Understanding variation – the springboard for process improvement
By Dr. Henry R. Neave
Edited and abridged by Mitch Beedie
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The Deming Forum Understanding variation – the springboard for process improvement
By Dr. Henry R. Neave Edited and abridged by Mitch Beedie
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There are four components, the main article: "Understanding Variation" and three linked expansions on the article.
- "Understanding variation"
- "Unknowns and unknowables"
- "Calculating control limits"
- "Six processes"
"Calculating control limits"
We can view a process as any task or set of actions, possibly involving decision-making, that generally repeats over time. Outputs and other important aspects of the process may be recorded and plotted over time. Control limits are calculated by a simple formula (easily implemented on a spreadsheet) using such data. The simplest processes to chart are those that only generate one measurement at a time: delivery time, number of mistakes per 100 invoices, weekly chemicals and water usage, etc. This is true for most service operations (as opposed to many manufacturing processes, where it is often conventional to obtain data “a few at a time” rather than individually).
Individual counts are drawn on an “X-chart”, an ordinary graph of these values but with three horizontal lines showing the average, and upper and lower control limits (UCL and LCL). These are calculated using the average measurement X(ave) and the average moving range MR(ave). The moving range is the size of the change between successive measurements, thus giving a "time-localised" measure of the variation in the data. The UCL and LCL are simply 2.66 x MR(ave) above and below X(ave) respectively.
Let’s take an example of 24 successive daily readings:
82 81 82 81 91 85 76 84 81 80 80 82 82 85 86 88 78 89 81 87 76 66 69 64.
These can be shown in a run chart:
What might we say about the behaviour of this process? There seem to be two periods of time over which it behaves very smoothly: the first few days, and days 9 to 16. In between these two periods (days 5 to 8) there is quite a wild "hiccup"; also, after day 16 the behaviour becomes quite erratic. So which of these features are "real", i.e. might have resulted from some definite causes? And which might realistically be dismissed as just randomness or natural variability - not worth trying to explain? A control chart helps us distinguish.
To find X(ave) we add the 24 values, making a total of 1936, and divide by 24 to give X(ave) = 80.67.
The moving ranges are the differences between adjacent values. The change from the first value (82) to the second value (81) is 82-81=1. The next two moving ranges are also equal to 1. But then from the fourth to the fifth value there is a jump of 91-81=10. From the fifth to the sixth, the change is 91-85=6, and so on. Note that it doesn't matter whether the change is up or down: simply subtract the smaller value from the larger value each time.
The final moving range is the change from the 23rd to the 24th value, 69-64=5 (note there are 23 moving ranges altogether, not 24!). The 23 moving ranges add up to 112, and dividing by 23 gives MR(ave)=4.87.
We can now calculate the control limits. The upper and lower control limits (UCL and LCL) are:
UCL = X(ave) + 2.66*MR(ave) = 80.67 + 2.66*4.87 = 80.67 + 12.95 = 93.62
LCL = X(ave) - 2.66*MR(ave) = 80.67 - 2.66*4.87 = 80.67 - 12.95 = 67.72.
In a spreadsheet, the data series could be held in cells B2 to Y2 and the moving ranges ABS(B2-C2) to ABS(X2-Y2) could be calculated into cells C3 to Y3. The average X(ave) calculated into cell Z2 is SUM(B2:Y2)/24, and the average moving range calculated into cell Z3 is SUM(C3:Y3)/23. The UCL is then Z2+2.66*Z3 and the LCL is Z2-2.66*Z3.
Drawing these control limits on the run chart:
What do we see? Everything is comfortably between the control limits (so, probably nothing we could "explain") except at the right-hand end of the graph: all of the last three values are hovering around the LCL, with two of them below it.
These final values cannot be explained by random variations within the process, so there must be another reason for them. There was. As we shall see in "Six Processes", the data are pulse-rates, and there was a change in medication starting on the 21st day.
