MGT 6421 Quality Management II
Control Charts for Variables
( Our objectives for this section are to learn how to use control charts to
monitor continuous data. We want to learn the assumptions behind
the charts, their application, and their interpretation.
( Since statistical control for continuous data depends on both the mean
and the variability, variables control charts are constructed to monitor
each. The most commonly used chart to monitor the mean is called
the chart. There are two commonly used charts used to monitor the X
variability: the R chart and the s chart.
( Procedure for using variables control charts:
1. Determine the variable to monitor.
2. At predetermined, even intervals, take samples of size n (usually
n=4 or 5).
3. Compute each sample, and plot them on their X and R (or s) for
respective control charts. Use the following relationships:
nn2()XX？X；；ii
i1i？1？X？s？ , R = X- X, . maxminnn？1
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MGT 6421 Quality Management II
4. After collecting a sufficient number of samples, k (k>20), compute the control limits for the charts (see the table on page 4 for the appropriate control limit calculations). The following additional calculations will be necessary:
kkk
XRs；；；jjj
jjj？1？1？1X？R？s？, , .
kkk
5. If any points fall outside of the control limits, conclude that the process is out of control, and begin a search for an assignable or special cause. When the special cause is identified, remove that point and return to step 4 to re-evaluate the remaining points.
6. If all the points are within limits, conclude that the process is in control, and use the calculated limits for future monitoring of the process.
( Because the limits of the chart are based on the variability of the X
process, we will first discuss the variability charts. I suggest that you first determine if the R chart (or s chart) shows a lack of control. If so, you cannot draw conclusions from the chart. X
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MGT 6421 Quality Management II
The R chart
( The R chart is used to monitor process variability when sample sizes
are small (n<10), or to simplify the calculations made by process
operators.
( This chart is called the R chart because the statistic being plotted is
the sample range.
;( Using the R chart, the estimate of the process standard deviation, , is ！
R.
d2
The s chart
( The s chart is used to monitor process variability when sample sizes
are large (n，10), or when a computer is available to automate the
calculations.
( This chart is called the s chart because the statistic being plotted is the
sample standard deviation.
;( Using the s chart, the estimate of the process standard deviation, , is ！
s.
c4
The Chart: X
( This chart is called the chart because the statistic being plotted is X
the sample mean. The reason for taking a sample is because we are
not always sure of the process distribution. By using the sample mean
we can "invoke" the central limit theorem to assume normality.
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MGT 6421 Quality Management II
Limits for Variables Control Charts Variability Standards Measure Chart Limits (，;and;！)
;Range Known ，;? A！ X
Range Not Known ? A ;RX X2
;Standard Known ，;? A！ X
Deviation
Standard Not Known ;? A sX X3Deviation
centerline=d！ 2Range Known R LCL=D！;;;1
UCL=D！ 2
centerline= RRange Not Known R LCL=D;;;R3
UCL=D R4
Standard centerline=c！ 4Deviation Known s LCL=B！;;;5
UCL=B！ 6
Standard centerline=s Deviation Not Known s LCL=Bs;;;3
UCL=Bs 4
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MGT 6421 Quality Management II
1( Example of Variables Control Chart
A large hotel in a resort area has a housekeeping staff that cleans and
prepares all of the hotel's guestrooms daily. In an effort to improve
service through reducing variation in the time required to clean and
prepare a room, a series of measurements is taken of the times to
service rooms in one section of the hotel. Cleaning times for five
rooms selected each day for 25 consecutive days appear below:
Day Room 1 Room 2 Room 3 Room 4 Room 5 Average Range St. Dev
1 15.6 14.3 17.7 14.3 15.0 15.4 3.4 1.41
2 15.0 14.8 16.8 16.9 17.4 16.2 2.6 1.19
3 16.4 15.1 15.7 17.3 16.6 16.2 2.2 0.85
4 14.2 14.8 17.3 15.0 16.4 15.5 3.1 1.27
5 16.4 16.3 17.6 17.9 14.9 16.6 3.0 1.19
6 14.9 17.2 17.2 15.3 14.1 15.7 3.1 1.40
7 17.9 17.9 14.7 17.0 14.5 16.4 3.4 1.69
8 14.0 17.7 16.9 14.0 14.9 15.5 3.7 1.71
9 17.6 16.5 15.3 14.5 15.1 15.8 3.1 1.24
10 14.6 14.0 14.7 16.9 14.2 14.9 2.9 1.16
11 14.6 15.5 15.9 14.8 14.2 15.0 1.7 0.69
12 15.3 15.3 15.9 15.0 17.8 15.9 2.8 1.13
13 17.4 14.9 17.7 16.6 14.7 16.3 3.0 1.39
14 15.3 16.9 17.9 17.2 17.5 17.0 2.6 1.00
15 14.8 15.1 16.6 16.3 14.5 15.5 2.1 0.93
16 16.1 14.6 17.5 16.9 17.7 16.6 3.1 1.26
17 14.2 14.7 15.3 15.7 14.3 14.8 1.5 0.65
18 14.6 17.2 16.0 16.7 16.3 16.2 2.6 0.98
19 15.9 16.5 16.1 15.0 17.8 16.3 2.8 1.02
20 16.2 14.8 14.8 15.0 15.3 15.2 1.4 0.58
21 16.3 15.3 14.0 17.4 14.5 15.5 3.4 1.37
22 15.0 17.6 14.5 17.5 17.8 16.5 3.3 1.59
23 16.4 15.9 16.7 15.7 16.9 16.3 1.2 0.51
24 16.6 15.1 14.1 17.4 17.8 16.2 3.7 1.56
25 17.0 17.5 17.4 16.2 17.9 17.2 1.7 0.64
sRX
15.94 2.70 1.14
1 This example is taken from Gitlow, H., Gitlow, S., Oppenheim, A., and Oppenheim, R.
(1989). Tools and Methods for the Improvement of Quality, Homewood, IL: Richard D.
Irwin, Inc.
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MGT 6421 Quality Management II
( For the R chart:
X( For the chart (with R)
_____________________________________________________________
( For the s chart
X( For the chart (with s)
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6
MGT 6421 Quality Management II
5
R chart 4
3
2
1
0
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2s chart 1.5
1
0.5
0
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18MGT 6421 Quality Management II
Chart X17
16
15
14
12345678910111213141516171819202122232425
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MGT 6421 Quality Management II
Individuals and Moving Range Charts
( Sometimes it is impossible or extremely costly to take samples.
Individuals and moving range charts can be constructed under these
circumstances.
( We cannot obtain a range or standard deviation with individual
measurements.
( The solution is to take moving ranges. The moving range is the
absolute difference between consecutive individual measurements.
The procedure then becomes identical to traditional variables control
charts, where n is assumed to be 2. The chart for central tendency is
called an individuals chart, and the chart for variability is called a
moving range chart.
( Limits:
MR chart: centerline=, LCL=0, UCL=3.267 MRMR
Individuals chart: centerline= X
MRX？3LCL=
1.128
MRX；3UCL=
1.128
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MGT 6421 Quality Management II
( Because only individual measurements are being taken, the
assumption of normality is more difficult to defend. For this reason,
the control limits are often based on 2 standard deviations rather than
3.
( We need to use some caution now because successive values on the
moving range chart are not independent.
( Example: I want to monitor my gas mileage to determine if there is
reason to believe the car needs to be serviced. Over the last 6 weeks I
have measured the following gas mileage values: 25.0, 24.6, 24.9,
23.1, 26.0, 24.0. (NOTE: We clearly need more points than these 6
to construct the appropriate control chart(s). I have included 6 points
just to reduce the computational burden.) Find the control limits for
the appropriate control chart(s).
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