1. Variables control charts

2. 𝑥 -chart Purpose: Watch for shifts in the mean
Each point represents average of n observations
Control limits based on distribution of sample mean
Center line = µ
Standard deviation = 𝜎 𝑥 = 𝜎 𝑛
Problem: µ, σ often unknown
Note: Not necessarily sensitive to changes in variability

3. R-chart Purpose: Monitor variability in the process
Each point represents the range of n observations
Control limits based on sampling distribution of the range
We need to explore this
Notes:
Often included with x-bar chart
Range is simpler to calculate than standard deviation

4. Question
What would you expect the sampling distribution of the range to look like?

5. Sampling distribution of the range
Shape is often not symmetric
Control limits will not be symmetric
Center is not at the true value of the range
How to define “true range” for normal? ∞? 6σ?
Variability decreases as sample size increases
So n will play an important role

6. What can happen? Change in mean Increase in the variability
Range will not necessarily change
Increase in the variability
Mean will not necessarily change
Range should increase
* So we need to look at x-bar chart in conjunction with R chart to make sure process is stable

7. Relationship b/w range and std. deviation
Where 𝑅 = 𝑅 1 + 𝑅 2 +…+ 𝑅 𝑚 𝑚 = the average range
Values for d2 are provided in Appendix VI
Why do this?
The range is easier to calculate than the standard deviation (especially in “real time,” as parts of coming off an assembly line), but we need to know σ for the distribution of the mean.

8. Recall: control limits for x-bar charts

9. What if µ and σ are not known?

10. What if µ and σ are not known?

11. What if µ and σ are not known?
So 𝐴 2 = 3 𝑑 2 𝑛 . Values are provided in Appendix VI.

12. Estimated control limits for x-bar charts
Center line = 𝑥 = 𝑥 𝑥 2 +…+ 𝑥 𝑚 𝑚 = historic process mean

13. Estimated control limits for R charts
Center line = 𝑅
- Values for D3 and D4 are provided in Appendix VI

14. Example
A small company supplies keys and locks to the automotive industry. For one of their products a “ward groove” is cut into the key. The depth of this groove is a key characteristic of the key.

15. Example
Data represents 20 subgroups of measures from this process (note: m=20, n=5)
Based on this data
Verify the provided UCL and LCL for both charts

16. Example (x-bar chart)

17. Example (x-bar chart)

18. Example (x-bar chart)

19. Example (R chart)

20. Example (R chart)

21. Example (R chart)

22. s chart Purpose: Monitor variability in the process
Alternative to the X-bar and R chart, preferred if sample size is moderately large (n > 12) or is variable across subgroups
Easy if computer based
Each point represents the sample standard deviation
Control limits based on the sampling distribution of the standard deviation

23. Sampling distribution of standard deviation
Shape is skewed for small sample sizes
Control limits will not be symmetric
Center is not at the true value of σ
s2 is an unbiased estimator of σ2, but s is not an unbiased estimator of σ
s estimates c4σ (c4 values provided in Appendix VI)
Variability decreases as sample size increases
So n will play an important role (affects c4)

24. Estimated control limits for s charts
Center line = 𝑠 = 𝑠 1 + 𝑠 2 +…+ 𝑠 𝑚 𝑚 = average value of s
- Values for B3 and B4 are provided in Table VI

25. Example-Verify UCL and LCL

26. Example (s chart)

27. Example (s chart)

28. Control chart for individuals (X Chart)
Purpose: Plot individual measurements rather than means
Each point represents an individual (i.e. average of one value)
Control limits
Centered around overall process mean
Problem: Don’t have a range or standard deviation
Notes: Often used when it is too difficult or costly to sample several measurements

29. Moving range Take the range of 2 consecutive measures:
Necessary for calculating control limits for X-chart
Can also create a Moving Range Chart to monitor process variance

30. Estimate of standard deviation
Recall: For R charts
Use same idea for this situation
(since n=2)
Where 𝑀𝑅 = average of the moving ranges over all subgroups of 2 consecutive observations.

31. Example
A metal plating process makes use of a nickel plating bath. The concentration of electroless nickel is important in the process. Each day at the start of a shift the concentration (in ounces/gallon) in the bath is analyzed.
Data for 25 measurements is given

32. Example
MR2 = | | = 0.03 MR3 = | | = 0.01 MR4 = | | = 0.20 MR5 = | | = 0.15 … 𝑀𝑅 =0.11

33. Control limits for X-chart
For X-bar chart use 3 standard deviations of the mean ( 𝜎 𝑥 )
For X chart we are working with a single value
Use 3 standard deviations of the population (σ)
Estimate with
3 𝜎 =3 𝑀𝑅 =2.66 𝑀𝑅

34. Control limits for X-chart
Center line = 𝑥 = process average
𝑈𝐶𝐿= 𝑥 𝑀𝑅
𝐿𝐶𝐿= 𝑥 −2.66 𝑀𝑅

35. Control limits for moving range chart
Recall-for R chart:
Since n = 2: D4 = and D3 = 0
Thus:
Center line = 𝑀𝑅
𝑈𝐶𝐿=3.267 𝑀𝑅
𝐿𝐶𝐿=0

36. Example: 𝑀𝑅 =0.11, 𝑥 =4.287 X chart: Moving range chart:
Centerline: 4.287
UCL = (2.66)(0.11) =
LCL = (2.66)(0.11) =
Moving range chart:
Centerline: 0.11
UCL = (3.267)(0.11) =
LCL = 0

37. Example

38. Example

39. Use when Items not in subgroups Items are continuously measured
Items come from the process on an individual basis
Subgrouping would be very artificial
 Items are continuously measured
Automated systems inspect every item
Have data on all items use it in the charting process

40. Use when Time between items is large
Processes that take long periods of time
Product is produced in batches that are relatively homogenous
Chemicals, foods, liquids, plastics
Very little variability within batches

41. Cautions Sensitivity to non-normal
Individuals chart assumes population is normally distributed
Does not average any values
If population is not normal control limits will not be appropriate. 

42. Cautions
Out of control signal will be seen much more often than it should be
Because average run length is less than for the x-bar chart
Average run length = average number of points plotted before process deemed out of control:
where p = chance process is out of control

43. Cautions Not good for detecting small process shifts (less than 1.5σ)
Need other charts (e.g. EWMA, CUSUM) for this

44. Questions for you

45. Exponentially Weighted Moving Average (EWMA)
Basic idea
Take point and average it with previous value
Then average the next point with the previous average
Points contribute to a running average, but amount of contribution decreases over time

46. EWMA

47. EWMA  is a weighting constant Typically between 0.1 and 0.25
Larger values tend to discount older observations faster
Process value or subgroup mean

48. Questions for you How to start zi calculations? Set z1 = x1
Calculate z1 per formula, using target value as z0

49. Control limits for EWMA chart
Based on L and  (more on these later)
Depends on the value of i
More narrow near start of process

50. Control limits for EWMA chart
If =1.0 and L=3 the chart becomes a typical x-bar or individuals control chart

51. Example Calculate the zi values of the nickel concentration data
Use λ = 0.2
Process has a target value of 4.2 ounces/gallon and a historical standard deviation of 0.1 ounces/gallon

52. Example i Concentration 1 4.33 4.226 14 4.18 4.282 2 4.30 4.241 151
4.33
4.226 14
4.18
4.282 2
4.30
4.241 15
4.25
4.276 3
4.31
4.255 16
4.39
4.298 4
4.11 17
4.299 5
4.26
4.233 18
4.42
4.323 6
4.17
4.220 19
4.41
4.340 7
4.44
4.264 20
4.16
4.304 8
4.19
4.249 21
4.29
4.301 9
4.27
4.253 22
4.14
4.269 10
4.32
4.267 23
4.21
4.257 11
4.273 24
4.256 12
4.34
4.287 25
4.293 13
4.307

53. Example Find the UCL and LCL at i=4 if λ=0.2 and L=3
Also find the UCL and LCL at i=20

54. Example
Find the UCL and LCL at i=4 if λ=0.2 and L=3

55. Example
Also find the UCL and LCL at i=20

56. Example

57. Example
Note:
𝑥 is 1σ higher than µ0; x-chart did not pick up this shift but the EWMA chart did.

58. Choice of Parameters
Chosen based on ARL

59. Choice of Parameters Chosen based on ARL
QC Manager specifies desired ARL & anticipated shift in mean

60. Choice of Parameters Chosen based on ARL
QC Manager specifies desired ARL & anticipated shift in mean

61. Choice of Parameters
Note: If the process does not shift the ARL is the same for all parameter choices
Note: If the  is small chart does not react quickly to shifts in the opposite direction.

62. Notes: The points of the EWMA chart are not independent.
Can not apply action rules
EWMA charts are generally more robust to the normality assumption than standard individuals charts
Better to use if data is skewed