## What is a Cumulative Sum Chart?

A cumulative sum chart (CUSUM) is a type of control chart used to detect the deviation of the individual values or subgroup mean from the adjusted target value, in other words monitor the deviation from the target value. CUSUM chart is an alternative to Shewhart control charts.

The basic advantage of CUSUM chart is that it is more sensitive to the small shift of the process mean when compared to the Shewhart charts (Individuals I-MR or Xbar charts).”

The cumulative sum chart and the exponentially weighted moving average (EWMA) charts are also monitors the mean of the process, but the basic difference is unlike Xbar charts they consider the previous value means at each point. Moreover these charts are considered as a reliable estimate when correct standard deviation exists.

## When Would you Use a Cumulative Sum Chart?

The purpose of cumulative sum chart (CUSUM) is to monitor the small shift in the process mean of the samples collects at a time intervals. These measurements of samples at a given time interval represents the subgroups.

Instead of calculating the subgroups mean independently, the CUSUM chart represents the information of current and previous samples. Hence the CUSUM chart is always better than the Xbar charts to detect the small shifts of process mean. The CUSUM chart is more effective when the samples size is one. These charts are basically used in process industries and in manufacturing.

**Comparison between CUSUM and Shewhart chart**

The purpose of both cumulative sum chart (CUSUM) and Shewhart charts are to detect the mean shits in the process, the basic differences are

- CUSUM chart considers all the samples up to current point and also considers the current sample for measurement whereas Shewhart chart is based on the single subgroup measurement
- CUSUM chart determination of out of control limit is based on decision interval or use of V-Mask method where as Shewhart chart it is based on control limits (upper and lower control limits)
- CUSUM chart the control limits are computed from average run length spec, where as Shewhart chart control limits are generally three sigma limits.

### Advantages of CUSUM chart

- CUSUM chart is the best way to detect the small shifts of process mean especially 0.5 to 2 SD from the target mean
- It is easy to identify visually the shifts in process mean

### Disadvantages of CUSUM chart

- Establishing and maintaining of CUSUM charts are more difficult
- CUSUM charts are slower in detect large process mean shift
- Since CUSUMs are correlated, it is tough to interpret the patterns.

## How do you make a Cumulative Sum Chart?

The cumulative sum chart (CUSUM) can be represented in visual method ie V-mask, this method was introduced by Barnard in 1959 to check whether process is out of control or not, but generally tabular (algorithmic) method will be used to monitor the process. Unlike other standard control charts, all previous measurements for CUSUM charts are included in the calculation for the latest plot. But, establishing and maintaining the CUSUM is difficult.

### V-Mask Method

V-mask looks like a sideways V. The V-mask chart is to check whether each marked sample falls within the boundaries of the V-mark. If any point falls outside of control limit that indicates a signal mean shift in the process. When each sample is plotted, the V-mask may shifted to the right. The below graph shows the V-mask and related formulas.

The behaviour of the V-Mask is determined by the distance k (which is the slope of the lower arm) and the rise distance h. The team could also specify d and the vertex angle (or, as is more common in the literature, q = 1/2 the vertex angle). For an alpha and beta design approach, we must specify

- α, the probability of concluding that a shift in the process has occurred, when in fact it did not.
- β, the probability of not detecting that a shift in the process mean has, in fact, occurred.
- δ(delta), the detection level for a shift in the process mean, expressed as a multiple of the standard deviation of the data points.

### Tabular Method

Tabular method is more easiest way than the V-Mask method, steps to make cumulative sum chart (CUSUM)

- First of all, estimate the standard deviation of the data from the moving range control chart σ= R̅/d
_{2} - Calculate the reference value or allowable slack, since CUSUM chart is used to monitor the small shifts, generally 0.5 to 1 sigma will be considered. K= 0.5 σ
- Compute decision interval H, generally ± 4 σ will be considered (some place ± 5 σ also be used)
- Calculate the upper and lower CUSUM values for each individual i value

- Upper CUSUM (UC
_{i})= Max[0, UC_{i-1}+x_{i }– Target value-k)

- Lower CUSUM (LC
_{i})= Min[0, LC_{i-1}+x_{i }– Target value+k)

- Draw all UC
_{i }& LC_{i}values in the graph and also draw decision intervals (UCL and LCL) - Check if any of the UC
_{i}values above the UCL and any of the LC_{i}values below the LCL - Finally, take necessary action to eliminate the special causes, if any of the points out of control limits

## Example of Using a Cumulative Sum Chart in a DMAIC Project

**Example: **CUSUM chart will be used in control phase of DMAIC. In drug manufacturing unit, potassium content is one of the important parameter. The target potassium content is 0.21% in wt. Therefore team has collected 25 batches from the production in a time interval to monitor the process mean shift

**Step1:** Estimate the standard deviation of the data from the moving range control chart σ= R̅/d_{2}

- n=2
- d2 = 1.128
- σ= R̅/d
_{2 }= 0.01914/1.128 = 0.0169

Refer common factors for various control charts

**Step 2: **Calculate the reference value or allowable slack K= 0.5 σ = 0.5*0.0169 = 0.00848

**Step 3: **Compute decision interval H, generally ± 4 σ = ± 4 * 0.0169= ± 0.0679

UCL = +0.0679 and LCL = -0.00679

**Step 4:** Calculate the upper and lower CUSUM values for each individual i value

Target value =0.21

Upper CUSUM (UC_{i})= Max[0, UC_{i-1}+x_{i }– Target value-k)

Lower CUSUM (LC_{i})= Min[0, LC_{i-1}+x_{i }– Target value+k)

For first sample: 0.235

- Upper CUSUM (UC
_{1})= Max[0, 0+0.235– 0.21-0.0084) =0.017 - Lower CUSUM (LC
_{1})= Min[0, 0+0.0235-0.21+0.00848) = 0

For second sample: 0.222

- Upper CUSUM (UC
_{1})= Max[0, 0.017+0.222– 0.21-0.0084) =0.02 - Lower CUSUM (LC
_{1})= Min[0, 0+0.0222-0.21+0.0084) = 0

Similarly find the Upper and Lower CUSUM for all the samples

**Step 5:** Draw all UC_{i }& LC_{i} values in the graph and also draw the decision intervals (UCL and LCL)

**Step 6:** Conclusion: So, from the above graph it is clearly evident that from sample 13 on-wards process is going out of control. Hence team has to identify the special cause and to take the appropriate corrective action.

**Compare CUSUM example with Shewhart Chart**

Furthermore, use sample example from the above and draw the Individuals I-MR

From the above graph it is clearly evident that both Individual and Moving range process is in control and predictable, but sample example CUSUM chart shows the process is going out of control. Hence CUSUM charts are more effective to monitor the small shifts in the process mean than Individuals I-MR or Xbar charts

## What Do You Need to Know for Your Exam?

### Green Belts

The IASSC Green Belt BOK requires you to know EWMAs as part of Statistical Process Control.

### Black Belts

The Villanova Black Belt BOK notes you are required to “Understand appropriate uses of short-run SPC, EWMA, CuSum, and moving average.”

The IASSC Black Belt BOK lists CUSUM Charts under what’s needed for 5.2 Statistical Process Control (SPC).