1. Introduction to Control Charts: XmR Chart
Farrokh Alemi, Ph.D.

2. 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.
Farrokh Alemi, Ph.D.

3. Elements of Control Chart
UCL
LCL
Observations
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
Farrokh Alemi, Ph.D.

4. 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.
Farrokh Alemi, Ph.D.

5. Assumptions of XmR chart
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.
Note to Skip & Mark: “Here” is a hyperlink to Alemi Web site with more
information on choosing control charts.
We could use this or ask him to give us a copy we could have
here. Or we could delete it.
Assumptions of XmR chart
Farrokh Alemi, Ph.D.

6. 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.
Farrokh Alemi, Ph.D.

7. 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
Farrokh Alemi, Ph.D.

8. 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
Farrokh Alemi, Ph.D.

9. Example Data
Farrokh Alemi, Ph.D.

10. Calculating the Average Moving Range
Add the differences and divide by n minus one to get the average moving range.
Mean R =  |(Xt - Xt-1)| / (n-1)
Farrokh Alemi, Ph.D.

11. 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
Farrokh Alemi, Ph.D.

12. Correction Factor Depends on Number of Time Periods
Farrokh Alemi, Ph.D.

13. Calculating Upper and Lower Control Limits
E for 8 time periods is 1.054
Farrokh Alemi, Ph.D.

14. 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
Farrokh Alemi, Ph.D.

15. 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
Farrokh Alemi, Ph.D.

16. 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
Farrokh Alemi, Ph.D.

17. Index of Contents Purpose of Control Chart Elements of Control Chart
Definition of Statistical 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
Farrokh Alemi, Ph.D.