Transcript Introduction to Control Charts: XmR Chart Farrokh Alemi, Ph.D. Purpose of Control Chart  To judge whether change has led to improvement.  To visually tell.

Introduction to Control
Charts: XmR 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
 X-axis shows time periods
UCL
50
 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
45
40
35
Observations
30
LCL
25
Farrokh Alemi, Ph.D.
1st
Qtr
2nd
Qtr
3rd
Qtr
4th
Qtr
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 XmR chart.
Assumptions of XmR chart
Farrokh Alemi, Ph.D.
Index
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.
Index
Moving Range
 An XmR 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
Time period
Observation
1
90
2
85
3
92
4
67
Farrokh Alemi, Ph.D.
5
98
6
83
7
94
Index
8
90
Calculating the Average
Moving Range
Time period
Observation
Moving range
1 2 3 4 5 6 7 8
90 85 92 67 98 83 94 91
5 7 25 31 15 11 3
Mean
87.50
13.86
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
Number of
Number of
time
time
periods E values periods
11
2
2.660
12
3
1.772
13
4
1.457
14
5
1.290
15
6
1.184
16
7
1.109
17
8
1.054
18
9
1.010
19
10
0.975
20
E values
0.945
0.921
0.899
0.881
0.864
0.849
0.836
0.824
0.813
0.803
d2 values
Based on Wheeler DJ. Advanced topics in
statisical process control, 1995 SPC Press
Inc, Knoxville TN 37919
Farrokh Alemi, Ph.D.
Index
Calculating Upper and Lower
Control Limits
Time period 1 2
Observations 90 85
Moving range
5
Upper control limit
Lower control limit
3
4
5
6
7
8
Average
92 67 98 83 94 91
87.50
7 25 31 15 11 3
13.86
= 87.50 + 1.054 * 13.86
= 87.50 - 1.054 * 13.86
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
110
100
Observations
90
80
UCL
LCL
70
60
1
Farrokh Alemi, Ph.D.
2
3
4
5
6
7
8
Time periods
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
110
100
Observations
90
UCL
80
LCL
70
60
1
Farrokh Alemi, Ph.D.
2
3
4
5
6
7
8
Time periods
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
 Example Data
Chart
 Elements of Control
Chart
 Definition of
Statistical Control
 Which Chart is Right?
 Assumptions
 Moving Range
 Calculating Moving
Range
 Calculating Average
Moving Range
 Calculating Control
Limits
 Plot the Control Chart
 Interpret the chart
 Share the Result
Farrokh Alemi, Ph.D.
Index