Introduction to Control Charts Xm R Chart Farrokh

Introduction to Control Charts: Xm. R Chart Farrokh 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. Farrokh Alemi, Ph. D. Index

Elements of Control Chart UCL ¨ 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. Observations LCL Index

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. Index

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 Xm. R chart. Assumptions of Xm. R chart Farrokh Alemi, Ph. D. Index

Assumptions of Xm. R 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. Index

Moving Range ¨ An Xm. R chart is based on the absolute differences between consecutive values, displayed as a “Moving Range” ¨ Even when observations come from nonnormal distributions, differences in consecutive values form a normal distribution as the number of observations increases Farrokh Alemi, Ph. D. Index

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. Index

Example Data Farrokh Alemi, Ph. D. Index

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. Index

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. Index

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

Calculating Upper and Lower Control Limits E for 8 time periods is 1. 054 Farrokh Alemi, Ph. D. Index

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. Index

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. Index

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. Index

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. Index