ON AN IMPROVED FRACTION NONCONFORMING CONTROL CHART
Jacob Alford, Graduate student, University of Alabama in Huntsville

Abstract. In their article published in the April 2013 edition of Control Engineering Practice, Silvia Joekes and Emanuel Pimentel Barbosa present the use of a correction factor, the Cornish-Fisher quantile correction, to improve the fraction nonconforming control chart (p-chart) when dealing with high quality processes. Introduction. The use of the standard Shewhart control chart for the fraction of nonconforming parts will produce a type I error–signaling that the process is out of control when, in fact, it is still in control–with probability 0 = 0.0027. When dealing with a process with both high quality and a large yield, the control limits for the Shewhart p-chart based on the normal distribution will become tighter, leaving a greater chance of a false alarm signal occurring. This journal entry proposes a modified version of the p-chart based on the CornishFisher expansion, which allows monitoring processes with very low values of p. This new corrected p-chart is shown using one and two terms of correction. Shewhart p-chart basics. Shewhart control charts use 3 limits for both upper and lower control limits, i.e.   3 . The use of Shewhart control charts is commonplace due to their ease of use: the process will only ever signal in control or out of control. This type of pass/fail or conforming/nonconforming relationship is used for attributes data: data that is not easily assigned some numerical value for the characteristic being assessed. The p-chart is an attribute chart that monitors the proportion of nonconforming units in a sample size of n items, where p is used to represent this ratio. If X nonconforming units are detected out of a sample size of n units, the sample proportion is generally ˆ ˆ represented as p , where p  X / n . X is a discrete random variable given as the number of items that provide an outcome of interest, making X binomially distributed with parameters n and p. When the true proportion nonconforming p is known, control limits for the 3s Shewhart control chart can be calculated as p(1  p) p(1  p) (1) UCL  p  3 , LCL  p  3 n n and the center line (CL) being the expected proportion

nonconforming, p. This is considered to be a good normal approximation to the binomial distribution when np(1-p )> 5 and p > 0.1. However, for small values of p, say less than 0.1, this is not a good approximation as the binomial distribution is an asymmetric, skewed distribution. The proposed corrected control limits are derived from calculating the cumulants of the distribution. Cornish-Fisher correction using cumulants. The cumulants of a random variable X are similar, in concept, to its moments. In fact, the cumulants generating function is simply the logarithm of the moment generating function. For the binomial distribution, ordinary moments are obtained using
'  h  E ( X h )  M ( h) (0) , where

M x (t )   e tk p k , k 0

n

(2)

n p k    p k (1  p) nk k   

(3)

The first four cumulants of the binomial distribution are given below:

1  where     , 2  2 , 3  3 , 4  4  2 3 4

(4)

  p,  2 

p(1  p) p(1  p)(1  2 p) (5) , 3  n n

Given these formulae for the cumulants of the binomial distribution, and after some simplification, the following adjusted control limits are achieved for using one correction term, which is using only the third cumulant.

UCL1  UCL 

4 (1  2 p) 3n 4 LCL1  LCL  (1  2 p) 3n

(6) (7)

According to the authors, these control limits produce a type-I error probability nearer the standard 0.0027 than the commonplace normal approximation Shewhart control chart when applied to high quality processes. The author suggests that these control limits can be

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successfully implemented with p values greater than approximately 0.01. For even higher quality processes than this (according to the authors, with a p value as low as 0.004) a second correction using the third and fourth cumulants can be used. The resultant control limits given for these higher quality processes are

false alarm risk, meaning that this process would produce many instances where an assignable cause would be sought out when, in fact, it did not exist and the process was indeed in control. Conclusions. The authors present an alternative to the standard p control chart which is easily implemented into statistical software packages and would help monitor high quality processes. This implementation could possibly eliminate unnecessary downtime caused by false alarms as well as the headache that goes along with such scenarios. References Joekes, Silvia, and Emanuel Pimentel Barbosa. "An improved attribute control chart for monitoring non-conforming proportion in high quality processes." Control Engineering Practice 21, no. 4 (April 2013): 407-412. Academic Search Premier, EBSCOhost (accessed April 7, 2013).

UCL2  UCL1 

p(1  p)  2

LCL2  LCL1 

p(1  p) 6n 2 n p(1  p)  2

(8)

6n

2

p(1  p) n

(9)

Authors’ “Rules” for p-chart choice. Taking into consideration the two new variations on the p-chart, the authors pose a set of guidelines for choosing which variant to use for any given situation. These are as follows: for np(1  p)  5 , normal chart without correction for np(1  p)  0.25 , one correction term for np(1  p)  0.08 , two correction terms Also, as taken from above, for p  0.01 , one correction term for p  0.004 , two correction terms These guidelines are given to provide a suggested course of action to better reflect the target Type-I error proportion 0.0027. Numerical results. Examples are given to illustrate the usefulness of these corrections. First, a p-chart is constructed with ˆ n  20 and p  0.015 . Samples were collected for both Phase I and Phase II and finally the Type-I error probabilities were calculated for each of the three types of p-charts: normal approximated, one correction factor, and two correction factor. The results are tabulated in Table 1. Table 1. Type of Chart Standard One Correction Factor Two Correction Factors  risk 0.035746 0.000202 0.003178

In this scenario, the two correction factor control chart would best reflect the expected type-1 error probability for 3 limits. The  risk exhibited with the Shewhart chart is over thirteen times larger than the expected

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