## This site is no longer maintained and has been left for archival purposes

## Text and links may be out of date

As background to the Poisson distribution, we should
compare the treatment of random count data with the
treatment of measurement data. Suppose that we did a
survey of the · so the standard deviation = square root of the
mean. The same point applies if we have a suspension of
blood cells in a counting chamber. Provided that these
cells do not attract or repel one another their count
will conform to Poisson distribution. If there is a mean
of 80 cells per square of the counting chamber, then
there will be a variance of 80, standard deviation of
8.94 (i.e. Ö 80) and 95%
confidence limits of
8.94 x 1.96 (the
1. Provided that the cells are randomly distributed
(no mutual attraction or repulsion) then their count
conforms to Poisson distribution, and this applies to all
the counts (of various types) that ever have been made or
that ever will be made. So we need not bother with
degrees of freedom - we use the 2. Provided that our count is reasonably high (say, above 30) then it can be treated as part of a Poisson distribution, and we do not even need replicates. So, a count of 30 in one square of a counting chamber (or a count of 80 pooled from, for example, 3 squares) is all we need. This count has: a mean of 80, variance (s standard deviation (s ) of Ö 80, standard error (s An improved estimate of confidence limits of the mean
can be obtained by introducing a correction factor. The
confidence limits of a count
Thus, for our count of 80, the 95% confidence limits are: 80 + 1.96
If you are starting to wonder where all that preamble is taking us, suppose that we count 100 cells in a certain volume of bacterial suspension (or blood), and 150 cells in the same volume of another suspension. Are these significantly different? Call the first count
[We have applied a correction factor of
0.5 here, as in Yates correction for c If we use our counts of 100 and 150 in the equation above, we get:
We compare this with the If the counts were obtained from different volumes
(termed
All the methods above can be applied to dilution
plating of bacteria or fungi on agar plates. For example,
if we used a soil dilution and counted 67 colonies on a
plate at the 10
Sometimes we might wish to test whether counts conform to a Poisson distribution. For example, motile cells can aggregate into clumps, non-motile cells can agglutinate by surface interactions, and cells can also repel one another by producing metabolites. We might wish to test whether these events are occurring, in order to investigate the mechanisms or their biological significance. The method is simple. Suppose that we incubate cells in a counting chamber for 30 minutes and then count the number of cells in several different squares of the chamber (of course, we can choose the size of our sampling unit by pooling counts for groups of 4 or 16 squares, etc. to get mean counts large enough (say, at least 30) to conform to Poisson expectation). We might find the following counts in five squares of the chamber: 50, 30, 80, 90, 10. For these five replicate counts we can obtain a mean (52) and variance in the normal way (see methods) by calculating:
If the data conformed to a Poisson distribution, then the mean of 52 would have a variance of 52. But our calculated variance is 1120. It seems that our counts do not conform to Poisson expectation - the cells are not randomly distributed in the counting chamber. There are different ways of testing this, which need
not be explained, but the simplest is to calculate S [ Now suppose that we had five counts: 49, 50, 50, 49, 50. We can calculate the mean (49.6), S
Now think about elephants! Poisson distributions don't apply only to cells or
bacterial counts (or postal vans). They apply equally to
elephants and animal behaviour. For example, if you
surveyed an area of a large game park and counted the
elephants in each square kilometre (or whatever area is
appropriate), would the data fit a Poisson distribution?
Would this be true at all times of the year? The results
you obtain would only tell you, in statistical terms,
whether the counts fit a Poisson distribution (i.e.
whether elephants are randomly distributed in space). But
the findings would suggest a lot about the behaviour of
elephants. Do they have large family groups? Do these
groups disperse at certain times of the year? Of course,
what this analysis can never tell us |

## This site is no longer maintained and has been left for archival purposes

## Text and links may be out of date