P-chart

p-chart
Originally proposed by	Walter A. Shewhart
Process observations
Rational subgroup size	n > 1
Measurement type	Fraction nonconforming in a sample
Quality characteristic type	Attributes data
Underlying distribution	Binomial distribution
Performance
Size of shift to detect	≥ 1.5σ
Process variation chart
Not applicable
Process mean chart
Center line	${\displaystyle {\bar {p}}={\frac {\sum _{i=1}^{m}\sum _{j=1}^{n}{\begin{cases}1&{\mbox{if }}x_{ij}{\mbox{ defective}}\\0&{\mbox{otherwise}}\end{cases}}}{mn}}}$
Control limits	${\displaystyle {\bar {p}}\pm 3{\sqrt {\frac {{\bar {p}}(1-{\bar {p}})}{n}}}}$
Plotted statistic	${\displaystyle {\bar {p}}_{i}={\frac {\sum _{j=1}^{n}{\begin{cases}1&{\mbox{if }}x_{ij}{\mbox{ defective}}\\0&{\mbox{otherwise}}\end{cases}}}{n}}}$

In statistical quality control, the p-chart is a type of control chart used to monitor the proportion of nonconforming units in a sample, where the sample proportion nonconforming is defined as the ratio of the number of nonconforming units to the sample size, n.

The p-chart only accommodates "pass"/"fail"-type inspection as determined by one or more go-no go gauges or tests, effectively applying the specifications to the data before they are plotted on the chart. Other types of control charts display the magnitude of the quality characteristic under study, making troubleshooting possible directly from those charts.

The binomial distribution is the basis for the p-chart and requires the following assumptions:

The control limits for this chart type are ${\displaystyle {\bar {p}}\pm 3{\sqrt {\frac {{\bar {p}}(1-{\bar {p}})}{n}}}}$ where ${\displaystyle {\bar {p}}}$ is the estimate of the long-term process mean established during control-chart setup. Naturally, if the lower control limit is less than or equal to zero, process observations only need be plotted against the upper control limit. Note that observations of proportion nonconforming below a positive lower control limit are cause for concern as they are more frequently evidence of improperly calibrated test and inspection equipment or inadequately trained inspectors than of sustained quality improvement.

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Wikipedia