# CUSUM chart

CUSUM chart (cumulative sum control chart) - a type of control chart that's using E. Page's sequential analysis technique [1] . That technique is an alternative to traditional individual charts, as it uses the cumulative sum of deviations from a target. It is typically used for detecting and monitoring small shifts in the process mean.

Due to chart's superior sensitivity, the primary purpose of using a CUSUM control chart is to detect small changes from the process target, as it improved the ability to detect shifts less than 1.5σ by charting a statistic that incorporates process' current and previous data values [2].

## Cumulative sum

Cumulative sum let us collect $$m$$ samples, each of size of $$n$$, and compute the mean of each sample. Then the CUSUM control chart is formed by plotting one of these quantities$S_m = \sum_{i=1}^m \bar x_i − \hat μ_0 \quad or \quad S′_m = \frac {1}{σ_ \bar x¯}\ \sum_{i=1}^m (\bar x_i − \hat μ_0)$ where $$\hat μ_0$$ is an estimation of the in-control mean and $$σ_ \bar x¯$$ is the known (or estimated) standard deviation of the sample means. The choice of which of these two quantities are plotted is standardly determined by the statistical software package. As long as the process remains in control centered at $$\hat μ_0$$, the CUSUM plot shows variation in a random pattern centered around zero. The charted CUSUM points will eventually drift upwards in case the process mean shifts upward and vice versa if the process mean decreases [3].

## CSUM charts usage and creation

Tabular CUSUM chart is useful in situations where sub-grouping is not desired or feasible, yet great sensitivity is necessary. CUSUM charts are also a good fit where the process average is expected to naturally shift or trend from the target and there is a need to make process adjustments in a timely manner to bring the process back on target [4].

• Upper control limit $$S_m^+ = max[0,(x_m - (\hat μ_0 + K) + S_(m-1)^+]$$
• Lower control limit $$S_m^- = max[0,(\hat μ_0 - K) - x_m + S_(m-1)^-]$$
• Plotted statistic $$S_m = \sum_{i=1}^m \bar x_i − \hat μ_0$$

## CSUM charts and Shewhart charts

A CUSUM chart plots the cumulative sums of the aberration of the sample values from a target value. The incorporation of using several samples in the cumulative sum results in greater sensitivity and speed for detecting shifts or trends over the traditional Shewhart charts [5].

## Footnotes

1. Page E., 1954
2. Wachs S., 2010
3. NIST/SEMATECH, 2012
4. Wachs S., 2010
5. Wachs S., 2010

## References

Author: Anna Strzelecka