1. STATISTICAL PROCESS CONTROL AND QUALITY MANAGEMENT

2. Quality Types of Quality Fitness for use, acceptable standard
Based on needs, expectations and customer requests
Types of Quality
Quality of design
Quality of conformance
Quality of performance

3. Quality of Conformance
Quality of Design
Differences in quality due to design differences, intentional differences
Quality of Conformance
Degree to which product meets or exceeds standards
Quality of Performance
Long term consistent functioning of the product, reliability, safety, serviceability, maintainability

4. Quality and Productivity
Improved quality leads to lower costs and increased profits

5. Statistics and Quality Management
Statistical analysis is used to assist with product design, monitor the production process, and check quality of the finished product

6. Checking Finished Product Quality
Random samples selected from batches of finished product can be used to check for product quality

7. Assisting With Production Design
A variety of experimental design techniques are available for improving the production process

8. Monitoring the Process
Control charts is used to monitor the process as it unfolds
Sampling from the production line to see if variation in product quality is consistent with expectations

9. The Control Chart
A special type of sequence plot which is used to monitor a process
Throughout the process measurements are taken and plotted in a sequence plot
Plot also contains upper and lower control limits indicating the expected range of the process when it is behaving properly

10. Examples

11. Variation There is no two natural items in any category are the same.
Variation may be quite large or very small.
If variation very small, it may appear that items are identical, but precision instruments will show differences.

12. 3 Categories of variation
Within-piece variation
One portion of surface is rougher than another portion.
Apiece-to-piece variation
Variation among pieces produced at the same time.
Time-to-time variation
Service given early would be different from that given later in the day.

13. Source of variation Equipment Material Environment Operator
Tool wear, machine vibration, …
Material
Raw material quality
Environment
Temperature, pressure, humadity
Operator
Operator performs- physical & emotional

14. Control Chart Viewpoint
Variation due to
Common or chance causes
Assignable causes
Control chart may be used to discover “assignable causes”

15. Control chart functions
Control charts are powerful aids to understanding the performance of a process over time.
PROCESS
Output
Input
What’s causing variability?

16. Control charts identify variation
Chance causes - “common cause”
inherent to the process or random and not controllable
if only common cause present, the process is considered stable or “in control”
Assignable causes - “special cause”
variation due to outside influences
if present, the process is “out of control”

17. Control charts help us learn more about processes
Separate common and special causes of variation
Determine whether a process is in a state of statistical control or out-of-control
Estimate the process parameters (mean, variation) and assess the performance of a process or its capability

18. Control charts to monitor processes
To monitor output, we use a control chart
we check things like the mean, range, standard deviation
To monitor a process, we typically use two control charts
mean (or some other central tendency measure)
variation (typically using range or standard deviation)

19. Types of Data Variable data
Product characteristic that can be measured
Length, size, weight, height, time, velocity
Attribute data
Product characteristic evaluated with a discrete choice
Good/bad, yes/no

20. Types of Control Charts
Control Chart For The Mean(X chart)
Control Chart For The Range(R chart)
Control Chart For A Proportion(p chart)
Control Chart For Attribute Measures(c chart)

21. Control Chart For The Sample Mean
Assuming that the sample mean is approximately normal with mean  and standard deviation , a control chart for the mean usually consists of three horizontal lines
The vertical axis is used to plot the magnitude of observed sample means while the horizontal axis represents time or the order of the sequence of observed means

22. Control Chart For The Sample Mean
The center line is at the mean,  and upper and lower control limits are at
and
Since  (standard deviation)is usually unknown the
term is usually replaced by an estimator based on the sample range

23. Control Chart For The Sample Mean
The formula is given by where = average value of the range = k = number of samples The values of A2 depend on the sample size

24. Control Chart For The Sample Mean
LCL =
UCL =

25. Control Chart For The Range
Designed to monitor variability in the product
Range easier to determine than standard deviation
Distribution of sample range assumes product measurement is normally distributed

26. Control Chart For The Range
Upper and lower control limits and center line obtained from and the values of D3, D4
according to the formulae
LCL =
UCL =
The values of D3, D4 are based on sample size

27. Control Chart For The Sample Proportion
Population proportion 
The sample mean is now a mean proportion given by
Where = total number of objects in sample with characteristic divided by total sample size

28. Control Chart For The Sample Proportion
Using the central limit theorem the control limits are given by:

29. Control Chart For The Sample Proportion
Since the true proportion  is usually unknown we replace it by the average proportion
If the sample size varies then the upper and lower limits will vary and the equations become:
where ni = sample size in sample i

30. Control Chart For Attribute Measures
Alternative method of counting good and bad items.
Defects are measured by merely counting the no. of defects.
Where c = total number of defects in a sample

31. Control Chart For Attribute Measures
Process average or central line c=
UCL:
LCL:

32. Example: Control Charts for Variable Data
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Example: Control Charts for Variable Data
Slip Ring Diameter (cm)
Sample X R
Example4.3
Statistics

33. Example: Control Charts for Variable Data
4/20/2017
Example: Control Charts for Variable Data
Slip Ring Diameter (cm)
Sample X R
Example4.3
Statistics

34. Calculation From Table above: Sigma X-bar = 50.09 Sigma R = 1.15
Thus;
X-Double bar = 50.09/10 = cm
R-bar = 1.15/10 = cm
Note: The control limits are only preliminary with 10 samples.
It is desirable to have at least 25 samples.

35. Trial control limit UCLx-bar = X-D bar + A2 R-bar
= (0.577)(0.115) = cm
LCLx-bar = X-D bar - A2 R-bar
= (0.577)(0.115) = cm
UCLR = D4R-bar
= (2.114)(0.115) = cm
LCLR = D3R-bar
= (0)(0.115) = 0 cm
For A2, D3, D4: see Table B, Appendix
n = 5

36. 3-Sigma Control Chart Factors
Sample size X-chart R-chart
n A2 D3 D4

37. X-bar Chart

38. R Chart

39. Subgroup 18 2 3 4 5 1
12.45
12.39
12.40
12.37
12.55
12.38
12.36
12.44
12.46
12.30 6
12.41 7
12.51 8 9 10
12.35 11 12 13 14 15

40. From a lot of 100 Sample No. Number of defects 1 10 2 9 3 8 4 11 5 7 612 13 14
From a lot of 100

41. From a lot of 100 Sample No. Number of defects Proportion 1 10 0.10 29
0.9 3 8
0.8 4 11
0.11 5 7
0.7 6 12
0.12 13
0.13 14
0.14
From a lot of 100

42. The Normal Distribution -3 -2 -1 +1 +2 +3 Mean 68.26% 95.44%
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-3 -2 -1  +2 +3
Mean
68.26%
95.44%
99.74%
The Normal
Distribution
 = Standard deviation
LSL
USL
-3
+3 CL
And at +/- 3 sigma, the most common choice of confidence/control limits in quality control application, the area is 99.97%.
Statistics 20

43. 34.13% of data lie between  and 1 above the mean ().
34.13% between  and 1 below the mean.
Approximately two-thirds (68.28 %) within 1 of the mean.
13.59% of the data lie between one and two standard deviations
Finally, almost all of the data (99.74%) are within 3 of the mean.

44. Normal Distribution Review
Define the 3-sigma limits for sample means as follows:
What is the probability that the sample means will lie outside 3-sigma limits?
Note that the 3-sigma limits for sample means are different from natural tolerances which are at

45. 4/20/2017
Common Causes
The first set of slides presents the various aspects of common and assignable causes. This slide and the two following show a normal process distribution and how it allows for expected variability which is termed common causes.
Statistics 2

46. Process Out of Control
The term out of control is a change in the process due to an assignable cause.
When a point (subgroup value) falls outside its control limits, the process is out of control.

47. Assignable Causes (a) Mean Average Grams 4/20/2017
The new distribution will look as shown in this slide.
Grams
Statistics 7

48. Assignable Causes (b) Spread Average Grams 4/20/2017
The new distribution has a much greater spread (higher standard deviation).
Grams
Statistics 9

49. Assignable Causes (c) Shape Average Grams 4/20/2017
A skewed (non-normal) distribution will result in a different pattern of variability.
Grams
Statistics 11

50. Control Charts Assignable causes likely UCL Nominal LCL 1 2 3 Samples
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Control Charts
Assignable causes likely
UCL
Nominal
However, the third sample plots outside the original distribution, indicating the likely presence of an assignable cause.
LCL
Samples
Statistics 24

51. Control Chart Examples
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Control Chart Examples
UCL
Nominal
Variations
LCL
And finally, this chart shows a process with two points actually outside the control limits, an easy indicator to detect but not the only one.
These rules, there are a few more, are commonly referred to as the Western Electric Rules and can be found in any advanced quality reference.
Sample number
Statistics 30

52. Control Limits and Errors
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Control Limits and Errors
Type I error:
Probability of searching for
a cause when none exists
(a) Three-sigma limits
UCL
Process
average
As long as the area encompassed by the control limits is less than 100% of the area under the distribution, there will be a probability of a Type I error. A Type I error occurs when it is concluded that a process is out of control when in fact pure randomness is present. Given +/- 3 sigma, the probability of a Type I error is = , a very small probability.
LCL
Statistics 32

53. Control Limits and Errors
4/20/2017
Control Limits and Errors
Type I error:
Probability of searching for
a cause when none exists
(b) Two-sigma limits
UCL
Process
average
When the control limits are changed to +/- 2 sigma, the probability of a Type I error goes up considerably, from to = While this is still a small probability, it is a change that should be carefully considered. In practice, this would mean more samples would be identified inappropriately as out-of-control. Even if they were subsequently ‘Okd’, there would be increased costs due to many more cycles through the four step improvement process shown previously.
LCL
Statistics 33

54. Control Limits and Errors
4/20/2017
Control Limits and Errors
Type II error:
Probability of concluding
that nothing has changed
(a) Three-sigma limits
UCL
Shift in process
average
Process
average
Returning to the 3 sigma limits, we can see the probability of making a Type II error, in this case failing to detect a shift in the process mean.
LCL
Statistics 34

55. Control Limits and Errors
4/20/2017
Control Limits and Errors
Type II error:
Probability of concluding
that nothing has changed
(b) Two-sigma limits
UCL
Shift in process
average
Process
average
By reducing the control limits to +/- 2 sigma, we see the probability of failing to detect the shift has been reduced.
LCL
Statistics 35

56. Achieve the purpose
Our goal is to decrease the variation inherent in a process over time.
As we improve the process, the spread of the data will continue to decrease.
Quality improves!!

57. Improvement

58. Examine the process
A process is considered to be stable and in a state of control, or under control, when the performance of the process falls within the statistically calculated control limits and exhibits only chance, or common causes.

59. Consequences of misinterpreting the process
Blaming people for problems that they cannot control
Spending time and money looking for problems that do not exist
Spending time and money on unnecessary process adjustments
Taking action where no action is warranted
Asking for worker-related improvements when process improvements are needed first

60. Process variation
When a system is subject to only chance causes of variation, 99.74% of the measurements will fall within 6 standard deviations
If 1000 subgroups are measured, 997 will fall within the six sigma limits.
-3 -2 -1  +2 +3
Mean
68.26%
95.44%
99.74%

61. Chart zones
Based on our knowledge of the normal curve, a control chart exhibits a state of control when:
Two thirds of all points are near the center value.
The points appear to float back and forth across the centerline.
The points are balanced on both sides of the centerline.
No points beyond the control limits.
No patterns or trends.

62. Acceptance Sampling

63. Acceptance sampling is a form of testing that involves taking random samples of “lots,” or batches, of finished products and measuring them against predetermined standards.
A “lot,” or batch, of items can be inspected in several ways, including the use of single, double, or sequential sampling.

64. Single Sampling
Two numbers specify a single sampling plan: They are the number of items to be sampled (n) and a pre specified acceptable number of defects (c). If there are fewer or equal defects in the lot than the acceptance number, c, then the whole batch will be accepted. If there are more than c defects, the whole lot will be rejected or subjected to 100% screening.

65. Double Sampling
Often a lot of items is so good or so bad that we can reach a conclusion about its quality by taking a smaller sample than would have been used in a single sampling plan. If the number of defects in this smaller sample (of size n1) is less than or equal to some lower limit (c1), the lot can be accepted. If the number of defects exceeds an upper limit (c2), the whole lot can be rejected. But if the number of defects in the n1 sample is between c1 and c2, a second sample (of size n2) is drawn. The cumulative results determine whether to accept or reject the lot. The concept is called double sampling.

66. Sequential Sampling
Multiple sampling is an extension of double sampling, with smaller samples used sequentially until a clear decision can be made. When units are randomly selected from a lot and tested one by one, with the cumulative number of inspected pieces and defects recorded, the process is called sequential sampling.

67. OPERATING CHARACTERISTIC (OC) CURVES
The operating characteristic (OC) curve describes how well an acceptance plan discriminates between good and bad lots. A curve pertains to a specific plan, that is, a combination of n (sample size) and c (acceptance level). It is intended to show the probability that the plan will accept lots of various quality levels.

68. Figure shows a perfect discrimination plan for a company that wants to reject all lots with more than 2 ½ % defectives and accept all lots with less than 2 ½ % defectives.

69. OC Curves for Two Different Acceptable Levels of Defects (c = 1, c = 4) for the Same Sample Size (n = 100).
So one way to increase the probability of accepting only good lots and rejecting only bad lots with random sampling is to set very tight acceptance levels.

70. OC Curves for Two Different Sample Sizes (n = 25, n = 100) but Same Acceptance Percentages (4%). Larger sample size shows better discrimination.

72. Sampling Terms
Acceptance quality level (AQL): the percentage of defects at which consumers are willing to accept lots as “good”
Lot tolerance percent defective (LTPD): the upper limit on the percentage of defects that a consumer is willing to accept
Consumer’s risk: the probability that a lot contained defectives exceeding the LTPD will be accepted
Producer’s risk: the probability that a lot containing the acceptable quality level will be rejected

73. THE OPERATING-CHARACTERISTIC (OC) CURVE
For a given a sampling plan and a specified true fraction defective p, we can calculate
Pa -- Probability of accepting lot
If lot is truly good, 1 - Pa = a (Producers' Risk)
If lot is truly bad, Pa = b (Consumer ‘s Risk)
A plot of Pa as a function of p is called the OC curve for a given sampling plan

74. The OC curve shows the features of a particular sampling plan, including the risks of making a wrong decision.

75. Construction of OC curve
In attribute sampling, where products are determined to be either good or bad, a binomial distribution is usually employed to build the OC curve. The binomial equation is
where n = number of items sampled (called trials)
p = probability that an x (defect) will occur on any one trial
P(x) = probability of exactly x results in n trials
In a Poisson approximation of the binomial distribution, the mean of the binomial, which is np, is used as the mean of the Poisson, which is λ; that is,
λ = np

76. Example
Probability of acceptance A shipment of 2,000 portable battery units for microcomputers is about to be inspected by a Malaysian importer. The Korean manufacturer and the importer have set up a sampling plan in which the risk is limited to 5% at an acceptable quality level (AQL) of 2% defective, and the risk is set to 10% at Lot Tolerance Percent Defective (LTPD) = 7% defective. We want to construct the OC curve for the plan of n = 120 sample size and an acceptance level of c ≤ 3 defectives. Both firms want to know if this plan will satisfy their quality and risk requirements.

78. Example
N=1000 n = 100 AQL=1% LTPD=5% ß=10% α = 5% C<=2 Does the plan meet the producer’s and consumer’s requirement.

79. AVERAGE OUTGOING QUALITY
In most sampling plans, when a lot is rejected, the entire lot is inspected and all of the defective items are replaced. Use of this replacement technique improves the average outgoing quality in terms of percent defective. In fact, given (1) any sampling plan that replaces all defective items encountered and (2) the true incoming percent defective for the lot, it is possible to determine the average outgoing quality (AOQ) in percent defective.

80. AVERAGE OUTGOING QUALITY
The equation for AOQ is

81. Example Selected Values of % Defective Mean of Poisson, λ = np
P (acceptance)
.01 1
0.920
.02 2
0.677
.03 3
0.423
.04 4
0.238
.05 5
0.125
.06 6
0.062

82. Example
The percent defective from an incoming lot in is 3%. An OC curve showed the probability of acceptance to be Given a lot size of 2,000 and a sample of 120, what is the average outgoing quality in percent defective?