# P chart

P chart | |
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**P chart**, also known as a fraction nonconforming control chart, is a graphical tool which was developed in industry to help with interpreting and reducing sources of variability in manufacturing processes. The definition of fraction nonconforming (defective) (*p*) is usually exemplified by the ratio of the number of nonconforming items in the population to the total number of items in that population. Investigated item may have one or more quality characteristics that are examined simultaneously. If at least one of the attributes does not conform to standard, the item is classified as nonconforming ^{[1]}.

## Genesis

In the 1920s, W.A. Shewhart developed the concept of statistical control chart for the purpose of improving the reliability of telephone transmission systems. The conception arose from observating the operators. Shewhart noticed that they were overreacting and making inappropriate changes in settings as a response to indicator variations that were simply random. Those decisions were wasteful and inducted more variation in the process, making the system less stable. Shewhart's theory of variation provides that quality is inversely proportional to variability. It also states that understanding the variability of some indicators may show the operator when and how reduce it ^{[2]}.

At that time, the concept of using the two-point moving range for measuring the dispersion of a set of individual values didn't occurred yet. Shewhart faced the problem how to create a process behavior chart for individual values based on counts. Then he decided to use theoretical limits based on a probability model. Therefore, he could estimate both the central line and the three-sigma distance with only one location statistic^{[3]}.

## Design of the P chart

There are three parameters of fraction nonconforming control chart that must be specified: **the sample size, the frequency of sampling**, and **the width of the control limits**. Preferably, we should have some general guidance for selecting those parameters.

### Formulas for the points on the Chart

Let's suppose we have *k* samples, each of *n _{i}* size. Let

*D*stand for the number of nonconforming units in the

_{i}*i*sample. The

^{th}*i*proportion

^{th}*p*is calculated as the following ratio

_{i}^{[4]}\[p_{i}=\frac{D_{i}}{n_{i}}\].

### P Chart Center Line

In the **P Charts** procedure, the *center line* proportion may be implement directly. It can also be estimated from a series of samples. If it is estimated from the samples we use the formula for the centerline proportion\[\bar{p}=\frac{\sum_{i=1}^{k}D_{i}}{\sum_{i=1}^{k}n_{i}}\].

You can reduce this formula if all the samples are the same size^{[5]}.\[\bar{p}=\frac{\sum_{i=1}^{k}D_{i}}{kn}=\frac{\sum_{i=1}^{k}p_{i}}{k}\]

### P Chart limits

To calculate the *lower and upper control limits* for the **P chart** use these formulas\[LCL=\bar{p}-m\sqrt{\frac{\bar{p}(1-\bar{p})}{n_{i}}}\]\[UCL=\bar{p}+m\sqrt{\frac{\bar{p}(1-\bar{p})}{n_{i}}}\]

where *m* stands for *multiplier* (usually set to 3) chosen to control the probability of false alarms (out-of-control signals
when the process is under control)^{[6]}.

There are two steps that one should take into concideration to avoid potential pitfalls^{[7]}:

- Ensure that there are enough observations taken for each sample,
- Be wary of differences in the number of observations from each sample.

## Footnotes

## References

- Duclos A., Voirin N. (2010),
*The p-control chart: a tool for care improvement*, “International Journal for Quality in Health Care”, Vol. 22, Nr 5 - Hou C. D., Shao Y. E., Haung S. (2013),
*A Combined MLE and Generalized P Chart Approach to Estimate the Change Point of a Multinomial Process*, "Applied Mathematics & Information Sciences", Vol. 7, Nr 4 - Montgomery D.C. (2012),
*Introduction To Statistical Quality Control, 7th Edition*, Wiley, Arizona State University - Nist/Sematech e-Handbook of Statistical Methods (2012),
*Proportions Control Charts*“Engineering Statistics handbook” - Pandurangan A. (2011),
*Fuzzy Multinomial Control Chart With Variable Sample Size*, "International Journal of Engineering Science and Technology", Vol. 3, Nr 9 - Ryan T.P. (2011),
*Statistical Methods for Quality Improvement*, Wiley, Smyrna - Shah S., Shridhar P., Gohil D. (2010),
*Control chart: A statistical process control tool in pharmacy*, "Asian Journal of Pharmaceutics", Vol. 4, Nr 3 - Wheeler D. J., (2011),
*What About p-Charts?*, Quality Digest

**Author:** Anna Kasprzyk