control charts for variables 2 -...

Control Charts for Variables 2

Montgomery, D.C. (2013) Statistical Quality Control: A Modern Introduction. 7th ed., Wiley.

Control Charts for 𝑥 and 𝐬

Generally, 𝑥 and 𝑠 charts are preferable to their more

familiar counterparts, 𝑥 and 𝑅 charts, when either

1. the sample size 𝑛 is moderately large – say, 𝑛 > 10 or 12,

or

2. the sample size 𝑛 is variable.


Case 1: A standard value is given for 𝜎

Parameters of the 𝑠 chart:

Parameters of the 𝑥 chart:


Case 2: No standard is given for 𝜎

Parameters of the 𝑠 chart:

Parameters of the 𝑥 chart:

𝑠 =1

𝑚 𝑠𝑖

𝑚

𝑖=1

𝑠𝑖: standard deviation of the 𝑖th sample

Example: Samples of 𝑛 = 4 items are taken from a process at regular intervals. A normally distributed quality characteristic is measured and 𝑥 and 𝑠 values are calculated for each sample. After 50 subgroups have been analyzed, we have

𝑥 𝑖

50

𝑖=1

= 1000 and 𝑠𝑖

50

𝑖=1

= 72

a) Compute the control limits for 𝑥 and 𝑠 control charts.

b) Assume that all points on both charts plot within the control limits. What are the natural tolerance limits of the process?

c) The specifications are 19 ± 4. Compute the process capability ratio 𝐶𝑝.

d) Assuming that if an item exceeds the upper specification limit it can be reworked, and if it is below the lower specification limit it must be scrapped, what percentage scrap and rework is the process now producing?

The 𝑥 and 𝐬 Control Charts with Variable Sample Size

𝑥 = 𝑛𝑖𝑥 𝑖𝑚𝑖=1

𝑛𝑖𝑛𝑖=1

, 𝑠 = 𝑛𝑖−1 𝑠𝑖

2 𝑚𝑖=1

𝑛𝑖−𝑚𝑚𝑖=1

1 2

𝐴3, 𝐵3 and 𝐵4 will depend on the sample size used in each individual subgroup.


• An alternative to using variable-width control limits on the x

and s control charts is to base the control limit calculations

on an average sample size 𝑛 .

• If the 𝑛𝑖 are not very different, this approach may be

satisfactory.

• Since the average sample size 𝑛 may not be an integer, a

useful alternative is to base these approximate control limits

on a modal (most common) sample size.

The Shewhart Control Chart for Individual Measurements

There are many situations in which the sample size used for process monitoring is 𝑛 = 1 (the sample consists of an individual unit). Some examples of these situations are as follows:

• Automated inspection and measurement technology is used, and every unit manufactured is analyzed so there is no basis for rational subgrouping.

• Data comes available relatively slowly, and it is inconvenient to allow sample sizes of 𝑛 > 1 to accumulate before analysis. The long interval between observations will cause problems with rational subgrouping. This occurs frequently in both manufacturing and nonmanufacturing situations.

• In process plants, such as papermaking, measurements on some parameter such as coating thickness across the roll will differ very little and produce a standard deviation that is much too small if the objective is to control coating thickness along the roll.

• Individual measurements are very common in many transactional, business, and service processes because there is no basis for rational subgrouping. Sometimes this happens because there are large time gaps between service activities.


• In many applications of the individuals control chart, we use the moving range of two successive observations as the basis of estimating the process variability.

• The moving range is defined as

𝑀𝑅𝑖 = 𝑥𝑖 − 𝑥𝑖−1

• It is also possible to establish a moving range control chart.


• Parameters of the moving range control chart:

UCL = 𝐷4𝑀𝑅

Center line = 𝑀𝑅

LCL = 𝐷3𝑀𝑅

• Parameters of the control chart for individual measurements:

UCL = 𝑥 + 3𝑀𝑅

𝑑2

Center line = 𝑥

LCL = 𝑥 − 3𝑀𝑅

𝑑2


Example: The mortgage loan

processing unit of a bank monitors

the costs of processing loan

applications. The quantity tracked

is the average weekly processing

costs, obtained by dividing total

weekly costs by the number of

loans processed during the week.

The processing costs for the most

recent 20 weeks are shown in the

table. Set up individual and

moving range control charts for

these data.


• Moving range control chart:

𝐷3 = 0 and 𝐷4 = 3.267 for 𝑛 = 2.

UCL = 𝐷4𝑀𝑅 = 3.267 7.79 = 25.45

Center line = 𝑀𝑅 = 7.79

LCL = 𝐷3𝑀𝑅 = 0

No points are out of control.

• Control chart for individual measurements:

𝑑2 = 1.128 for 𝑛 = 2.

UCL = 𝑥 + 3𝑀𝑅

𝑑2= 300.5 + 3

7.79

1.128= 321.22

Center line = 𝑥 = 300.5

LCL = 𝑥 − 3𝑀𝑅

𝑑2= 300.5 − 3

7.79

1.128= 279.78

There are no out of control observations on the individuals control chart.


Data on mortgage application processing costs for weeks 21–40:

Phase II Operation and Interpretation of the Charts


Phase II Operation and Interpretation of the Charts


An upward shift in cost has occurred around week 39, since there is an obvious “shift in process level” pattern on the chart for individuals followed by another out-of-control signal at week 40.

The moving range chart also reacts to this level shift with a single large spike at week 39. This spike on the moving range chart is sometimes helpful in identifying exactly where a process shift in the mean has occurred.

Clearly one should look for possible assignable causes around weeks 39.

Possible causes could include an unusual number of applications requiring additional manual underwriting work, or possibly new underwriters working in the process, or possibly temporary underwriters replacing regular employees taking vacations.

Phase II Operation and Interpretation of the Charts:

control charts for variables 2 -...

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