# Solved – Can variance be equal to mean

meanpoisson distributionvariance

I understand that variance can be zero if all data points are equal.

Can anybody state an example, where sample variance is equal to sample mean and what – if any – is its significance to distributions like the Poisson.

I'll assume you only want samples consisting of non-negative integers (otherwise its obviously not Poisson). Indeed, without those restrictions, it's pretty trivial - though largely meaningless, because the variance-ratio changes as you change scale. (It makes more sense with counts though.)

I'll also assume you want the $n-1$ form of the sample variance (the unbiased form).

Some examples:

n=2: a) 0 1
b) 1 3
c) 3 6
d) 6 10 (you may discern a pattern here, and yes, it holds in general for n=2)

n=3: a) 0 1 2
b) 2 4 6
c) 4 6 9

n=4: a) 1 1 2 4
b) 3 3 3 7

n=7: 1 1 2 3 4 5 5

n=11: 1 1 1 2 3 3 3 4 5 5 6


These took me a few minutes to construct.

It doesn't really have much significance other than giving a lack of evidence against the Poisson (at least on the basis of the ratio of variance to mean). In no way does it tell you that the data is Poisson.

Edit: For example, here's a sample that is pretty obviously not Poisson (in the sense that the chance that you could end up with a sample like that from a Poisson is reaally small):

n=21:  2 2 2 2 2 2 2 2 2 2 4 6 6 6 6 6 6 6 6 6 6


For starters, all the values are even!

Here's another that's pretty clearly not Poisson:

n=9: 3 10 10 10 10 10 10 14 14


Edit: here's a biggish sample that's not so plainly inconsistent with Poisson:

n=101:
1  2  4  5  6  6  6  6  6  6  7  7  7  7  7  7  7  7  7  7  7  7  7  8  8
8  8  8  8  9  9  9  9  9  9  9  9  9  9  9  9  9  9  9  9 10 10 10 10 10
10 10 10 10 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11
11 11 12 12 12 12 12 12 12 12 13 13 13 13 14 14 14 14 14 14 15 16 16 17 18
22


... though it's a bit too kurtotic to really be very consistent with a Poisson.