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# Cusum

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 Title: Cusum Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Cusum

CUSUM chart
Originally proposed by E. S. Page
Process observations
Rational subgroup size n = 1
Measurement type Cumulative sum of a quality characteristic
Quality characteristic type Variables data
Underlying distribution Normal distribution
Performance
Size of shift to detect ≤ 1.5σ
Process variation chart
Not applicable
Process mean chart
Center line The target value, T, of the quality characteristic
Upper control limit C_i^+ = max \lbrack 0, x_i - \left ( T + K \right ) + C_{i - 1}^+\rbrack
Lower control limit C_i^- = max \lbrack 0, \left ( T - K \right ) - x_i + C_{i - 1}^-\rbrack
Plotted statistic C_i = \sum_{j=1}^i \bar x_j - T

In statistical quality control, the CUSUM (or cumulative sum control chart) is a sequential analysis technique developed by E. S. Page of the University of Cambridge. It is typically used for monitoring change detection.[1] CUSUM was announced in Biometrika, in 1954, a few years after the publication of Wald's SPRT algorithm.[2]

Page referred to a "quality number" \theta, by which he meant a parameter of the probability distribution; for example, the mean. He devised CUSUM as a method to determine changes in it, and proposed a criterion for deciding when to take corrective action. When the CUSUM method is applied to changes in mean, it can be used for step detection of a time series.

A few years later,

• "Engineering Statistics Handbook - Cusum Control Charts"