{"id":"control-chart","model_id":"Control chart","name":"Control chart","summary":"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.","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."],"safety_notes":["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."],"sources":[{"url":"https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc31.htm","title":"6.3.1. What are Control Charts?","accessed":"2026-10-25"}],"verification":{"status":"source_checked","checked_at":"2026-10-25","values":1,"method":"llm_source_check"},"updated_at":"2026-10-25","type":"operational","spec":{"kind":"methodology","related":[]},"url":"https://machineschema.com/operational/control-chart","license":"https://creativecommons.org/licenses/by/4.0/"}