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# Control Charts for Variables

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Control Charts for Variables

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()XXXii

i1i1Xs , R = X- X, . maxminnn1

Variables Control Charts - 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:

kkk

XRsjjj

jjj111XRs, , .

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

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 (n10), 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.

Variables Control Charts - 3

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

Variables Control Charts - 4

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)

Variables Control Charts - 6

6

MGT 6421 Quality Management II

5

R chart 4

3

2

1

0

123456789101112131415161718192021222324252.5

2s chart 1.5

1

0.5

0

12345678910111213141516171819202122232425

Variables Control Charts - 7

18MGT 6421 Quality Management II

Chart X17

16

15

14

12345678910111213141516171819202122232425

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: centerline=, LCL=0, UCL=3.267 MRMR

Individuals chart: centerline= X

MRX3LCL=

1.128

MRX3UCL=

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).

Variables Control Charts - 10

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