# XmR Chart

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```					XmR Chart

Farrokh Alemi, Ph.D.
Purpose of Control Chart
 Real or random.
 Tell a story of changes in outcomes of the
process.

Farrokh Alemi, Ph.D.
Moving Range

Difference of Difference of Difference of
2             3              4
Time    Observed consecutive consecutive consecutive
period    value      values        values         values
1       199
2       201          2
3       197          4             4
4       197          0             4              4
5       200          3             3              4
6       195          5             5              5
7       193          2             7              7

Farrokh Alemi, Ph.D.
Moving Range

Difference of Difference of Difference of
2             3              4
Time    Observed consecutive consecutive consecutive
period    value      values        values         values
1       199
2       201          2
3       197          4             4
4       197          0             4              4
5       200          3             3              4
6       195          5             5              5
7       193          2             7              7

Farrokh Alemi, Ph.D.
Moving Range

Difference of Difference of Difference of
2             3              4
Time    Observed consecutive consecutive consecutive
period    value      values        values         values
1       199
2       201          2
3       197          4             4
4       197          0             4              4
5       200          3             3              4
6       195          5             5              5
7       193          2             7              7

Farrokh Alemi, Ph.D.
Moving Range

Difference of Difference of Difference of
2             3              4
Time    Observed consecutive consecutive consecutive
period    value      values        values         values
1       199
2       201          2
3       197          4             4
4       197          0             4              4
5       200          3             3              4
6       195          5             5              5
7       193          2             7              7

Farrokh Alemi, Ph.D.
Which Chart is Right?

Assumptions of
XmR chart
Assumptions
1. There is one observation per time period
2. Observations are measured in an
“interval” scale
3. Observations are independent of each
other

Farrokh Alemi, Ph.D.
Selection of Time Period
205

200
Observations

195

190       Pre-intervention
185

180

175
1    2   3   4   5      6    7    8    9    10 11 12 13 14 15 16
Time

Farrokh Alemi, Ph.D.
Selection of Time Period
205

200
Observations

195

190       Pre-intervention
185

180

175
1    2   3   4   5      6    7    8    9    10 11 12 13 14 15 16
Time

Farrokh Alemi, Ph.D.
Calculate Average Moving Range
for 2 Consecutive Values

Observation at                   Number of
time “t”                         observations

Average range =  |Xt - Xt-1| / (n-1)

Absolute value of
difference of two
consecutive observations

Farrokh Alemi, Ph.D.
Formula for Calculating Upper
and Lower Control Limits
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.
E Depends on Number of Time
Periods in the Range
Number of         Number of
time              time
periods E values  periods          E values   d2 values
11              0.945
Typically         2     2.660       12             0.921
3     1.772       13             0.899
we look at        4     1.457       14             0.881
two               5     1.290       15             0.864
consecutive       6     1.184       16             0.849
7     1.109       17             0.836
periods           8     1.054       18             0.824
9     1.010       19             0.813
10    0.975       20             0.803
Based on Wheeler DJ. Advanced topics in
statisical process control, 1995 SPC Press
Inc, Knoxville TN 37919

Farrokh Alemi, Ph.D.
Example
 Diabetes patient
measured weight for
16 weeks
 First 7 weeks were
pre-intervention
 Has the patient’s
weight changed?

Farrokh Alemi, Ph.D.
1. Check Assumptions
One observation per time period
Interval scale
Independent observations

Farrokh Alemi, Ph.D.
2. Select Pre- Or Post-Intervention
Period
205

200
Observations

195

190       Pre-intervention
185

180

175
1    2   3   4   5      6    7    8    9    10 11 12 13 14 15 16
Time

Farrokh Alemi, Ph.D.
3. Calculation of Moving Range
Weight in
Time period      Pounds         Range of 2 consecutive values
1             199                      Not available
2             201                            2
3             197                            4
4             197                            0
5             200                            3
6             195                            5
7             193                            2
8             198
9             196
10            196
11            193
Not relevant to control limits from pre-
12            190
intervention period
13            194
14            189
15            185
16            188
Average         197.43
Farrokh Alemi, Ph.D.             2.67
=abs(B3-B2)

Farrokh Alemi, Ph.D.
Farrokh Alemi, Ph.D.
Farrokh Alemi, Ph.D.
4. Calculation of Control Limits
 UCL = 197.43 + 2.66 * 2.67 = 204.52
 LCL = 197.43 - 2.66 * 2.67 = 190.33

Farrokh Alemi, Ph.D.
5. Plot the Control Chart
 Plot the x and y axis, plot the observations
 Plot the limits as line with no marker
– Plot solid line for pre-intervention period
210
205
200
Observations
Pounds

195
UCL
190
LCL
185
180

175
1   2   3 4   5   6    7   8 9 10 11 12 13 14 15 16

Time period

Farrokh Alemi, Ph.D.
6. Interpret the Chart
 Points outside the limits show real changes in
outcomes of the process
210

205
200
Observations
195
Pounds

UCL
190
LCL
185
180

175
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Time period

Farrokh Alemi, Ph.D.
7. Distribute the Chart
 Distribute the chart by electronic media, as
part of company newsletter, or as an
element of a story board display
 Keep following in mind:
– Show that you have verified assumptions
– Check that your chart is accurately labeled
– Include your interpretation of the finding

Farrokh Alemi, Ph.D.
Take Home Lesson

How and When to Construct an XmR
chart

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