The Road to Six Sigma - Control Charts

Part 4 of a 4-Part Series

We come now to what started this “Road to Six Sigma” series in the first place, Dr. Shewhart’s simple control tool, the control chart. The control chart is really two time-based charts plotted on the same form. For simplicity’s sake, we will focus on control charts for variable individual readings because this chart is usually most useful in understanding and improving business processes.A control chart for individuals is used when a sample size of one is the only sample available. Examples are daily sales numbers, monthly earnings reports, daily site traffic, or other metrics for which a multiple sample size is not available. This is in contrast to a manufacturing process where one can randomly choose ten representative products from the production stream, measure them all, and average the results.

A chart produced from such data would be called an X-Bar and R chart. X-Bar is the average of the ten measurements within the sample and R is the range between the highest and lowest individual readings within the sample. The purpose of such a chart is to measure the variations within the sample of ten specimens and compare those variations to the variations between samples. If the within sample variations are consistent with the between sample variations, the process is in control and the variability is predictable.

In contrast, an individuals chart does not allow a sample size of more than one. How do we then calculate a “range”, R. An individuals chart uses the measured value of the variable at interval “t” as the X measurement, and the absolute value of the difference between X_t and X_t-1 the previous interval as the range R. Let’s see how we can calculate the data necessary to plot an individuals control chart. The following data is derived from actual weekly sales data for an on-line business.

Weekly Sales X and R chart

Week	Weekly Sales X
1	104,679
2	115,537
3	134,696
4	177,393
5	205,437
6	184,038
7	105,863
8	163,746
9	183,134
10	205,348
11	265,599
12	197,901
13	113,093
14	219,758
15	192,949
16	174,363
17	80,148
18	212,387
Process Average	168,671

For the previous 18 weeks, the weekly sales has varied from a high of 265,599 to a low of 80,148. The average weekly sales, or X-Bar, is 168,671. With this data we can calculate the moving ranges between readings. Remember that the range is the absolute value (always positive) of the difference between successive readings. This calculated moving range data looks like:

Weekly Sales X and R Chart

Week	Weekly Sales X	Moving Range R
1	104,679	-
2	115,537	10,858
3	134,696	19,159
4	177,393	42,697
5	205,437	28,044
6	184,038	21,399
7	105,863	78,175
8	163,746	57,883
9	183,134	19,388
10	205,348	22,214
11	265,599	60,251
12	197,901	67,698
13	113,093	84,808
14	219,758	106,665
15	192,949	26,809
16	174,363	18,586
17	80,148	94,215
18	212,387	132,239
Process Average	168,671	52,417

For the period of the analysis, the average range, R-Bar, has been 52,417. Clearly, this process is highly variable, but is it in control? In order to determine that, we must calculate some control limits. In this case, control limits are an approximation of the statistical observations of a normal distribution. Any data point more than three standard deviations, designated +/- 3 sigma from the process average is out of control, and the process is not predictable. The 3-sigma limits are approximated by:

UCL = X-Bar + 2.66 x R-Bar

and

LCL = X-Bar – 2.66 x R-Bar

With the data at hand, the upper control limit is 168,671 + 2.66 x 52,417 or 308,100

The lower control limit is 168,671 – 2.66 x 52,417 or 29,241.

When plotted on a chart this looks like the one on the right. We can immediately see that the process is in control based on the three sigma rule, but is that sufficient? Shewhart later modified his control scheme to account for other types of non-normal distribution such as time series trends, process shifts and other special causes of variability that did not violate the three sigma rule.

To conduct a more refined analysis we need to divide the area between the control limits into six control zones, limited by +/- 1-sigma, +/- 2-sigma, and +/- 3-sigma. Since we already have the +/- 3-sigma limits, we can obtain the +/- 1-sigma limits by dividing the difference between the process average and the control limits by three

X-Bar + 2.66 x R-Bar – X-Bar = 2.66x R-Bar/3 = 46,476

Adding the plus and minus 1 and 2-sigma lines to our chart looks like:

Shewhart’s modification added a set of control rules, violation of any of which would constitute an out of control condition. These rules are:

• Rule 1 - any point outside the control limits (more than 3 sigma)
• Rule 2 – two out of three successive points more than 2 sigma away from the process average on the same side
• Rule 3 - four out of five successive points more than 1 sigma away from the process average on the same side
• Rule 4 - eight successive points on the same side of the process average

Even with these more detailed guidelines, we can not find any rule violations in this process, thus the process is in control.

Control charts are a good way to determine if a test change to a process, say a promotional offer, has a statistically significant effect on the process. To show how this might work, look at the following data from the end of the period analyzed above.

wk19	246,644
wk20	233,876
wk21	301,726
wk22	181,823
wk23	208,339
wk24	189,499
wk25	156,770
wk26	265,408
wk27	205,144
wk28	167,705
wk29	213,889
wk30	128,115
wk31	211,445
wk32	182,777
wk33	236,409
wk34	237,402
wk35	252,436
wk36	192,923
wk37	320,541
wk38	240,444

Add this data to the control chart data used above, plot the control chart using the process average and control limits already calculated plot the X and R individuals chart and answer the following questions:

1. Is the process in control?
2. If not, what are the rule violations?
3. Is this a good or a bad situation?

E-Mail me with the correct answers to these questions (john@bizmanualz.com) and I’ll take 15% off your purchase of one of the following manuals:

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This concludes our series on using statistical process control tools in business settings.

Learn more about developing policies, procedures and processes, or about improving your organization by attending the next How to Create Well Defined Procedures and Processes or Statistical Process Control classes. To address other training needs, please visit the Bizmanualz Training Website.

Related Articles:

1. Continuous Improvements with Control Charts
2. Correlation and Control Charts
3. The Road to Six Sigma - Understanding Correlation Between Variables
4. The Road to Six Sigma: Cause-Effect and Scatter Diagrams
5. The Road to Six Sigma: Applying Statistical Process Control Tools

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Posted on Tuesday, October 31st, 2006 under Business Process Improvement

This work is licensed under a Creative Commons Attribution 3.0 United States License.