The mean and standard deviation of the binomial distribution are np = 1.2 and
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
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.
1.5 Types of Control Charts
Broadly speaking, there are two types of control charts, i.e. attribute control charts and variable control charts.
1.5.1 Attribute Control Charts
When articles or units are studied on the basis of qualitative measures as go/no go, yes/no, satisfied/not satisfied, positive/negative, etc., then an attribute control chart is suitable to monitor the unusual changes. These are the charts commonly known as p‐chart used for monitoring proportion of nonconforming items in a sample or proportion of defective items in a sample, and np‐chart is used to monitor the number of nonconforming items in a sample of size n.
The control limits of these control charts are similar particularly when the sample size is fixed. The limits of p‐chart charts may be constructed by using the binomial probability distribution with parameter p. As we know that the mean of the binomial distribution is np and its standard deviation is
And the control limits for the np‐chart can be constructed as
Another attribute control chart that is commonly used in the control chart literature is the c‐chart used for monitoring the number of nonconformities in each sample or the number of defects in each sample. The limits of c‐chart charts may be constructed by using the Poisson probability distribution with parameter c. As we know that the mean of the Poisson distribution is c and its standard deviation is
Yet another attribute control chart commonly used in the literature of the SPC is the μ‐chart. It is used to monitor the nonconformity items per unit or the nonconforming items per unit. The distribution of the c‐chart and the μ‐chart is the same except the scale of the units, which is changed by n in the μ‐chart. The control limits of μ‐chart can be established as
1.5.2 Variable Control Charts
When the characteristics under study are measured on the continuous scale, such as the height, weight, time, and temperature, then such variables are classified as the continuous variables. The commonly used control charts are the average or the dispersion charts or the combination of both the charts. The
