Introduction to Statistical Process Control. Muhammad Amir Aslam

Introduction to Statistical Process Control

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 images = 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.

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 images . Thus

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 images . Thus

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 images ‐chart is the most common chart developed with the association of the R‐chart (also known as images ‐ and R‐chart). The development of images ‐ and S‐chart or S²‐chart is also very common. These charts have been discussed thoroughly in the latter chapters. In general the distribution of all the variable control charts under study is assumed to be normally distributed. The three‐sigma control limits are thus computed with the false alarm rate, the probability that an observation falling outside the control limits is 0.0027. The average run length of the three‐sigma control limits can be calculated as 1/0.0027 = 370.3704, which means that on average the process indicates an out‐of‐control process after 370 samples if the distribution of observations follows exactly the normal distribution. This is the most commonly used level in evaluating and comparing the process in the control chart literature. The concept of three‐sigma control limits is commonly accepted by the industry, which is easy to understand, and applied by the quality control personnel. The variable control limits can be developed using the pre‐specified constant values given in Appendix B.

1.6 Meaning of Process Control

The term process control is used to keep the process in control. In statistical terminology a graphical method of presenting a sequence of samples on a chart is called control chart, a tool used for monitoring different changes in the process. A typical control chart has been given in Figure 1.6.

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