# Is my variance compatible with a Poisson distribution

I have data coming from a genomics experiment which consists of 100s of thousands of observations (RNA sequencing reads) from 18 different indeviduals. Some of the samples come from one tissue some from another. The number of reads (observations) for each sample is different.

For each sequencing read I have determined whether it fits a particular criteria (whether its alignment to the genome contains a gap). Thus the data looks something like:

|patient|tissue|gapped|total|percent|
|-------+------+------+-----+-------|
|     1 |    1 |  100 |10000|     1 |
|     1 |    2 |  400 |25000|   1.6 |
¦       ¦      ¦      ¦     ¦       ¦


I wish to test if the rate of gapped alignments is significantly different between tissues. The obvious model is Poisson with an offset. The model seems to fit reasonably well (studying residual vs fitted and scale-location plots etc).

As a summary I plotted a violin plot of the log percents, which with this many observations, you'd expect to be gaussian, and indeed they are. But I noticed that the variance is smaller than the mean, which shouldn't be the case if the data is Poisson right? The sample variance is around 20-50% of the mean.

1. How far from the mean might the sample variance of multiple observations from a Poisson distribution be?

2. Is this a problem? Am I missing something?