DOC PREVIEW
ISU IE 361 - Control Charts for Counts

This preview shows page 1-2-23-24 out of 24 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

IE 361 Module 13Control Charts for Counts ("AttributesData")Prof.s Stephen B. Vardeman and Max D. MorrisReading: Section 3.3, Statistical Quality Assurance M ethods for Engineers1In this module, we discuss the Shewhart control charts for so-called "fractionnonconforming" and "mean nonconformities per unit" contexts. These are theShewhart p and np charts and the Shewhart u (and c)charts. Thesetoolsare easy enough to explain and use, but are typically really NOT very effectivein modern applications where acceptable nonconformity rates are so small asto be stated in "parts per million."p Charts (and np Charts)The scenario under which a p chart or (np chart) is potentially appropriateis one where periodically groups of n items (or outcomes) from a process are2look ed at andX = the number of outcomes among the n that are "nonconforming"is observed. This is illustrated in the cartoon in Figure 1, where dark ballsrepresent nonconforming outcomes.Figure 1: Cartoon Representing n Process Outcomes, X of Which are Non-conforming3In this kind of circumstance, the notationˆp =Xn= the sample fraction nonconformingis standard, and• control charts for ˆp are called p charts• control charts for X (= nˆp) are called np chartsIf the process producing items/outcomes is physically stable, a reasonable prob-ability model for X (met in Stat 231) is thebinomial (n, p)4distribution, wherep = the current probability that any particular outcome is nonconforming(the mental fiction here is that the particular n outcomes observed are a randomsample of a huge pool of outcomes, a fraction p of which a re nonconforming).Stat 231 facts about the binomial distribution are thatμX= np and σX=qnp (1 − p)so that (since ˆp =³1n´X)μˆp= p and σˆp=sp (1 − p)nThese facts in turn lead to standards given (p chart) control limits for ˆpLCLˆp= p − 3sp (1 − p)nand UCLˆp= p +3sp (1 − p)n5and standards given (np chart) control lim its for XLCLX= np − 3qnp (1 − p) and UCLX= np − 3qnp (1 − p)Example 13-1 Below are some artificially generated (using p = .03) n =400binomial data (except for sample 9,wherealargervalueofp was used) andthe corresponding values of ˆp.Sample 12345678910X 15 11 18 9 13 11 10 19 24 7ˆp .0375 .0275 .0450 .0225 .0325 .0275 .0250 .0475 .0600 .0175Sample11 12 13 14 15 16 17 18 19 20X 9131771019118 8 7ˆp .0225 .0325 .0425 .0175 .0250 .0475 .0275 .0200 .0200 .01756Standards given control limits for ˆp here areLCLˆp= p − 3sp (1 − p)n= .03 − 3s.03 (1 − .03)400= .0044andUCLˆp= .03 + 3s.03 (1 − .03)400= .0556A JMP plot of the corresponding control chart is in Figure 2 and shows clearlythat sample 9 produces an out-of-control signal. That sample simply does notfit the "stable process with p = .03" model that stands behind the controllimits.7Figure 2: Standards Given (p = .03) p Chart for the Artificial Data8Retrospective control limits for X or ˆp are obtained (as always) by making aprovisional assumption of process stability and estimating process parameters(here, the value p). The most sensible estimate obtainable from r valuesXi= nˆpibased on respective sample sizes niisˆppooled=total outcomes nonconformingtotal outcomes observed=X1+ X2+ ···+ Xrn1+ n2+ ···+ nr(This is the arithmetic mean of the ˆpiincaseswherethesamplesizeiscon-stant.)Example 13-1 continued Returningtotheartificial data, there are a total of246 nonconforming outcomes indicated among the 20 (400) = 8000 outcomesrepresented. Soˆppooled=2468000= .03089and retrospective control limits for ˆp areLCLˆp= .0308 − 3s.0308 (1 − .0308)400= .0049andUCLˆp= .0308 + 3s.0308 (1 − .0308)400= .0567Figure 3 show that the retrospective limits do not produce a picture muchdifferent from the standards given ones used to make Figure 2. The 20 samplesdo not fit with a "constant p"model.10Figure 3: Retrospective p Chart for the Artificial Data11Notice that a lower control limit for a p chart CAN be positive. This sometimesseems counter-intuitive, but only to those who fail to remember that controlcharts 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 isalerted to unexpected/fortuitous process improvement. One wants to followup on such lucky circumstances, looking for an assignable cause to incorporateinto the process on a permanent basis.In some cases, sample sizes vary period to period. When that happens, eachnew period may need a new calculation of control limits for that sample size.NOTICE that big sample sizes carry relatively larger amounts of informationabout current process conditions and have control limits that are tighter arounda 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 surethat something has changed in the process.)12Example 13-2 Below is a comparison between (p = .03) standards givenlimits for ˆp for n =400and n =100.nCLˆpLCLˆpUCLˆp100 .03 none .0812400 .03 .0044 .0556u Charts (and c Charts)The scenario under which a u chart or (c chart) is potentially appropriate is onewhere periodically k inspection units of product or production from a processarelookedatandX = the total number of "nonconformities" found across those k units13Figure 4: Cartoon of k Inspection Units of Process Output and X "Noncon-formities"is observed. This is illustrated in the cartoon in Figure 4, where "x"’s representnoncon formities.14In this kind of circumstance, the notationˆu =Xk= the sample rate of nonconformities per unitis standard, and• control charts for ˆu are called u charts• control charts for the special case where k =1and thus ˆu = X are calledc chartsIf the process producing items/outcomes is physically stable and produces non-conformities at a rate of λ per unit, a reasonable probability model for X (metin Stat 231) is thePoisson (kλ)15distribution (the Poisson distribution with mean μX= kλ). A Stat 231 factabout the Poisson distribution is that its mean and variance are the same, thatis thatσX=√μXSo in the present contextμX= kλ and σX=√kλandinturnthatμˆu= λ and σˆu=sλkThese facts lead to standards given (u chart) control limits for ˆuLCLˆu= λ − 3sλkand UCLˆu= λ +3sλk16andforthecaseofk =1(so that ˆu = X) standards given (c chart) controllimits for XLCLX= λ − 3√λ and LCLX= λ +3√λExample 13-3 Below are some artificially generated (using λ


View Full Document

ISU IE 361 - Control Charts for Counts

Download Control Charts for Counts
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Control Charts for Counts and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Control Charts for Counts 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?