## Box 33 The Poisson distribution

The Poisson distribution is appropriate for counts because the number of objects within a quadrat will follow a Poisson distribution if the objects are distributed randomly in space. The probability distribution function for the Poisson distribution is:

where X is the random variable (e.g. the number of plants in the quadrat), l is a parameter that is equal to both the mean number of plants in the quadrat and the variance in this number, and e is the constant equal to 2.71828 ... The expression x! ('x factorial') is equal to 1 x 2 x 3 x ...x x, with 0! equal to 1.

Thus, the probability of having no plants in a quadrat is equal to e—1 (1x and x! are both equal to 1 when x = 0), the probability of one plant is e_11, the probability of two is e_112/2, the probability of three is e_113/6, etc.

Therefore, if the quadrat is 1 m2, Fig. 3.1 shows the expected distribution of counts of the number of plants per quadrat (x) for different densities of plants (l) ranging from 0.5—3 plants m—2. 3 4 5 6 7 Count per quadrat

Fig. 3.1 Three Poisson distributions describing variation in the number of plants in 1-m2 quadrats for three different plant densities: 0.5 (black), 1.5 (grey) and 3.0 (white) plants m—2.

### 3 4 5 6 7 Count per quadrat

Fig. 3.1 Three Poisson distributions describing variation in the number of plants in 1-m2 quadrats for three different plant densities: 0.5 (black), 1.5 (grey) and 3.0 (white) plants m—2.

Box 3.4