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The Poisson distribution is another important discrete probability distribution used for calculating the characteristics of the control chart, which identifies a given number of defects per unit; for instance, the number of stones in a piece of glass of the given size or the number of defects in the manufacturing of items, etc. This distribution has only one parameter μ.

Table 1.1 Probabilities of number of defective items using binomial distribution.

No. of defective items	Probability	No. of defective items	Probability
0	0.27850098	6	0.00037604
1	0.37977406	7	0.00002930
2	0.23304317	8	0.00000150
3	0.08474297	9	0.00000005
4	0.02022275	10	0.00000000
5	0.00330918	Total	1.00000000

The mean and standard deviation of the binomial distribution are np = 1.2 and = 1.264911, respectively.

Table 1.2 Probabilities of number of defective items using Poisson distribution.

No. of defective items	Probability	No. of defective items	Probability
0	0.110803158	6	0.017448405
1	0.243766948	7	0.005483784
2	0.268143643	8	0.001508041
3	0.196638672	9	0.000368632
4	0.108151269	10	0.000081099
5	0.047586559	11 and more	0.000019789

The mean and standard deviation of this distribution are μ = 2.5 and = 1.58, respectively.

The frequency distribution of the Poisson distribution can be defined as

where μ = mean number of defects, μ > 0, e = 2.71828… and x = number of occurrences, x = 0, 1, 2, 3, …

Suppose that the average number of stones in a glass of a particular shape was 2.5. Find the probabilities of different number of stones in the glass. Evaluating the Poisson distribution in this case, we get the results shown in Table 1.2.