p-chart

Sampling requires some careful consideration. If the organization elects to use 100% inspection on a process, the production rate determines an appropriate sampling rate which in turn determines the sample size.[2]^: 277 If the organization elects to only inspect a fraction of units produced, the sample size should be chosen large enough so that the chance of finding at least one nonconforming unit in a sample is high—otherwise the false alarm rate is too high. One technique is to fix sample size so that there is a 50% chance of detecting a process shift of a given amount (for example, from 1% defective to 5% defective). If δ is the size of the shift to detect, then the sample size should be set to ${\displaystyle n\geq \left({\frac {3}{\delta }}\right)^{2}{\bar {p}}(1-{\bar {p}})}$.[2]^: 278 Another technique is to choose the sample size large enough so that the p-chart has a positive lower control limit or ${\displaystyle n>{\frac {3^{2}(1-{\bar {p}})}{\bar {p}}}}$.

In the case of 100% inspection, variation in the production rate (e.g., due to maintenance or shift changes) conspires to produce different sample sizes for each observation plotted on the p-chart. There are three ways to deal with this:

Technique	Description
Use variable-width control limits[2]: 280	Each observation plots against its own control limits: ${\displaystyle {\bar {p}}\pm 3{\sqrt {\frac {{\bar {p}}(1-{\bar {p}})}{n_{i}}}}}$, where ni is the size of the sample that produced the ith observation on the p-chart
Use control limits based on an average sample size[2]: 282	Control limits are ${\displaystyle {\bar {p}}\pm 3{\sqrt {\frac {{\bar {p}}(1-{\bar {p}})}{\bar {n}}}}}$, where ${\displaystyle {\bar {n}}}$ is the average size of all the samples on the p-chart, ${\displaystyle {\frac {\sum _{i=1}^{m}n_{i}}{m}}}$
Use a standardized control chart[2]: 283	Control limits are ±3 and the observations, ${\displaystyle {\hat {p}}_{i}}$, are standardized using ${\displaystyle Z_{i}={\frac {{\hat {p}}_{i}-{\bar {p}}}{\sqrt {\frac {{\bar {p}}(1-{\bar {p}})}{n_{i}}}}}}$, where ni is the size of the sample that produced the ith observation on the p-chart

Some practitioners have pointed out that the p-chart is sensitive to the underlying assumptions, using control limits derived from the binomial distribution rather than from the observed sample variance. Due to this sensitivity to the underlying assumptions, p-charts are often implemented incorrectly, with control limits that are either too wide or too narrow, leading to incorrect decisions regarding process stability.[3] A p-chart is a form of the Individuals chart (also referred to as "XmR" or "ImR"), and these practitioners recommend the individuals chart as a more robust alternative for count-based data.[4]