Title: cumulative sum control chart
Long Title: IUPAC Gold Book - cumulative sum control chart
DOI: 10.1351/goldbook.08071
Status: current

Definition
Control chart where the cumulative sum of deviations of successive measured quantity values from a reference quantity value is plotted to detect shifts in the level of the measurand.


Notes
1) The ordinate of each plotted point represents the algebraic sum of the previous ordinate and the most recent deviation from the reference, target, or control value.
2) The best discrimination of changes in the level is achieved when the reference quantity value is equal to the overall mean value.
3) The chart can be used for control, diagnostic, or predictive purposes.
4) When used for process control, it can be interpreted graphically using a mask (for example, V-mask) superimposed on the graph. An out-of-control criterion is when the path of the cumulative sum intersects or touches the boundary of the mask.
5) More suited to spreadsheet analysis, the following quantities are calculated for N measured quantity values \(x_{i} (i=1; …, N)\) \(S_{\rm{hi}}(i) = {\rm{max}}(0,S_{\rm{hi}}(i - 1) + x_{i} - \mu - k)\); \(S_{\rm{hi}}(0) = 0\) \(S_{\rm{lo}}(i) = \rm{max}(0,S_{\rm{lo}}(i - 1) - {x}_{i} + \mu - k)\); \(S_{\rm{lo}}(0) = 0\). \(\mu\) is a suitably chosen target value and \(k\) is a reference quantity value such that only shifts away from the target value greater than \(k\) will add to the cumulative sum. \(k\) is usually taken as half the shift in the process mean that is required to be detected divided by the standard deviation (\(\sigma\)) obtained under repeatability conditions of measurement. An out-of-control criterion is when \(S_{\rm{hi}}(i)\) or \(S_{\rm{lo}}(i)\) becomes greater than a limiting value \(h\) (usually \(h = 4\sigma\)). (Note that the reference quantity value \(k\) is not a coverage factor).
6) CUSUM charts are particularly sensitive to reveal a small measurement bias.

Related Terms
- Control chart: https://goldbook.iupac.org//terms/view/08068
- out-of-control criterion: https://goldbook.iupac.org//terms/view/08092
- repeatability conditions of measurement: https://goldbook.iupac.org//terms/view/08040

Source
- PAC, 2021, 93, 997. 'Metrological and quality concepts in analytical chemistry (IUPAC Recommendations 2021)' on page 1024 (https://doi.org/10.1515/pac-2019-0819)

Other Outputs
- html: https://goldbook.iupac.org/terms/view/08071/html
- json: https://goldbook.iupac.org/terms/view/08071/json
- xml: https://goldbook.iupac.org/terms/view/08071/xml

Citation: Citation: 'cumulative sum control chart' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.08071
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Disclaimer: The International Union of Pure and Applied Chemistry (IUPAC) is continuously reviewing and, where needed, updating terms in the Compendium of Chemical Terminology (the IUPAC Gold Book). Users of these terms are encouraged to include the version of a term with its use and to check regularly for updates to term definitions that you are using.
Accessed: 2026-05-30T08:06:19+00:00