# The Tabular Cumulative Summation (CUSUM) Chart

Dr. Shewhart’s original control charts have been around for nearly 100 years. Since then, many different quality control texts have been written which highlight modifications to his 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, which is new to ProFicient 4.4.

CUSUM charts do not plot raw data values or averages. Instead, plot points on a CUSUM chart are data values roughly representative of the cumulatively summed, subgroup-to-subgroup deviations from a specified target or production mean.

#### Why Use a CUSUM Chart?

Basically, it boils down to sensitivity. The primary advantage of the tabular CUSUM chart is that it is more sensitive to small changes in the mean, especially when compared with IX-MR control charts. Some might argue that X-bar and Range or X-bar and s control charts could be used with large subgroup sizes to increase sensitivity. While that may be true, not everyone has the luxury of leveraging large sample sizes. Plus, unlike traditional control charts, the Tabular CUSUM plot points are influenced by previous data values, thus ensuring that historical information is accounted for in the current plot point.

Another reason CUSUM charts are popular is their ability to mitigate risk. For example, consider companies who tempt catastrophe when values stray too far from production targets. These organizations cannot afford to wait until plot point falls outside +/-3σ control limits before generating an alarm and taking action. Instead, they need greater sensitivity to small process changes.

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 Tabular CUSUM Charts Created?

ProFicient’s Tabular CUSUM charts are based upon those popularized in statistical texts such as Dr. Douglas Montgomery’s book, Introduction to Statistical Quality Control, 6th Edition. ProFicient’s Tabular CUSUM features include the ability to specify options such as:

1. The positive mean shift to be detected
2. The negative mean shift to be detected
3. The Decision Interval
4. Fast Initial Response (FIR)

The result is a Tabular CUSUM chart with two different plot lines – one for detecting the mean shift above Target and another for detecting a mean shift below Target. When compared with traditional control charts, the tabular CUSUM chart is unique in how it looks and in how plot points are calculated.

#### A Tabular CUSUM Example

A steel manufacturer makes large I-beams for the construction industry. I-beams are large, expensive to manufacture and are made infrequently. One of the critical features, the “A-Dimension,” must be closely controlled and is difficult to measure. If the A-Dimension sustains a shift of more than 1σ above its overall target, then the strength of the product could be compromised and I-beams 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 oneσ occurs.

Historically, the A-Dimension 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 A-Dimension values. They are found in the table below. The asterisks found next to subgroups 21-28 indicate that the mean changed from 50.00 to 50.75 – approximately a oneσ increase.

 Subgroup # A-Dimension Subgroup # A-Dimension 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

A traditional IX-MR control chart of the A-Dimension data is found below. The blue vertical line at subgroup 21 indicates the beginning of an increase of 1σ in the mean A-Dimension value.

Although there is a sustained 1σ increase in the A-Dimension for the final 8 values, no data values fall outside of the IX-MR chart’s control limits. The small subgroup size makes it difficult for the IX chart to identify the shift.

However, employing the Tabular CUSUM chart delivers a different result. The chart below shows the Tabular CUSUM chart for the data in the table above.

#### 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, which is part of its attraction to operators and administrators alike. The two different plotted lines allow the steel manufacturer to simultaneously view cumulative summations for both above and below the Target. While plot points are calculated on the top chart, the CUSUM calculations have no effect on the Moving Range plot points.

If there is no 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. Note that the lower CUSUM line hovers around the zero line for all 28 subgroups. This is because the sigma shift occurred above the Target, not below. Therefore, the Tabular CUSUM chart also specifies not only a significant change in sigma, but also the direction of the shift.

The steel company was especially concerned with an increase in the mean value above the Target. Unlike the IX-MR control chart, the CUSUM chart has triggered an alarm, indicating an increase in the mean of more than 1σ. Collectively, plot points after the 20th 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. Unlike independent plot points on an IX-MR control chart, each CUSUM plot point is influenced by previously plotted values. Not so with IX-MR charts. This fact helps tabular CUSUM charts provide additional sensitivity to small changes in the mean.

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