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 X chart. There are two commonly used charts used to monitor the
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 X and R (or s) for each sample, and plot them on their
respective control charts. Use the following relationships:
n
n
 ( Xi  X ) 2
 Xi
X
i 1
n
,
R = Xmax - Xmin,
Variables Control Charts - 1
s
i 1
n 1
.
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:
k
 Xj
X
j 1
k
k
k
Rj
,
R
j 1
k
 sj
,
s
j1
k
.
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 X chart are based on the variability of the
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 X chart.
Variables Control Charts - 2
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 X Chart:

This chart is called the X chart because the statistic being plotted is
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.
Variables Control Charts - 3
MGT 6421
Quality Management II
Limits for Variables Control Charts
Variability
Measure
Standards
(and)
Chart
Range
Known
X
Limits

± A
Range
Not Known
X
 X ± A2 R
Standard
Deviation
Known
X

± A
Standard
Deviation
Not Known
X
X ± A3 s
R
centerline=d2
LCL=D1
UCL=D2
R
centerline= R
LCL=D3 R 
UCL=D4 R
s
centerline=c4
LCL=B5
UCL=B6
s
centerline= s
LCL=B3 s 
UCL=B4 s
Range
Range
Standard
Deviation
Standard
Deviation
Known
Not Known
Known
Not Known
Variables Control Charts - 4
MGT 6421

Quality Management II
Example1 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
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Room 1
15.6
15.0
16.4
14.2
16.4
14.9
17.9
14.0
17.6
14.6
14.6
15.3
17.4
15.3
14.8
16.1
14.2
14.6
15.9
16.2
16.3
15.0
16.4
16.6
17.0
Room 2
14.3
14.8
15.1
14.8
16.3
17.2
17.9
17.7
16.5
14.0
15.5
15.3
14.9
16.9
15.1
14.6
14.7
17.2
16.5
14.8
15.3
17.6
15.9
15.1
17.5
Room 3
17.7
16.8
15.7
17.3
17.6
17.2
14.7
16.9
15.3
14.7
15.9
15.9
17.7
17.9
16.6
17.5
15.3
16.0
16.1
14.8
14.0
14.5
16.7
14.1
17.4
Room 4
14.3
16.9
17.3
15.0
17.9
15.3
17.0
14.0
14.5
16.9
14.8
15.0
16.6
17.2
16.3
16.9
15.7
16.7
15.0
15.0
17.4
17.5
15.7
17.4
16.2
1
Room 5
15.0
17.4
16.6
16.4
14.9
14.1
14.5
14.9
15.1
14.2
14.2
17.8
14.7
17.5
14.5
17.7
14.3
16.3
17.8
15.3
14.5
17.8
16.9
17.8
17.9
Average
15.4
16.2
16.2
15.5
16.6
15.7
16.4
15.5
15.8
14.9
15.0
15.9
16.3
17.0
15.5
16.6
14.8
16.2
16.3
15.2
15.5
16.5
16.3
16.2
17.2
Range
3.4
2.6
2.2
3.1
3.0
3.1
3.4
3.7
3.1
2.9
1.7
2.8
3.0
2.6
2.1
3.1
1.5
2.6
2.8
1.4
3.4
3.3
1.2
3.7
1.7
X
R
s
15.94
2.70
1.14
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.
Variables Control Charts - 5
St. Dev
1.41
1.19
0.85
1.27
1.19
1.40
1.69
1.71
1.24
1.16
0.69
1.13
1.39
1.00
0.93
1.26
0.65
0.98
1.02
0.58
1.37
1.59
0.51
1.56
0.64
MGT 6421
Quality Management II

For the R chart:

For the X chart (with R)
_____________________________________________________________

For the s chart

For the X chart (with s)
Variables Control Charts - 6
MGT 6421
Quality Management II
R chart
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
s chart
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Variables Control Charts - 7
MGT 6421
Quality Management II
X Chart
18
17
16
15
14
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Variables Control Charts - 8
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:
Individuals chart:
centerline= M R , LCL=0, UCL=3.267 M R
centerline= X
LCL= X  3
MR
1.128
UCL= X  3
MR
1.128
Variables Control Charts - 9
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).
Variables Control Charts - 10