Introduction to Statistical Process Control. Muhammad Amir Aslam
calculate the area Z ≥ − 1.23, we proceed by identifying the area on the normal curve as given in Figure 1.4.
To calculate the area between that −2.00 ≤ Z ≤ 1.50, we proceed by identifying the area on the normal curve as shown in Figure 1.5.
The standard normal distribution has the following important properties:
1 The cumulative area for the Z‐score equal to −3.49 is close to 0.
2 The cumulative area for the Z‐score increases as the value of Z‐score increases.
3 The cumulative area for the Z‐score equal to 0 is 0.50.
4 The cumulative area for the Z‐score equal to 3.49 is close to 1.
Figure 1.5 Standard normal curve.
Source: https://www.google.com/search?q=Standard+normal+curve&safe=strict&rlz=1C1CHBD_enSA905SA905&sxsrf=ALeKk038CFj1c5F9mxFymaEMSjV1xUEkzA:1592774965148&tbm=isch&source=iu&ictx=1&fir=PAgPMxS8fXpb_M%253A%252CrrsoLwiuhUAKeM%252C_&vet=1&usg=AI4_−kQqYcq7FaH5CrTe620‐F‐8cvWw6Bg&sa=X&ved=2ahUKEwjdr4SQ7ZPqAhVRPJoKHQTVC50Q_h0wAXoECAcQBg&biw=1280&bih=631#imgrc=PAgPMxS8fXpb_M:
Student's t‐Distribution
Another most commonly used probability distribution is the Student's t‐distribution, which was discovered by the English statistician William Sealy Gossett (1876–1937), when he published his paper with the pseudonym, “Student.” This distribution is used for the estimation of the population mean when a sample of v assumed to be normally distributed from that population. If x is a random variable, then the heavy tailed symmetrical t‐distribution is defined as
with mean zero and variance
As v increases, the Student's t‐distribution tends to a normal (0, 1) distribution.
Gamma Distribution
Yet another important probability distribution commonly used in the literature of the control charts for non‐normal random variables is the Gamma distribution. The probability distribution of the Gamma distribution can be defined as
with the scale parameter β > 0 and the shape parameter α > 0.
The mean and variance of the Gamma distribution are
and
respectively (Montgomery, 2009).
Discrete Probability Distributions
Binomial Probability Distribution
Let a process consists of a set of n independent trials. Here the term independent means that any outcome is not affected by the previous outcome whether it had occurred or not. Here we define any outcome as either success or failure. Suppose that the probability of success is denoted by p, p belongs to the interval (0,1), and the probability of failure is denoted by q = 1 − p, then the binomial probability distribution can be defined as
where n is the total number of independent trials and x is a binomial random variable ranging from 0 to n. This distribution has only two parameters n and p. The distribution is symmetric when
The binomial probability distribution is the most commonly used distribution in the control chart literature to model the number cases in a sample of n items when the proportion in the population is known. For example, if the proportion of defective item in any mass production unit is 0.12, then find the complete binomial probability distribution for n = 10 (Table 1.1).
Poisson Probability Distribution
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
Table 1.2 Probabilities of number of defective items using Poisson distribution.
|
No. of defective items
|