IE 361 Module 13. Control Charts for Counts ("Attributes Data") Reading: Section 3.3 of Statistical Quality Assurance Methods for Engineers

Size: px

Start display at page:

Download "IE 361 Module 13. Control Charts for Counts ("Attributes Data") Reading: Section 3.3 of Statistical Quality Assurance Methods for Engineers"

Virginia Welch
6 years ago
Views:

1 IE 361 Module 13 Control Charts for Counts ("Attributes Data") Reading: Section 3.3 of Statistical Quality Assurance Methods for Engineers Prof. Steve Vardeman and Prof. Max Morris Iowa State University Vardeman and Morris (Iowa State University) IE 361 Module 13 1 / 23

2 Control Charting Counts In this module, we discuss the Shewhart control charts for so-called "fraction nonconforming" and "mean nonconformities per unit" contexts. These are the Shewhart p and np charts and the Shewhart u (and c) charts. These tools are easy enough to explain and use, but are typically really NOT very effective in modern applications where acceptable nonconformity rates are often so small as to be stated in "parts per million." Vardeman and Morris (Iowa State University) IE 361 Module 13 2 / 23

3 Control Charts for Fraction Nonconforming The scenario under which a p chart or (np chart) is potentially appropriate is one where periodically groups of n items (or outcomes) from a process are looked at and X = the number of outcomes among the n that are "nonconforming" is observed. This is illustrated in the figure below, where dark balls represent nonconforming outcomes. Figure: Cartoon Representing n Process Outcomes, X of Which are Nonconforming Vardeman and Morris (Iowa State University) IE 361 Module 13 3 / 23

4 Probability Basis for Control Limits In this kind of circumstance, the notation ˆp = X n = the sample fraction nonconforming is standard, and control charts for ˆp are called p charts control charts for X (= n ˆp) are called np charts If the process producing items/outcomes is physically stable, a reasonable probability model for X (met in Stat 231) is the distribution, where binomial (n, p) p = the current probability that any particular outcome is nonconforming (the mental fiction here is that the particular n outcomes observed are a random sample of a huge pool of outcomes, a fraction p of which are nonconforming). Vardeman and Morris (Iowa State University) IE 361 Module 13 4 / 23

5 Standards Given Control Limits Stat 231 facts about the binomial distribution are that µ X = np and σ X = np (1 p) so that (since ˆp = ( 1 n ) X ) µ ˆp = p and σ ˆp = p (1 p) These facts in turn lead to standards given (np chart) control limits for X LCL X = np 3 np (1 p) and UCL X = np + 3 np (1 p) and standards given (p chart) control limits for ˆp n LCL ˆp = p 3 p (1 p) n and UCL ˆp = p + 3 p (1 p) n Vardeman and Morris (Iowa State University) IE 361 Module 13 5 / 23

6 p Chart Example Example 13-1 Below are some artificially generated (using p =.03) n = 400 binomial data (except for sample 9, where a larger value of p was used) and the corresponding values of ˆp. Sample X ˆp Sample X ˆp Sample X ˆp Vardeman and Morris (Iowa State University) IE 361 Module 13 6 / 23

7 p Chart Example Example 13-1 continued (Standards Given Control Limits) Standards given control limits for ˆp here are and p (1 p) LCL ˆp = p 3 n.03 (1.03) =.03 3 = (1.03) UCL ˆp = = Vardeman and Morris (Iowa State University) IE 361 Module 13 7 / 23

p Chart Example Example 13-1 continued (Standards Given Chart) A plot of the corresponding control chart is in the figure below and shows clearly that sample 9 produces an out-of-control signal.

8 p Chart Example Example 13-1 continued (Standards Given Chart) A plot of the corresponding control chart is in the figure below and shows clearly that sample 9 produces an out-of-control signal. That sample simply does not fit the "stable process with p =.03" model that stands behind the control limits. Figure: Standards Given (p =.03) p Chart for the Artificial Data Vardeman and Morris (Iowa State University) IE 361 Module 13 8 / 23

9 Retrospective Control Limits Retrospective control limits for X or ˆp are obtained (as always) by making a provisional assumption of process stability and estimating process parameters (here, the value p). The most sensible estimate obtainable from r values X i = n ˆp i based on respective sample sizes n i is ˆp pooled = total outcomes nonconforming total outcomes observed = X 1 + X X r n 1 + n n r (This is the arithmetic mean of the ˆp i in cases where the sample size is constant.) Vardeman and Morris (Iowa State University) IE 361 Module 13 9 / 23

10 p Chart Example Example 13-1 continued (Retrospective Limits) Returning to the artificial data on panel 6, there are a total of 246 nonconforming outcomes indicated among the 20 (400) = 8000 outcomes represented. So ˆp pooled = =.0308 Retrospective control limits for ˆp are thus and.0308 (1.0308) LCL ˆp = = (1.0308) UCL ˆp = = Vardeman and Morris (Iowa State University) IE 361 Module / 23

p Chart Example Example 13-1 continued (Retrospective Chart) The figure below shows that the retrospective limits do not produce a picture much different from the standards given ones used to

11 p Chart Example Example 13-1 continued (Retrospective Chart) The figure below shows that the retrospective limits do not produce a picture much different from the standards given ones used to make panel 8. The 20 samples do not fit with any "constant p" model. Figure: Retrospective p Chart for the Artificial Data Vardeman and Morris (Iowa State University) IE 361 Module / 23

12 Positive Lower Control Limits Notice that a lower control limit for a p chart CAN be positive. (Example 13-1 shows this.) This sometimes seems counter-intuitive, but only to those who fail to remember that control charts are about detecting process change, NOT about product acceptability. When a point plots below a lower control limit on a p or np chart, one is alerted to unexpected/ fortuitous process improvement. One wants to follow up on such lucky circumstances, looking for an assignable cause to incorporate into the process on a permanent basis. Vardeman and Morris (Iowa State University) IE 361 Module / 23

13 Varying Sample Size In some cases, sample sizes vary period to period. When that happens, each new period may need a new calculation of control limits for that sample size. NOTICE that big sample sizes carry relatively larger amounts of information about current process conditions and have control limits for ˆp that are tighter around a center line than small sample sizes. (With more information in a sample, less deviation from the expected value for ˆp is required before one is fairly sure that something has changed in the process.) Example 13-2 Below is a comparison between (p =.03) standards given limits for ˆp for n = 400 and n = 100. n CL ˆp LCL ˆp UCL ˆp none Vardeman and Morris (Iowa State University) IE 361 Module / 23

14 Control Charts for Mean Nonconformities per Unit The scenario under which a u chart or (c chart) is potentially appropriate is one where periodically k inspection units of product or production from a process are looked at and X = the total number of "nonconformities" found across those k units is observed. This is illustrated in the following figure, where "x" s represent nonconformities. Figure: Cartoon of k Inspection Units of Process Output and X "Nonconformities" Vardeman and Morris (Iowa State University) IE 361 Module / 23

15 Control Charts for Mean Nonconformities per Unit In this kind of circumstance, the notation û = X k = the sample rate of nonconformities per unit is standard, and control charts for û are called u charts control charts for the special case where k = 1 and thus û = X are called c charts Vardeman and Morris (Iowa State University) IE 361 Module / 23

16 Probability Basis for Control Limits If the process producing items/outcomes is physically stable and produces nonconformities at a rate of λ per unit, a reasonable probability model for X (met in Stat 231) is the Poisson (kλ) distribution (the Poisson distribution with mean µ X = kλ). A Stat 231 fact about the Poisson distribution is that its mean and variance are the same, that is that So in the present context σ X = µ X µ X = kλ and σ X = kλ and in turn that µû = λ and σû = λ k Vardeman and Morris (Iowa State University) IE 361 Module / 23

17 Standards Given Control Limits These probability facts lead to standards given (u chart) control limits for û λ λ LCLû = λ 3 k and UCL û = λ + 3 k and for the case of k = 1 (so that û = X ) standards given (c chart) control limits for X LCL X = λ 3 λ and UCL X = λ + 3 λ Vardeman and Morris (Iowa State University) IE 361 Module / 23

18 u Chart Example Example 13 3 Below are some artificially generated (using λ = 2.0) Poisson data, X. We may consider these to be either counts or rates for a constant (k = 1) number of inspection units. Sample X Sample X Standards given control limits for X would then be CL X = 2 and UCL X = λ + 3 λ = = 6.24 and no lower control limit would be used since is negative. Clearly, none of the values X in the table would plot above the upper control limit and there is no indication of "lack of control"/deviation from the λ = 2.0 nonconformities per unit picture of process performance under which the above limits are developed. Vardeman and Morris (Iowa State University) IE 361 Module / 23

19 u Chart Example Example 13-3 continued (Varying Amounts of Inspection) To extend this example a bit, suppose that an additional 5 inspection Periods are involved, but now the number of inspection units observed per period is no longer constant (at "k = 1"), and in fact that the table below is relevant. Sample k X û = X k UCLû = k LCLû none none none none none Vardeman and Morris (Iowa State University) IE 361 Module / 23

u Chart Example Example 13-3 continued (Shewhart u Chart) The following figure is a JMP control chart for the whole data set, and there is no reason in these data to doubt that the nonconformity rate

20 u Chart Example Example 13-3 continued (Shewhart u Chart) The following figure is a JMP control chart for the whole data set, and there is no reason in these data to doubt that the nonconformity rate is constant at λ = 2.0 per inspection unit. (Note that JMP in ambiguous fashion indicates that LCL = 0. It is better practice and provides clearer intent to say that there is no lower control limit.) Figure: Standards Given (λ = 2.0) u Chart for the Artificial Data Vardeman and Morris (Iowa State University) IE 361 Module / 23

21 Retrospective Control Limits Retrospective control limits for û are obtained (as always) by making a provisional assumption of process stability and estimating process parameters (here, the value λ). The most sensible estimate obtainable from r values X i based on respective numbers of inspection units k i is ˆλ pooled = total nonconformities observed total inspection units = X 1 + X X r k 1 + k k r Example 13-3 continued There are a total of 53 nonconforming outcomes indicated among the inspection units. So ˆλ pooled = = 1.98 and there is negligible difference between the standards given chart on panel 20 and the retrospective control chart that would be made using this. Vardeman and Morris (Iowa State University) IE 361 Module / 23

22 p versus u Charts Students sometimes have a diffi cult time identifying whether a given application calls for a p chart or for a u chart. Both ˆp and û are kinds of "rates," but they are intrinsically different. For one thing, it is perfectly possible for û to exceed 1.0, while ˆp is always between 0 and 1. The figure below lays the two basic scenarios side by side. Figure: Comparison of "Fraction Nonconforming" and "Mean Nonconformities per Unit" Scenarios Vardeman and Morris (Iowa State University) IE 361 Module / 23

23 p versus u Charts In the p chart/"fraction nonconforming" context, n discrete items or outcomes are individually each either acceptable or nonconforming, and ˆp is the sample fraction nonconforming. In the u chart/"mean nonconformities per unit" context, k inspection units of process output potentially suffer nonconformities, and û is the sample rate at which these are observed per unit. It is worth saying again that Table 3.11, page 107 of SQAME, summarizes the control limit formulas for both the "attributes data" of this module and for the "measurements data" of Module 11. Vardeman and Morris (Iowa State University) IE 361 Module / 23

IE 361 Module 4. Metrology Applications of Some Intermediate Statistical Methods for Separating Components of Variation

IE 361 Module 4. Metrology Applications of Some Intermediate Statistical Methods for Separating Components of Variation IE 361 Module 4 Metrology Applications of Some Intermediate Statistical Methods for Separating Components of Variation Reading: Section 2.2 Statistical Quality Assurance for Engineers (Section 2.3 of Revised