1. Determine the central line and control limit
X-bar Chart
A chart that tracks the changes in the means of the samples by plotting the means that were taken from a process. g X j i å = 1
Total number of samples
Sample number
The average of the means of the samples = g i X

2. Determine the central line and control limit
R Chart
A chart that tracks the change in the variability by plotting the range within each sample. The range is the difference between the lowest and highest values in that sample. g R i å = 1
Total number of samples
Difference between the highest and lowest values in sample i
Average of the measurement differences R for all samples = g Ri R

3. Determine the central line and control limit
UCLx = x + 3x
LCLx = x - 3x =
UCLR = x + 3R
LCLR = x - 3R =
UCLx = x + A2R
LCLx = x - A2R =
UCLR = D4R
LCLR = D3R

4. Note: All factors are based on the normal distribution.
Source: E. L. Grant, Statistical Quality Control, 6th ed. pp. 57–59 (Copyright © 1998 by The McGraw-Hill Companies, Inc. 1988). Reprinted by permission of McGraw-Hill, Inc.

5. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
West Allis Industries
The next series of slides presents Example 5.1. The series builds in steps to the conclusion of the Example showing the development of key equations along the way. The first example is of variables charting using x-bar and R-charts.
Continuous process
Produce 400 pieces per day
Quality characteristic is the diameter of bolts
Krajewaksi, operation management 7th, 2004 1

6. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Subgroup Sample
Number R x
Special Metal Screw _
This slide and the next six show in a step-by-step fashion how the raw data is used to derive ranges and means for each observation.
Example 5.1
Krajewaksi, operation management 7th, 2004 4

7. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Sample Sample
Number R x
Special Metal Screw _
– =
Example 5.1
Krajewaksi, operation management 7th, 2004 9

8. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Sample Sample
Number R x
Special Metal Screw _
– =
Example 5.1
Krajewaksi, operation management 7th, 2004 9

9. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Sample Sample
Number R x
Special Metal Screw _
– = (
)/4 =
Example 5.1
Krajewaksi, operation management 7th, 2004 9

10. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Special Metal Screw
Sample Sample
Number R x
R =
x = _
To complete the exercise, the R-bar and grand mean values are calculated and added at the bottom of the appropriate columns. =
Example 5.1
Krajewaksi, operation management 7th, 2004 11

11. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Control Charts – Special Metal Screw
R-Charts
R =
UCLR = D4R
LCLR = D3R
The data set is replaced by a worksheet showing the given values for R-bar and the appropriate equations for the R-chart. The following segment will show x-double bar.
Example 5.1
Krajewaksi, operation management 7th, 2004 13

12. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Table 5.1 Control Chart Factors
Factor for UCL Factor for Factor
Size of and LCL for LCL for UCL for
Sample x-Charts R-Charts R-Charts
(n) (A2) (D3) (D4)
The data set is replaced by a worksheet showing the given values for R-bar and the appropriate equations for the R-chart. The following segment will show x-double bar.
Example 5.1
Krajewaksi, operation management 7th, 2004 13

13. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Control Charts—Special Metal Screw
R-Charts
R = D4 = 2.282
D3 = 0
UCLR = D4R
LCLR = D3R
Removing the Table, the values are now shown on the worksheet.
Example 5.1
Krajewaksi, operation management 7th, 2004 16

14. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Control Charts—Special Metal Screw
R-Charts
R = D4 = 2.282
D3 = 0
UCLR = D4R
LCLR = D3R
In this slide the values are substituted into the UCL equation and the resulting value is shown.
UCLR = (0.0021) = in.
Example 5.1
Krajewaksi, operation management 7th, 2004 17

15. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Control Charts—Special Metal Screw
R-Charts
R = D4 = 2.282
D3 = 0
UCLR = D4R
LCLR = D3R
In a similar fashion, this slide shows the substitution and calculation for the LCL. The arrows will disappear after a few seconds leaving only the calculations.
UCLR = (0.0021) = in.
LCLR = 0 (0.0021) = 0 in.
Example 5.1
Krajewaksi, operation management 7th, 2004 18

16. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Control Charts—Special Metal Screw
R-Charts
R = D4 = 2.282
D3 = 0
UCLR = D4R
LCLR = D3R
This slide can be used to review and discuss the calculations.
UCLR = (0.0021) = in.
LCLR = 0 (0.0021) = 0 in.
Example 5.1
Krajewaksi, operation management 7th, 2004 19

17. Range Chart - Special Metal Screw
This slide shows the blank control chart for the R-chart with the mean and control limit values already plotted. This is a screen shot from the OM Explorer software.
The process Range is comfortably in control. (The picky person might suggest there is not enough variability in the process, but with so few observations it is difficult to draw such conclusions.)
Krajewaksi, operation management 7th, 2004 20

18. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Control Charts—Special Metal Screw
X-Charts
R =
x = =
UCLx = x + A2R
LCLx = x - A2R =
We return to the calculations, this time for the x-bar control limits.
Example 5.1
Krajewaksi, operation management 7th, 2004 22

19. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Table 5.1 Control Chart Factors
Factor for UCL Factor for Factor
Size of and LCL for LCL for UCL for
Sample x-Charts R-Charts R-Charts
(n) (A2) (D3) (D4)
Control Charts—Special Metal Screw
X-Charts
R =
x = =
UCLx = x + A2R
LCLx = x - A2R =
We return to the calculations, this time for the x-bar control limits.
Example 5.1
Krajewaksi, operation management 7th, 2004 22

20. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Control Charts—Special Metal Screw
x-Charts
R = A2 = 0.729
x = =
UCLx = x + A2R
LCLx = x - A2R =
The A2 value is now shown with the other required values for the calculations.
Example 5.1
Krajewaksi, operation management 7th, 2004 24

21. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Control Charts—Special Metal Screw
x-Charts
R = A2 = 0.729
x = =
UCLx = x + A2R
LCLx = x - A2R =
Here the required values are substituted in the UCL equation and the value is calculated. The arrows on this slide will disappear after a few seconds and the second calculation will appear.
UCLx = (0.0021) = in.
Example 5.1
Krajewaksi, operation management 7th, 2004 25

22. Krajewaksi, operation management 7th, 2004
Control Charts
for Variables
Control Charts—Special Metal Screw
x-Charts
R = A2 = 0.729
x = =
UCLx = x + A2R
LCLx = x - A2R =
This slide completes the substitution and calculation for the LCL.
UCLx = (0.0021) = in.
LCLx = – (0.0021) = in.
Example 5.1
Krajewaksi, operation management 7th, 2004 26

23. Control chart (Ex 5-2)