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A new type of plot: Process Behavior Chart (X-Chart)

A new type of plot: Process Behavior Chart (X-Chart) as devised by Donald J. Wheeler would be very useful.

Rationale: Process Behavior Charts allow to differentiate between routine (random) variation and special variation. The first should be interpreted as noise and ignored, and only the latter requires our attention and action. In addition to business and engineering uses, there are many important personal use applications such as:

  • body weight control,
  • blood pressure control,
  • blood sugar control,
  • cholesterol levels,
  • workout outcomes, etc.

Without Process Behavior Charts people often undertake a lot of unnecessary action (they impulsively react to routine variation) and miss the opportunity to take action when it is needed (i.e. in the context of special variation).

Project file: Process_Behavior_Chart__X-chart_.lml

An example of an X-Chart (from Understanding Variation by Donald J. Wheeler):

X-Chart_for_Manhours

Steps to make an X-Chart:

  1. Make a time series plot (line+symbol).
  2. Compute and plot the Central Line (CL).
  3. Compute and plot the Upper Natural Process Limit (UNPL).
  4. Compute and plot the Lower Natural Process Limit (LNPL).

To compute the central line and the two limits you need to:

  1. Compute the Average or the Median of individual values.
  2. Compute the average or median of moving ranges, where: MovingRange(n) = abs[x(n) - x(n-1)].

There are two ways to compute natural process limits (the first based on the average, the second based on the median):

CL = the average or the median of individual values
UNPL = the average + 2,66 x AverageMovingRange
LNPL = the average - 2,66 x AverageMovingRange

or:

UNPL = the average + 3,14 x MedianMovingRange
LNPL = the average - 3,14 x MedianMovingRange

Below is a Process Behavior Chart made in LabPlot based on the above data.

PBChart

The Central Line and Natural Process Limits in my plot made in LabPlot are the same for the whole range of data. This would be OK, if the process was stable. However, if the process is not stable then you need to find out, when then process ceases to be stable and compute new Central Lines and Natural Control Limits for all new stable periods within the whole range of data and for clarity reasons disconnect the main plot line accordingly, i.e. between different stable periods. See the attached screenshot above from Wheeler's book. Unfortunately, I can't detect and plot multiple Central Lines and Natural Process Limits in LabPlot, because the range lines in LabPlot apply only to the whole range of data contained in a plot. It's very interesting to compare these two plots.

How to detect, if the process is not stable anymore? There are some easy rules to use. Three of such rules (recommended by Donald J. Wheeler) are presented below):

THREE RECOMMENDED DETECTION RULES (by Donald J. Wheeler in Making Sense of Data)

[x] Detection Rule One: Points Outside the Limits
A single point outside the computed limits should be taken as a signal of the presence of an assignable cause which has a dominant effect.

[x] Detection Rule Two: Runs About the Central Line
Eight successive values on the same side of the central line will be taken as an indication of the presence of an assignable cause which has a weak but sustained effect. Ignore the points on the central line.

[x] Detection Rule Three: Runs Near the Limits
Three out of four successive values in the upper 25%, or three out of four successive values in the lower 25%, of the region between the limits may be taken as a signal of the presence of an assignable cause which has a moderate but sustained effect.

Detection Rules One, Two, and Three form a coherent set of detection rules that collectively achieve maximum power. Using more detection rules will not boost the sensitivity of the chart, but it will have the effect of increasing the number of false alarms excessively. Many people have been taught to use a rule for six or seven points going up or down. This rule has repeatedly been found to be of little use except to increase the number of false alarms. It should not be used.

OTHER RULES:
[ ] Detection Rule Four: Runs Beyond Two-Sigma
When two out of three successive values fall more than two-sigma above the central line, or more than two-sigma below the central line, they may be interpreted as a signal of the presence of an assignable cause which has a moderate but sustained effect.

[ ] Detection Rule Five: Runs Beyond One-Sigma
When four out of five successive values fall more than one-sigma above the central line, or more than one-sigma below the central line, they may be interpreted as a signal of the presence of an assignable cause which has a moderate but sustained effect.

Detection Rules One, Two, Four, and Five also form a coherent set of detection rules that achieve maximum power. Using more detection rules will not boost the sensitivity of the chart, but will merely increase the number of false alarms by an excessive amount.

The Pattern Detection Guideline: Any pattern consisting of two, three, or more points, which is repeated eight times in succession, is highly unlikely to be due to routine variation.


Each of the above rules should be applied on user's request (e.g. use the choice boxes [x]). By default, a single central line and control limits should be plotted (based on the average or median measures).

Once you have computed the average and standard deviation statistic for your data you can expect (acc. to Donald J. Wheeler's "Empirical Rule"):

  • About 60% to 75% of the data within one standard deviation of the average.
  • Usually 90% to 98% of the data within two standard deviations of the average.
  • Approximately 99% to 100% of the data within three standard deviations of the average.
Edited by Dariusz Laska