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Introduction to Control Charts: XmR Chart
Farrokh Alemi, Ph.D.
IndexFarrokh Alemi, Ph.D.
Purpose of Control Chart
To judge whether change has led to improvement.
To visually tell a story of changes in a key measure over time.
IndexFarrokh Alemi, Ph.D.
Elements of Control Chart X-axis shows time periods Y-axis shows the
observation values UCL line shows the upper
control limit LCL line shows the lower
control limit 95% of data should fall
within UCL and LCL Values outside the control
limits are likely to be statistically significant
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1st Qtr
2ndQtr
3rdQtr
4thQtr
UCL
LCL
Observations
IndexFarrokh Alemi, Ph.D.
Definition of Statistical Control Variation occurs in any outcome of interest over
time.– In a stable situation, some variation will occur just by
chance, but it will be predictable over time. Statisticians call this “common cause” variation or “within control limits.”
– If there is a significant change, data points will show up outside the range expected for chance variation alone. Statisticians call this “special cause” variation or “outside control limits.”
A control chart allows us to detect statistically important changes.
IndexFarrokh Alemi, Ph.D.
Which Chart is Right?
For different outcomes different control charts are appropriate.
Click here to see which chart is appropriate for the outcome you have in mind.
This presentation focuses on one type of chart named XmR chart.
Assumptions of XmR chart
IndexFarrokh Alemi, Ph.D.
Assumptions of XmR charts
There is one observation per time period. Patients’ case mix or risk factors do not change in
important ways over the time periods. Observations are measured in an “interval” scale,
i.e. the observation values can be meaningfully added or divided.
Observations are independent of each other, meaning that knowledge of one observation does not tell much about what the next value will be.
IndexFarrokh Alemi, Ph.D.
Moving Range
An XmR chart is based on the absolute differences between consecutive values, displayed as a “Moving Range”
Even when observations come from non-normal distributions, differences in consecutive values form a normal distribution as the number of observations increases
IndexFarrokh Alemi, Ph.D.
Calculating the Moving Range
The number of observations is “n.” The absolute value of the difference
between consecutive values is the moving range, “R”
An example follows
IndexFarrokh Alemi, Ph.D.
Example Data
Time period 1 2 3 4 5 6 7 8Observation 90 85 92 67 98 83 94 90
IndexFarrokh Alemi, Ph.D.
Calculating the Average Moving Range
Time period 1 2 3 4 5 6 7 8 Mean Observation 90 85 92 67 98 83 94 91 87.50Moving range 5 7 25 31 15 11 3 13.86
Add the differences and divide by n minus one to get the average moving range.
Mean R = |(Xt - Xt-1)| / (n-1)
IndexFarrokh Alemi, Ph.D.
Calculating Upper and Lower Control Limits
If E is a correction constant, then:
Upper Control Limit = Average of observations + E * Average of moving range
Lower Control Limit = Average of observations - E * Average of moving range
IndexFarrokh Alemi, Ph.D.
Correction Factor Depends on Number of Time Periods
Number of time
periods E values
Number of time
periods E values d2 values11 0.945
2 2.660 12 0.9213 1.772 13 0.8994 1.457 14 0.8815 1.290 15 0.8646 1.184 16 0.8497 1.109 17 0.8368 1.054 18 0.8249 1.010 19 0.81310 0.975 20 0.803
Based on Wheeler DJ. Advanced topics in statisical process control, 1995 SPC Press
Inc, Knoxville TN 37919
IndexFarrokh Alemi, Ph.D.
Calculating Upper and Lower Control Limits
Time period 1 2 3 4 5 6 7 8 AverageObservations 90 85 92 67 98 83 94 91 87.50Moving range 5 7 25 31 15 11 3 13.86
= 87.50 + 1.054 * 13.86
= 87.50 - 1.054 * 13.86
Upper control limit
Lower control limit
E for 8 time periods is 1.054
IndexFarrokh Alemi, Ph.D.
Plot the Control Chart
Plot the x and y axis Plot the observations Plot the upper control
limit Plot the lower control
limit Variation among
observations that fall between control limits is likely due to chance
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1 2 3 4 5 6 7 8
Time periods
Observations
UCL
LCL
IndexFarrokh Alemi, Ph.D.
Interpret the Control Chart Points outside the limits
represent real changes in the outcome of interest
The observation at time period 4 falls below the LCL; it is unlikely that this is due to random chance events
The next step is to determine the possible causes of this significantly different observation
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1 2 3 4 5 6 7 8
Time periods
Observations
UCL
LCL
IndexFarrokh Alemi, Ph.D.
Share the Results With Others
Control charts are effective ways to visually tell a story
Distribute the chart by electronic media, as part of a newsletter, or as an element of a story board display
Show that you have verified any assumptions, check that your chart is accurately labeled, and include your interpretation of the finding
IndexFarrokh Alemi, Ph.D.
Index of Contents Purpose of Control Ch
art Elements of Control C
hart Definition of Statistica
l Control Which Chart is Right? Assumptions Moving Range Calculating Moving
Range
Example Data Calculating Average
Moving Range Calculating Control
Limits Plot the Control Chart Interpret the chart Share the Result
