2. Individual values VS Averages
Comparison of individual values compared to averages

3. Individual values VS Averages
Calculations of the average for both the individual values and for the subgroup averages are the same. However the sample standard deviation is different.
Besterfield Quality Control 8th Ed

4. Besterfield Quality Control 8th Ed
Central Limit Theorem
If the population from which samples are taken is not normal, the distribution of sample averages will tend toward normality provided that the sample size, n, is at least 4. This tendency gets better and better as the sample size gets larger.
Besterfield Quality Control 8th Ed

5. Illustration of central limit theorem

6. Central Limit Theorem Dice illustration of central limit theorem
Besterfield Quality Control 8th Ed

7. Control Limits & Specifications
Figure 5-21 Relationship of limits, specifications, and distributions
Besterfield Quality Control 8th Ed

8. Control Limits & Specifications
The control limits are established as a function of the average (ค่า control limits เป็นค่าที่คำนวนจากค่าเฉลี่ยที่ได้จากการเก็บข้อมูล)
Specifications are the permissible variation in the size of the part and are, therefore, for individual values (ค่า Specifications เป็นค่าที่กำหนดขึ้นเพื่อใช้เป็นขอบเขตของการกระจายสำหรับค่าของข้อมูลแต่ละตัว)
The specifications or tolerance limits are established by design engineers or customers to meet a particular function
Besterfield Quality Control 8th Ed

9. Process Capability & Tolerance
The process spread will be referred to as the process capability and is equal to 6σ
The difference between specifications is called the tolerance
When the tolerance is established by the design engineer without regard to the spread of the process, undesirable situations can result
Besterfield Quality Control 8th Ed

10. Process Capability & Tolerance
Three situations are possible:
Case I: When the process capability is less than the tolerance 6σ<USL-LSL
Case II: When the process capability is equal to the tolerance 6σ=USL-LSL
Case III: When the process capability is greater than the tolerance 6σ >USL-LSL
Besterfield Quality Control 8th Ed

11. Process Capability & Tolerance
Case I: When the process capability is less than the tolerance 6σ<USL-LSL
Case I 6σ<USL-LSL
Besterfield Quality Control 8th Ed

12. Process Capability & Tolerance
Case II: When the process capability is less than the tolerance 6σ=USL-LSL
Case I 6σ=USL-LSL
Besterfield Quality Control 8th Ed

13. Process Capability & Tolerance
Case III: When the process capability is less than the tolerance 6σ>USL-LSL
Case I 6σ>USL-LSL
Besterfield Quality Control 8th Ed

14. Capability Index
Process capability and specifications or tolerance are combined to form the capability index, Cp.

15. Besterfield Quality Control 8th Ed
Capability Index
The capability index does not measure process performance in terms of the nominal or target value. This measure is accomplished by Cpk.
Besterfield Quality Control 8th Ed

16. Besterfield Quality Control 8th Ed
Capability Index
The Cp value does not change as the process center changes
Cp = Cpk when the process is centered
Cpk is always equal to or less than Cp
A Cpk = 1 (and Cpk = 1.33) is a de facto standard. It indicates that the process is producing product that conforms to specifications
A Cpk < 1 indicates that the process is producing product that does not conform to specifications
Besterfield Quality Control 8th Ed

17. Besterfield Quality Control 8th Ed
Capability Index
A Cp < 1 indicates that the process is not capable
A Cpk = 0 indicates the average is equal to one of the specification limits
A negative Cpk value indicates that the average is outside the specifications
Besterfield Quality Control 8th Ed

18. Cpk Measures Cpk = negative number Cpk = zero Cpk = between 0 and 1
Cpk = 1 (and Cp = 1)
Cpk > 1

19. 1-How to estimate Process Capability
This following method of calculating the process capability assumes that the process is stable or in statistical control:
Take 25 (g) subgroups of size 4 for a total of 100 measurements
Calculate the range, R, for each subgroup
Calculate the average range, = ΣR/g
Calculate the estimate of the population standard
deviation using:
Process capability will equal 6σ0

20. 2-How to estimate Process Capability
The process capability can also be obtained by using the standard deviation:
Take 25 (g) subgroups of size 4 for a total of 100 measurements
Calculate the sample standard deviation, s, for each subgroup
Calculate the average sample standard deviation, = Σs/g
Calculate the estimate of the population standard
deviation
Process capability will equal 6σo

21. Capability Index (EX. 5-4)
A new process is started, and the sum of the sample standard deviations for 25 groups of size 4 is 105. Approximate the process capability.
Determine the capability index before (σo = 0.038) and after (σo = 0.03) improvement using specification of 6.40 ± 0.15.
What is the Cpk value after improvement for Question 1 when the process center is 6.40? When the process center is 6.30?
A new process is started, and the sum of the sample standard deviations for 25 subgroups of size 4 is 750. If the specifications are 700 ± 80, what is the process capability index? What action would you recommend?

22. Additional control charts
Standard deviation chart (or s chart)
This chart is nearly the same X-bar and R chart. However, for subgroup sizes ≥ 10, an s chart is more accurate than an R Chart
Moving average and Moving range chart
This chart is used to combine a number of individual values and plot them on the chart. This technique is quite common in the chemical industry, where only one reading (datum) is possible at a time ( Can chemical engineering students give an example?)
Exponential Weighted Moving-Average (EWMA) chart
The EWMA chart gives the greatest weight to the most recent data and less weight to all previous data. It primary advantage is the ability to detect small shifts in the process average; however, it does not react as quickly to large shifts as the X-bar chart.

23. S Control Chart
For subgroup sizes ≥10, an s chart is more accurate than an R Chart. Trial control limits are given by:

24. Revised Limits for S control chart

25. S chart (Ex. 5-5) Subgroup No. Measurement Average S.D. Date Time X1
X-bar S
Comment 1
12/26
8:50
6.35
6.40
6.32
6.37
0.034 2
11:30
6.46
6.36
6.41
0.045 3
1:45
6.34
0.028 4
3:45
6.69
6.64
6.68
6.59
New, temporary operator 5
4:20
6.38
6.44
0.042 6
12/27
8:35
6.42
6.43
0.041 7
9:00
0.024 8
9:40
6.33 9
1:30
6.48
6.47
6.45
0.018 10
2:50 11
12/28
8:30
6.39
0.014 12
1:35
0.020 13
2:25
0.051 14
2:35
0.032 15
3:35
6.50
0.036

26. S chart (Ex. 5-5) Subgroup No. Measurement Average Range Date Time X1
X-bar S
Comment 16
12/29
8:25
6.33
6.35
6.29
6.39
6.34
0.042 17
9:25
6.41
6.4
6.36
0.056
Damaged oil line 18
11:00
6.38
6.44
6.28
6.58
6.42
0.125 19
2:35
6.37
0.025 20
3:15
6.56
6.55
6.45
6.48
6.51
0.054
Bad material 21
12/30
9:35
6.40
0.036 22
10:20
0.029 23
11:35
0.024 24
2:00
6.43 25
4:25

27. Moving average and Moving range chart
This chart is used to combine a number of individual values and plot them on the chart. This technique is quite common in the chemical industry, where only one reading (datum) is possible at a time
Value
Three-period moving sum
X-bar R 35 - 26 28
=89
( )/3 =29.6
35-26 = 9 32 86
28.6 6 36 96 8 .
Sx-bar =
SR =

28. Exponential Weighted Moving-Average (EWMA) chart

29. Exponential Weighted Moving-Average (EWMA) chart

30. Exponential Weighted Moving-Average (EWMA) chart (EX. 5-6)

31. Exponential Weighted Moving-Average (EWMA) chart

32. Quiz 5-1

33. Quiz 5-2
Control charts for X-bar and R are to be established on a certain dimension part, measured in millimeters. Data were collected in subgroup sizes of 6 and are given below. Determine the trial central and control limits. Assume assignable causes and revise the central line and limits.
Subgroup Number
X-bar R 1
20.35
0.34 14
20.41
0.36 2
20.40 15
20.45 3
20.36
0.32 16
20.34 4
20.65 17
0.37 5
20.20 18
20.42
0.73 6
0.35 19
20.50
0.38 7
20.43
0.31 20
20.31 8
20.37 21
20.39 9
20.48
0.30 22
0.33 10 23 11
0.29 24 12
20.38 25 13
20.4

34. Quiz 5-3
Control charts for X-bar and s are to be established on the Brinell hardness of hardened tool steel in kilograms per square millimeter. Data for subgroup sizes of 8 are shown below. Determine the trail central lime and control limits for the X-bar and s charts. Assume that the out-of-control points have assignable causes. Calculate the revised limits and central line.
Subgroup Number
X-bar
S.D. 1
540 26 14
551 24 2
534 23 15
522 29 3
545 16
579 4
561 27 17
549 28 5
576 25 18
508 6
523 50 19
569 22 7
571 20
574 8
547 21
563 33 9
584 10
552
548 11
541
556 12
553 13
546

35. Quiz 5-4 Use data in Quiz 5-2 to establish EWMA chart, using  = 0.1
Subgroup Number
X-bar R 1
20.35
0.34 14
20.41
0.36 2
20.40 15
20.45 3
20.36
0.32 16
20.34 4
20.65 17
0.37 5
20.20 18
20.42
0.73 6
0.35 19
20.50
0.38 7
20.43
0.31 20
20.31 8
20.37 21
20.39 9
20.48
0.30 22
0.33 10 23 11
0.29 24 12
20.38 25 13
20.4