**The Cusum control chart is even more efficient than the EWMA control chart but its implementation and interpretation are more complex. Its use is therefore to be reserved to specific cases where precision is essential.**

## Introduction

The Cusum control chart (cumulative Sum translated by map of accumulated sums) was developed in 1954 by E. S. Page^{1}. It will ” *accumulate* ” the standard deviation of each levy. It is used in various industries, especially in the chemical industry, and has been improved as time progresses to increase its sensitivity^{2}. It is, in the same way as the EWMA control chart, particularly suitable for the search for weak and slow drifts, therefore mainly for continuous processes.

The Cusum control chart is even more efficient than the **EWMA control chart** But its implementation and interpretation are more complex. Its use is therefore to be reserved to specific cases where precision is essential.

The principle of the Cusum control chart is, in the same way as the **EWMA control chart**, to take into account earlier, even minimal, drifts. It is the sum of the total in relation to the target. If the process moves away, the accumulation of variances will increase and exceed a limit. We will follow two curves simultaneously:

**the SH**It is used to detect a lag on the positive side of the target._{i}curve:**the curve SL**It is used to detect a lag on the negative side of the target._{i}:

The CUSUM control chart is suitable for quantitative measurements as well as for non-compliant proportions. All of the formulas presented below are written for the measurements. For a control chart to attributes, it will suffice to replace the term of mean by the proportion, the formulas remaining the same. |

## 1 – Choose the sensitivity settings of the control chart

The first step of this particular method is to identify the K and H parameters that are related to the sensitivity of the control chart :

**K:**The sensitivity parameter k is a penalty imposed so as to limit the number of false alerts. This constant depends on the magnitude of the gap that you want to detect. The smaller the k, the lower the drifts, but the higher the risk of false alarms.**h:**The parameter H represents the control limits.

The table below (standard NF-X06-031-4) gives us according to the level of precision we want, the value of our parameters.

Number of samples for a false alarm |
||||
---|---|---|---|---|

Adjustmentδ * √n | 100 | 370 | 500 | 1000 |

0,5 | k = 0,25 h = 5,6 POM1 = 19,3 | k = 0,25 h = 8,01 POM1 = 28,8 | k = 0,25 h = 8,585 POM1 = 31,1 | k = 0,25 h = 9,93 POM1 = 36,4 |

0,75 | k = 0,375 h = 4,33 POM1 = 19,3 | k = 0,375 h = 6 POM1 = 19,3 | k = 0,375 h = 6,39 POM1 = 16,6 | k = 0,375 h = 7,3 POM1 = 19,1 |

1 | k = 0,5 h = 3,502 POM1 = 7,4 | k = 0,5 h = 4,77 POM1 = 9,9 | k = 0,5 h = 5,07 POM1 = 10,5 | k = 0,5 h = 5,758 POM1 = 11,9 |

1,5 | k = 0,75 h = 2,48 POM1 = 4 | k = 0,75 h = 3,34 POM1 = 5,2 | k = 0,75 h = 3,54 POM1 = 5,4 | k = 0,75 h = 4 POM1 = 6,1 |

2 | k = 1 h = 1,874 POM1 = 2,6 | k = 1 h = 2,516 POM1 = 3,3 | k = 1 h = 2,665 POM1 = 3,4 | k = 1 h = 3,01 POM1 = 3,8 |

2,5 | k = 1,25 h = 1,46 POM1 = 1,87 | k = 1,25 h = 1,986 POM1 = 2,29 | k = 1,25 h = 2,105 POM1 = 2,39 | k = 1,25 h = 2,379 POM1 = 2,61 |

3 | k = 1,5 h = 1,132 POM1 = 1,44 | k = 1,5 h = 1,604 POM1 = 1,72 | k = 1,5 h = 1,708 POM1 = 1,79 | k = 1,5 h = 1,943 POM1 = 1,95 |

**Using the table**

We wish to have at most a false alarm every 500 samples. It is also desired to detect an adjustment of ± 1 standard deviation in 3 or 4 maximum samples. For this:

- The column with a POM0 (operational periods controlled) of 500 is selected in the table.
- In this column, a POM1 value (uncontrolled operational periods) is sought that approximates the more than 3 or 4. We find a cell with 3.4.
- We then identify our values K = 1 and H = 2.665.
- By making the reverse path, we deduce that N is 2.56, so we take 3.

## 2-Deduct the points

We calculate the different points of the 2 curves that we are going to follow. On the other hand its principle, the central card value is always 0. The respective formulas of the two curves are as follows:

**SH _{I} = Max (0; X_{Icross} -(target + K) + SH_{i-1})**

**SL _{I} = Min (0; (target-K)-X_{Icross} + SL_{I-1})**

With:

**x**average of the data of our samples._{Icross}:**k =**k * σ**H =**h * σ**n:**the number of data per sample.**σ:**Standard deviation of all of our data.

## 3 – Calculating control limits

The control limits are dependent on the precision level h that we have determined above. The formula is as follows:

**UCL = h**

**LCL =-H**

## 4-Interpretation

The interpretation is the same as for the **EWMA** control chart. So if a point comes out of the high or low limit, you have to act to refocus the process.

In the same way, it will be necessary in case of re-centring to 0 the control chart.

Finally, if the CUSUM control chart is very efficient to detect slow drifts, it is less than the SHEWHART control chart to detect fast drifts. The ideal is therefore to use all three control chart : CUSUM, average **X _{cross}** (or individual values

**I**) and range

**R**.

## Source

1-E.S. Page (1954) – Continuous inspection schemes

2 – J. M. Lucas, R. B. Croser (1982) – Fast Initial response for Cusum quality control schemes

D. M. Hawkins, D. H. Olwell (1998) – Cumulative sum charts and charting for quality improvement

A. Rose (2005) – Apply statistical control of processes to the manufacture of film-made compresses.

A. Groups (2013) – Applied Quality control

D. Durat (2008) – Quality of measurement in production

Standard NF X 06-031-4