XmR Chart

Farrokh Alemi, Ph.D.

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

Time period

Observed value

Difference of 2

consecutive values

1 1992 201 23 197 4 44 197 0 4 45 200 3 3 46 195 5 5 57 193 2 7 7

Farrokh Alemi, Ph.D.

Moving Range

Time period

Observed value

Difference of 2

consecutive values

Difference of 3

consecutive values

1 1992 201 23 197 4 44 197 0 4 45 200 3 3 46 195 5 5 57 193 2 7 7

Farrokh Alemi, Ph.D.

Moving Range

Time period

Observed value

Difference of 2

consecutive values

Difference of 3

consecutive values

Difference of 4

consecutive values

1 1992 201 23 197 4 44 197 0 4 45 200 3 3 46 195 5 5 57 193 2 7 7

Farrokh Alemi, Ph.D.

Moving Range

Time period

Observed value

Difference of 2

consecutive values

Difference of 3

consecutive values

Difference of 4

consecutive values

1 1992 201 23 197 4 44 197 0 4 45 200 3 3 46 195 5 5 57 193 2 7 7

Most

common

approach

Which Chart is Right?

Assumptions of XmR chart

Farrokh Alemi, Ph.D.

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

175

180

185

190

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200

205

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

Time

Ob

serv

atio

ns

Pre-intervention

Farrokh Alemi, Ph.D.

Selection of Time Period

175

180

185

190

195

200

205

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

Time

Ob

serv

atio

ns

Pre-intervention

Farrokh Alemi, Ph.D.

Calculate Average Moving Range for 2 Consecutive Values

Average range |Xt - Xt-1| / (n-1)

Absolute value of difference of two consecutive observations

Number of observations

Observation at time “t”

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 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

Typically we look at two consecutive periods

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

175

180

185

190

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200

205

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

Time

Ob

serv

atio

ns

Pre-intervention

Farrokh Alemi, Ph.D.

Time periodWeight in

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

Not relevant to control limits from pre-intervention period

9 196

10 196

11 193

12 190

13 194

14 189

15 185

16 188

Average 197.43 2.67

3. Calculation of Moving Range

Farrokh Alemi, Ph.D.

=abs(B3-B2)

Farrokh Alemi, Ph.D.

=Average(C3:C8)

Farrokh Alemi, Ph.D.=Average(B2:B8)

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

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210

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

Time period

Po

un

ds Observations

UCL

LCL

Farrokh Alemi, Ph.D.

6. Interpret the Chart

Points outside the limits show real changes in outcomes of the process

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210

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

Time period

Pou

nds

Observations

UCL

LCL

175

180

185

190

195

200

205

210

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

Time period

Pou

nds

Observations

UCL

LCL

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

Take Home Lesson

How and When to Construct an XmR chart