Tabular Cumulative Summation (CUSUM) Chart
Control charts have been around for nearly 100 years and were first created by Dr. Walter Shewhart at the Western Electric Company in the 1920's. Published in 1931, his groundbreaking book, Economic Control of Quality of Manufactured Product, set the standard for modern statistical quality control methods.
Today, many quality control texts have been written which highlight modifications to Shewhart's techniques. Most of these enhancements have been born of the need to better support modern manufacturing methods and challenges. One such chart is the Tabular Cumulative Summation chart, or simply, the CUSUM chart.
What is a CUSUM chart?
Like control charts, CUSUM charts are used to plot data in timeseries. The charts are meant to alert users to significant changes in process performance. CUSUM charts do not plot raw data values, averages, ranges or standard deviations. Instead, plot points on a CUSUM chart are data values roughly representative of the cumulatively summed, subgrouptosubgroup deviations from a specified target or production mean. The primary advantage of the tabular CUSUM chart is that the chart is more sensitive to small changes in the mean, especially when compared with IXMR control charts.
Why Use a CUSUM Chart?
Basically, it boils down to sensitivity. X̄ and Range and X̄ and S control charts are commonly understood, powerful process control tools. Their power lies in plotting average values which become more normally distributed as the subgroup size, n, increases. The larger the subgroup size, the more sensitive X̄ charts become in detecting significant shifts in the mean. Increasing control chart sensitivity is as simple as increasing subgroup size.
But not everyone has the luxury of creating large subgroups. Consider manufacturing organizations who specialize in short production runs or who rely on destructive testing to quantify product quality. In these cases, very little data is available, or it is costprohibitive to collect. Therefore, large subgroups are not an option.
If subgrouping cannot be leveraged, then an Individual X and Moving Range (IXMR) control chart might seem like a good alternative to X̄ and Range control charts. However, because plotted points on IX charts are individual data values (not averages), the IX values are less sensitive to small changes in the overall mean, especially when compared with X̄ charts.
Another reason CUSUM charts are popular is their ability to mitigate risk. For example, consider companies who tempt catastrophe when critical data values stray too far away from production targets. These organizations cannot afford to wait until plot point falls outside +/3σ control limits before generating an alarm. Instead, they need greater sensitivity to small process changes.
Each of the situations above indicates why CUSUM charts are useful. When small changes in the mean (1  1.5σ) must be identified and when very little data is available for process control purposes, the CUSUM chart is an excellent solution. As such, the CUSUM chart is popular in the aerospace, metallurgical, chemical and continuous processing industries.
How Are CUSUM Charts Created?
When compared with traditional control charts, the tabular CUSUM chart is unique, and certain parameters and statistics must be used in its creation.
Parameter 
Description 

The plot point is used exclusively for plotting the upper line on the tabular CUSUM chart. The upper line is used for identifying changes in the process above the stated target value, . 

The point is used exclusively for plotting the lower line on the tabular CUSUM chart. The lower line is used for identifying changes in the process below the chosen target value, . 

Process Mean or Target value. 

Sample standard deviation, or Long Term (LT) standard deviation. 

The positive shift in Process Mean that the CUSUM chart is meant to detect, in standard deviation units. Typically between 0.5σ and 1.5σ. 

The negative shift in Process Mean that the CUSUM chart is meant to detect, in standard deviation units. Typically between 0.5σ and 1.5σ. 

The "Upper Reference" value specified for plotted Positive CUSUM line. Positive CUSUM values are accumulated only when the deviation from the target value exceeds specified K value. K value is half the magnitude of the mean shift value to detect. 

The "Lower Reference" value specified for plotted Negative CUSUM line. Negative CUSUM values are accumulated only when the deviation from the target value exceeds specified K value. K value is half the magnitude of the mean shift value to detect. 

Decision Parameter. Factor for determining Decision Interval, H.
Generally, h is defined as 5, although sometimes 4 is utilized. 

H is the "Decision Interval," which acts as a control limit. 

Optional. Fast Initial Response or "head start" is typically half of H and is treated as the initial CUSUM value when no data exists. 
Figure 1 Table of parameters and statistics necessary to construct the tabular CUSUM chart.
A Tabular CUSUM Example
A steel manufacturer makes large Ibeams for the construction industry. Ibeams are large, expensive to manufacture and are made infrequently. One of the critical features, the "ADimension," must be closely controlled and is difficult to measure.
If the ADimension average sustains a shift of more than 1σ above its overall target, then the strength of the product could be compromised and product could be scrapped, resulting in significant financial loss. To prevent these issues, the company wants line operators to be notified when a shift of at least one σ has occurred in the ADimension's average.
Historically, the ADimension has mean value of close to 50.00 and a historical standard deviation of around 0.70. Data has been collected for the last 28 ADimension values. They are found in the table below. The asterisks found next to subgroups 2128 indicate that the mean changed from 50.00 to 50.75  approximately a one σ increase.
Subgroup # 
ADimension 

Subgroup # 
ADimension 
1 
50.453 

15 
49.895 
2 
50.682 

16 
50.014 
3 
49.686 

17 
49.373 
4 
49.572 

18 
50.523 
5 
51.333 

19 
51.111 
6 
50.280 

20 
50.044 
7 
49.240 

21* 
51.601 
8 
50.478 

22* 
50.479 
9 
49.263 

23* 
49.089 
10 
50.046 

24* 
50.632 
11 
49.540 

25* 
50.373 
12 
49.270 

26* 
51.682 
13 
50.316 

27* 
50.521 
14 
49.512 

28* 
51.639 
Figure 2 Table of ADimension data showing an increase of one σ in the last 8 subgroups.
A traditional IXMR control chart of the ADimension data is found in Figure 3. The blue vertical line indicates the beginning of a sustained increase of 1σ in the mean ADimension value.
Figure 3 IXMR control chart of ADimension data.
Although there is a sustained 1σ increase in the ADimension for the final 8 values on the IXMR control chart, no data values fall outside control limits. This shift is difficult for the IX chart to pick up because of the small subgroup size and the chart's inherent inability to sense small changes in the mean.
To demonstrate the advantages of the tabular CUSUM chart, the same data found in the table from Figure 2 will be used in the following example.
Creating the Tabular CUSUM chart
These are several steps for creating a tabular CUSUM chart:

 The data from Figure 2 must be converted and plotted into two different CUSUM plot points:
a. values indicating the cumulative summation of values below the target
b. values indicating the cumulative summation of values above the target
 Identification of the target,
 Selection of the positive shift in the mean to detect,
 Selection of the negative shift in the mean to detect,
 Selection of Decision parameter, H
First, the CUSUM plot point values for the steel Ibeam example are calculated and found in Figure 4.
Subgroup # 
ADimension 

CUSUM Plot
Point Upper

CUSUM Plot
Point Lower

1 
50.453 

0.065 
0.000 
2 
50.682 

0.359 
0.000 
3 
49.686 

0.000 
0.022 
4 
49.572 

0.000 
0.158 
5 
51.333 

0.946 
0.000 
6 
50.280 

0.838 
0.000 
7 
49.240 

0.000 
0.468 
8 
50.478 

0.091 
0.000 
9 
49.263 

0.000 
0.445 
10 
50.046 

0.000 
0.107 
11 
49.540 

0.000 
0.275 
12 
49.270 

0.000 
0.714 
13 
50.316 

0.000 
0.106 
14 
49.512 

0.000 
0.301 
15 
49.895 

0.000 
0.114 
16 
50.014 

0.000 
0.000 
17 
49.373 

0.000 
0.335 
18 
50.523 

0.135 
0.000 
19 
51.111 

0.858 
0.000 
20 
50.044 

0.515 
0.000 
21* 
51.601 

1.728 
0.000 
22* 
50.479 

1.819 
0.000 
23* 
49.089 

0.520 
0.619 
24* 
50.632 

0.764 
0.000 
25* 
50.373 

0.749 
0.000 
26* 
51.682 

2.044 
0.000 
27* 
50.521 

2.177 
0.000 
28* 
51.639 

3.428 
0.000 
Figure 4 CUSUM plot points calculated for both upper and lower CUSUM lines.
Plotted Points
Following are the assumptions and calculations necessary for completing the CUSUM chart for the steel Ibeam example:
 = Target Value = Actual Process mean = 50.048
 = Estimated Standard Deviation = 0.6796
 = Positive mean shift to detect (z) = 1.0σ
 = Negative mean shift to detect (z) = 1.0σ
 = Upper Reference = = = 0.3398
 = Lower Reference = = = 0.3398
 h = Decision Parameter = 5
 H = Decision Interval = h = 5 × = 5 × 0.6796 = 3.398
Begin creating the CUSUM chart by calculating the Upper and Lower CUSUM plot points using this equation:
Each plot point is the larger of either one of two results:
 Zero or
 The sum of and K, subtracted from the current subgroup data value, x_{i}, then added to the previous plot point (denoted by ).
Examples of Plot Point Calculations for the Upper and Lower CUSUM Line:
Figure 5 Calculated plot points for the first 3 subgroups on the CUSUM chart.
Once both the Upper and Lower CUSUM values have been calculated, they are plotted on the same chart (see Figure 6).
Figure 6 Steel IBeam ADimension data plotted on a tabular CUSUM chart.
Tabular CUSUM Chart Interpretation
With the notable exception of the dual plot point lines on the upper chart, the tabular CUSUM chart looks much like a Shewhart control chart. And that is part of its attraction. The two different plotted lines allow the steel manufacturer to simultaneously view cumulative summations for both above and below the Target. While the plot points are calculated on the top chart, the CUSUM calculations have no effect on the Moving Range plot points in the chart below. If there is not a significant shift in the mean, CUSUM plot points tend to gather around the center line, zero. This is true for the first 20 plot points.
Recall that the first twenty subgroups have a mean of 50.048 and an estimated standard deviation, , of .6796. From the 21^{st} through the 28^{th} subgroup, the mean has changed to 50.75. This is an increase of a little more than 1σ in the ADimension mean.
The steel company was especially concerned with an increase in the mean value above the Target. Unlike the IXMR control chart in Figure 3, the CUSUM chart has triggered an alarm, indicating an increase in the mean of more than 1σ. Collectively, plot points after the 20^{th} subgroup fall farther above the centerline when compared with data values prior to the change. This is further evidence of a sustained increase in the average.
Additionally, the last 4 plot points steadily increase until the last plot point falls outside the upper control limit (H value = 5 = 5(.6796) = 3.398). Unlike independent plot points on an IXMR control chart, each CUSUM plot point is influenced by previously plotted values. Not so with IXMR charts. This fact helps tabular CUSUM charts provide additional sensitivity to small changes in the mean.
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