# Control chart

Operations · `Control chart` · verification: source checked

A control chart is a graphical display of a quality characteristic plotted against sample number or time. It carries a centre line at the mean of the in-control process plus an upper control limit (UCL) and a lower control limit (LCL), set so that almost all points fall inside them while the process stays in control. A point outside the limits, or a non-random pattern inside them, signals that the process is probably out of control and should be investigated.

## Specification

| Property | Value |
|---|---|
| Kind | methodology |

## Safety

- A data point outside the control limits is assumed to mean the process is probably out of control, and an investigation is warranted to find and eliminate the cause or causes.
- If the underlying distribution is skewed, the 3-sigma upper limit falls short of the 0.001 limit and the lower 3-sigma limit falls below it, increasing the risk of searching for assignable causes of positive variation when none exists.
- For a Poisson distribution with np = 0.8, using 3-sigma limits raises the risk of exceeding the upper limit by chance from 0.001 to 0.009 and reduces the lower-limit risk from 0.001 to 0.

## Study notes

- Univariate control charts display one quality characteristic; multivariate control charts display a statistic that summarises or represents more than one quality characteristic.
- In the U.S. it is acceptable practice to base control limits on a multiple of the standard deviation, usually 3, giving 3-sigma limits; the term is used whether the standard deviation is the population parameter, an estimate, or a standard value.
- In the U.K. statisticians generally prefer probability limits rather than 3-sigma limits.
- For a normal distribution the 3-sigma limits correspond to a probability of 0.00135 in one direction and 0.0027 in both directions, making them the practical equivalent of 0.001 probability limits.
- Two out of one thousand is an arbitrary figure; limits could be set at one out of a hundred or larger depending on the risk management of the quality control programme is willing to take. In general, limits approximating the 0.002 standard are customary.
- For a Poisson distribution with np = 0.8, the mean and variance both equal np, so the upper 3-sigma limit is 0.8 + 3*sqrt(0.8) = 3.48 and the lower limit is 0.
- In control means all points lie between the control limits and form a random pattern; systematic behaviour, such as 25 of 30 points above the centre line followed by 5 below, still indicates a problem.
- Statistical methods for detecting sequences or non-random patterns can be applied to the interpretation of control charts.

## Sources

1. [6.3.1. What are Control Charts?](https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc31.htm) (accessed 2026-10-25)

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