294 BASIC PRINCIPLES of MEASUREMENT Probability distributions for Six Sigma This section discusses the following probability distributions often used in Six Sigma:. Binomial distribution. Poisson distribution. Hypergeometric distribution. Normal distribution. Exponential distribution. Chi-square distribution. Student s t distribution. F distribution BINOMIAL DISTRIBUTION Assume that a process is producing some proportion of non-conforming units, which we will call p. If we are basing p onasamplewefindp by dividing the number of non-conforming units in the sample by the number of items sampled. The equation that will tell us the probability of getting x defectives in asampleofn units is shown by Equation 9.7. PðxÞ ¼C n xp x ð1 pþ n x ð9:7þ This equation is known as the binomial probability distribution. In addition to being useful as the exact distribution of non-conforming units for processes in continuous production, it is also an excellent approximation to the cumbersome hypergeometric probability distribution when the sample size is less than 10% of the lot size. Example of applying the binomial probability distribution A process is producing glass bottles on a continuous basis. Past history shows that1%ofthebottleshaveoneormoreflaws.ifwedrawasampleof10unitsfrom theprocess,whatistheprobabilitythattherewillbe0non-conformingbottles? Using the above information, n = 10, p =.01, and x = 0. Substituting these values into Equation 9.7 gives us Pð0Þ ¼C 10 0 0:01 0 ð1 0:01Þ 10 0 ¼ 1 1 0:99 10 ¼ 0:904 ¼ 90:4% Anotherwayofinterpretingtheaboveexampleisthatasamplingplan inspect 10 units, accept the process if no non-conformances are found has a 90.4% probability ofacceptinga processthatis averaging 1%non-conforming units.
Overview of statistical methods 295 Example of binomial probability calculations using Microsoft Excel 1 Microsoft Excel has a built-in capability to analyze binomial probabilities. To solve the above problem using Excel, enter the sample size, p value, and x value as shown in Figure 9.10. Note the formula result near the bottom of the screen. Figure 9.10. Example of nding binomial probability using Microsoft Excel. Poisson distribution Another situation encountered often in quality control is that we are not just concerned with units that don t conform to requirements, instead we are concerned with the number of non-conformances themselves. For example, let s say we are trying to control the quality of a computer. A complete audit of the finished computer would almost certainly reveal some non-conformances, even though these non-conformances mightbeofminorimportance(forexample, a decal on the back panel might not be perfectly straight). If we tried to use the hypergeometric or binomial probability distributions to evaluate sampling plans for this situation, we would find they didn t work because our lot or pro-
296 BASIC PRINCIPLES of MEASUREMENT cess would be composed of 100% non-conforming units. Obviously, we are interested not in the units per se, but in the non-conformances themselves. In other cases, it isn t even possible to count sample units per se. For example, the number of accidents must be counted as occurrences. The correct probability distribution for evaluating counts of non-conformances is the Poisson distribution. The pdf is given in Equation 9.8. PðxÞ ¼ x e x! ð9:8þ In Equation 9.8, m is the average number of non-conformances per unit, x is the number of non-conformances in the sample, and e is the constant approximately equal to 2.7182818. P(x) gives the probability of exactly x occurrences in the sample. Example of applying the Poisson distribution A production line is producing guided missiles. When each missile is completed, an audit is conducted by an Air Force representative and every nonconformance to requirements is noted. Even though any major nonconformance is cause for rejection, the prime contractor wants to control minor non-conformances as well. Such minor problems as blurred stencils, small burrs, etc., are recorded during the audit. Past history shows that on the average each missile has 3 minor non-conformances. What is the probability that the next missile will have 0 non-conformances? We have m ¼ 3, x ¼ 0. Substituting these values into Equation 9.8 gives us Pð0Þ ¼ 30 e 3 0! ¼ 1 0:05 1 ¼ 0:05 ¼ 5% In other words, 100% 5% ¼ 95% of the missiles will have at least one nonconformance. The Poisson distribution, in addition to being the exact distribution for the number of non-conformances, is also a good approximation to the binomial distribution in certain cases. To use the Poisson approximation, you simply let ¼ np in Equation 9.8. Juran (1988) recommends considering the Poisson approximation if the sample size is at least 16, the population size is at least 10 times the sample size, and the probability of occurrence p on each trial is less than 0.1. The major advantage of this approach is that it allows you to use the tables of the Poisson distribution, such as Table 7 in the Appendix. Also, the approach is useful for designing sampling plans.
Overview of statistical methods 297 Example of Poisson probability calculations using Microsoft Excel Microsoft Excel has a built-in capability to analyze Poisson probabilities. To solve the above problem using Excel, enter the average and x values as shown in Figure 9.11. Note the formula result near the bottom of the screen. Figure 9.11. Example of nding Poisson probability using Microsoft Excel. HYPERGEOMETRIC DISTRIBUTION Assume we have received a lot of 12 parts from a distributor. We need the parts badly and are willing to accept the lot if it has fewer than 3 non-conforming parts. We decide to inspect only 4 parts since we can t spare the time to check every part. Checking the sample, we find 1 part that doesn t conform to the requirements. Should we reject the remainder of the lot? This situation involves sampling without replacement. We draw a unit from the lot, inspect it, and draw another unit from the lot. Furthermore, the lot is
298 BASIC PRINCIPLES of MEASUREMENT quite small, the sample is 25% of the entire lot. The formula needed to compute probabilities for this procedure is known as the hypergeometric probability distribution, and it is shown in Equation 9.9. PðxÞ ¼ CN m n x C m x C N n ð9:9þ In the above equation, N is the lot size, m is the number of defectives in the lot, n is the sample size, x is the number of defectives in the sample, and P(x) is the probability of getting exactly x defectives in the sample. Note that the numerator term Cn x N m gives the number of combinations of non-defectives while Cx m is the number of combinations of defectives. Thus the numerator gives the total number of arrangements of samples from lots of size N with m defectives where the sample n contains exactly x defectives. The term Cn N in the denominator is the total number of combinations of samples of size n from lots of size N, regardless of the number of defectives. Thus, the probability is a ratio of the likelihood of getting the result under the assumed conditions. For our example, we must solve the above equation for x = 0 as well as x = 1, since we would also accept the lot if we had no defectives. The solution is shown as follows. Pð0Þ ¼ C12 3 4 0 C0 3 C4 12 ¼ 126 1 ¼ 0:255 495 Pð1Þ ¼ C12 3 4 1 C1 3 C4 12 ¼ 84 3 ¼ 252 495 495 ¼ 0:509 Pð1 orlessþ¼pð0þþpð1þ Adding the two probabilities tells us the probability that our sampling plan will accept lots of 12 with 3 non-conforming units. The plan of inspecting 4 parts and accepting the lot if we have 0 or 1 non-conforming has a probability of.255 +.509 =.764, or 76.4%, of accepting this bad quality lot. This is the consumer s risk for this sampling plan. Such a high sampling risk would be unacceptable to most people. Example of hypergeometric probability calculations using Microsoft Excel Microsoft Excel has a built-in capability to analyze hypergeometric probabilities. To solve the above problem using Excel, enter the population and sample values as shown in Figure 9.12. Note the formula result near the bottom of the
Overview of statistical methods 299 Figure 9.12. Example of nding hypergeometric probability using Microsoft Excel. screen (0.509) gives the probability for x ¼ 1. To find the cumulative probability you need to sum the probabilities for x ¼ 0 and x ¼ 1etc. NORMAL DISTRIBUTION The most common continuous distribution encountered in Six Sigma work is, by far, the normal distribution. Sometimes the process itself produces an approximately normal distribution, other times a normal distribution can be obtained by performing a mathematical transformation on the data or by using averages. The probability density function for the normal distribution is given by Equation 9.10. f ðxþ ¼ p 1 ffiffiffiffiffi e ðx Þ2 =2 2 ð9:10þ 2 If f ðxþ is plotted versus x, the well-known bell curve results. The normal distribution is also known as the Gaussian distribution. An example is shown in Figure 9.13.