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Adjusted P-Chart Scoring Process for Percentage Data

Quality monitoring tool provides more accurate computation of percentage for comparison and ranking

Wed, 02/11/2015 - 11:08
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The purpose of this article is to point out a problem when using percentages for subgroups over time, or for members in a larger group, where the size of the denominator varies and probabilities are being estimated. Also to introduce a solution: adjusted p-chart scores (APC), a new way to score or compute percentages (e.g., in the p-chart setting).

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Regular percentages are not a fair comparison when the denominators are different. Although 75 percent corresponds to three out of four, and 75 out of 100, and 3,000 out of 4,000, shooting three out of four free throws, for example, does not necessarily correspond to the same consistency as someone able to shoot 75 out of 100. Some degree of luck, or random error, is involved. The amount is directly related to the size of the denominator. Someone shooting three out of four free throws one time may shoot only one out of four (25%) the next time, while having the same underlying level of performance or ability. There is more range of error related to a small denominator, less with a larger denominator. 

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Comments

Submitted by NT3327 on Wed, 02/11/2015 - 09:35

P' Charts

Some might be interested in the article "Improved Control Charts for Attributes" by David Laney (Quality Engineering, 2002), in which he deals with varying sample sizes in attribute control charts. 

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Submitted by Tom Pyzdek on Fri, 02/13/2015 - 10:12

Not ready for prime-time

A few observations:

  1. This approach uses control charts (p-charts) to make comparisons between groups. P-charts are intended to be used for time-ordered data, not to make group-to-group comparisons. ANOM charts for p would be a better choice for the problem addressed here. Among other things, ANOM charts also adjust for variation in sample size.
  2. The approach described in this article should be submitted to a peer-reviewed journal before being recommended for widespread use to a general audience.
  3. Assuming that this approach bears up under peer review scrutiny, a Laney p-chart would almost certainly be better (see http://tinyurl.com/oq784c5 and http://tinyurl.com/pasyc73.) I.e., this approach looks like a heuristic for a problem that the Laney p-chart already solved. 
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Submitted by psthomas01 on Wed, 03/04/2015 - 15:51

In reply to Not ready for prime-time by Tom Pyzdek

Author's reply

I appreciate your feedback, thanks.  My comments:  1.  APC charts are used when k = t or time.  APC scores apply when k is more general, the binomial distribution is assumed.  Which has but one parameter to estimate.  I understand ANOM charts to estimate both mean and variance separately, which does not apply.  Although ANOM does use the idea of charting over non-temporal variables, my article does not.

2.  I had several PH.D. Mathematicians and Statisticians review the article before it was published.  Quality Digest had a well known quality expert provide feedback as well.

3.  I understand the Laney chart is used when the binomial distribution does NOT apply -- when there is overdispersion or underdispersion, i.e., when the width of the control limits is too tight or too loose.  This is not the problem I address.  I identify and offer a solution to the bias problem demonstrated with regular percentages when comparing subgroups over time (or members across a group).

4.  Perhaps the Z chart transformed back to the P' chart scale is what you mean?  Please let me know if this solves the sorting problem I present in my article.  I found one example online where the P' chart had WIDER limits when subgroup size varied, but the limits varied.  The APC scores adjust the actual "percentage" computation, not the width of the limits compared to binomial determined ones.  Further, I am working on a more Robust APC method, which will certainly have advantages over the Laney tool.

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Submitted by JFThomas on Wed, 12/14/2016 - 06:10

Subgrouping

I enjoyed the article.  I have data which has grouping as shown below.  Do you know a control chart that takes account of the grouping as well?  Thanks

group, id, number of items, number reviewed

1, 1, 567, 1000

1, 2, 543, 1020

1, 3, 456, 999

2, 4, 569, 1007

2, 5, 545, 1040

2, 6, 486, 1300

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Submitted by Tom Pyzdek on Wed, 12/14/2016 - 09:14

I'm not sure what you mean by

I'm not sure what you mean by "grouping." If the data are time-ordered samples from a process then a standard p-chart would work. A Laney p'chart might be better http://bit.ly/2hFuvwl.

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