. There is a long tail to the right (positive) side.

So the skewness is positive, and the amount of skewness is shown below the graph.

In the interests of science, you can add roadworks and potholes to the stretch of road to make accidents more common (increase λ).

Click on the graph to move λ to a particular value.

Is it always either

mean-median-mode  ?

(Note that when λ is a whole number, the distribution has two modes.)