1. Statistical Process Control (SPC)
Variability is inherent in every process
Natural or common causes
Special or assignable causes
Provides a statistical signal when assignable causes are present
Detect and eliminate assignable causes of variation
Points which might be emphasized include:
- Statistical process control measures the performance of a process, it does not help to identify a particular specimen produced as being “good” or “bad,” in or out of tolerance.
- Statistical process control requires the collection and analysis of data - therefore it is not helpful when total production consists of a small number of units
- While statistical process control can not help identify a “good” or “bad” unit, it can enable one to decide whether or not to accept an entire production lot. If a sample of a production lot contains more than a specified number of defective items, statistical process control can give us a basis for rejecting the entire lot. The issue of rejecting a lot which was actually good can be raised here, but is probably better left to later.

2. Natural Variations Natural variations in the production process
These are to be expected
Output measures follow a probability distribution
For any distribution there is a measure of central tendency and dispersion

3. Assignable Variations
Variations that can be traced to a specific reason (machine wear, misadjusted equipment, fatigued or untrained workers)
The objective is to discover when assignable causes are present and eliminate them

4. Each of these represents one sample of five boxes of cereal
Samples
To measure the process, we take samples and analyze the sample statistics following these steps
Each of these represents one sample of five boxes of cereal
(a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight
Frequency
Weight #
Figure S6.1

5. The solid line represents the distribution
Samples
The solid line represents the distribution
(b) After enough samples are taken from a stable process, they form a pattern called a distribution
Frequency
Weight
Figure S6.1

6. Samples
(c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape
Weight
Central tendency
Variation
Shape
Frequency
Figure S6.1

7. Samples
(d) If only natural causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable
Prediction
Weight
Time
Frequency
Figure S6.1

8. Samples
Prediction ?
(e) If assignable causes are present, the process output is not stable over time and is not predicable
Weight
Time
Frequency
Figure S6.1

9. Control Charts
Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes
Students should understand both the concepts of natural and assignable variation, and the nature of the efforts required to deal with them.

10. Types of Data Variables Attributes
Characteristics that can take any real value
May be in whole or in fractional numbers
Continuous random variables
Defect-related characteristics
Classify products as either good or bad or count defects
Categorical or discrete random variables
Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process.

11. Control Charts for Variables
For variables that have continuous dimensions
Weight, speed, length, strength, etc.
x-charts are to control the central tendency of the process
R-charts are to control the dispersion of the process

12. Setting Chart Limits For x-Charts when we know s
Upper control limit (UCL) = x + zsx
Lower control limit (LCL) = x - zsx
where x = mean of the sample means or a target value set for the process
z = number of normal standard deviations
sx = standard deviation of the sample means
= s/ n
s = population standard deviation
n = sample size

13. Setting Control Limits
Hour 1
Sample Weight of Number Oat Flakes
1 17
2 13
3 16
4 18
5 17
6 16
7 15
8 17
9 16
Mean 16.1
s = 1
Hour Mean Hour Mean
n = 9
For 99.73% control limits, z = 3
UCLx = x + zsx = (1/3) = 17 ozs
LCLx = x - zsx = (1/3) = 15 ozs

14. Setting Control Limits
Control Chart for sample of 9 boxes
Variation due to assignable causes
Out of control
Sample number
| | | | | | | | | | | |
17 = UCL
15 = LCL
16 = Mean
Variation due to natural causes
Out of control

16. Setting Chart Limits For x-Charts when we don’t know s
Upper control limit (UCL) = x + A2R
Lower control limit (LCL) = x - A2R
where R = average range of the samples
A2 = control chart factor found in Table S6.1
x = mean of the sample means

17. Control Chart Factors Sample Size Mean Factor Upper Range Lower Range
n A2 D4 D3
Table S6.1

18. Setting Control Limits
Process average x = ounces
Average range R = .25
Sample size n = 5

19. Setting Control Limits
Process average x = ounces
Average range R = .25
Sample size n = 5
UCLx = x + A2R
= (.577)(.25) =
= ounces
From Table S6.1

20. Setting Control Limits
Process average x = ounces
Average range R = .25
Sample size n = 5
UCL =
Mean = 16.01
LCL =
UCLx = x + A2R
= (.577)(.25) =
= ounces
LCLx = x - A2R =
= ounces

21. R – Chart Type of variables control chart
Shows sample ranges over time
Difference between smallest and largest values in sample
Monitors process variability
Independent from process mean

22. Setting Chart Limits For R-Charts Upper control limit (UCLR) = D4R
Lower control limit (LCLR) = D3R
where
R = average range of the samples
D3 and D4 = control chart factors from Table S6.1

24. Setting Control Limits
Average range R = 5.3 pounds
Sample size n = 5
From Table S6.1 D4 = 2.115, D3 = 0
UCL = 11.2
Mean = 5.3
LCL = 0
UCLR = D4R
= (2.115)(5.3)
= 11.2 pounds
LCLR = D3R
= (0)(5.3)
= 0 pounds

25. Mean and Range Charts (a)
These sampling distributions result in the charts below
(Sampling mean is shifting upward but range is consistent)
x-chart
(x-chart detects shift in central tendency)
UCL
LCL
R-chart
(R-chart does not detect change in mean)
UCL
LCL
Figure S6.5

26. Mean and Range Charts (b)
These sampling distributions result in the charts below
(Sampling mean is constant but dispersion is increasing)
x-chart
(x-chart does not detect the increase in dispersion)
UCL
LCL
R-chart
(R-chart detects increase in dispersion)
UCL
LCL
Figure S6.5

27. Automated Control Charts

28. Control Charts for Attributes
For variables that are categorical
Good/bad, yes/no, acceptable/unacceptable
Measurement is typically counting defectives
Charts may measure
Percent defective (p-chart)
Number of defects (c-chart)

29. Control Limits for p-Charts
Population will be a binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics
UCLp = p + zsp ^
p(1 - p) n
sp = ^
Instructors may wish to point out the calculation of the standard deviation reflects the binomial distribution of the population
LCLp = p - zsp ^
where p = mean fraction defective in the sample
z = number of standard deviations
sp = standard deviation of the sampling distribution
n = sample size ^

30. p-Chart for Data Entry p = = .04 sp = = .02 1 6 .06 11 6 .06
Sample Number Fraction Sample Number Fraction
Number of Errors Defective Number of Errors Defective
Total = 80
p = = .04 80
(100)(20)
(.04)( )
100
sp = = .02 ^

31. p-Chart for Data Entry UCLp = p + zsp = .04 + 3(.02) = .10^
LCLp = p - zsp = (.02) = 0 ^
.11 –
.10 –
.09 –
.08 –
.07 –
.06 –
.05 –
.04 –
.03 –
.02 –
.01 –
.00 –
Sample number
Fraction defective
| | | | | | | | | |
UCLp = 0.10
LCLp = 0.00
p = 0.04

32. Possible assignable causes present
p-Chart for Data Entry
.11 –
.10 –
.09 –
.08 –
.07 –
.06 –
.05 –
.04 –
.03 –
.02 –
.01 –
.00 –
Sample number
Fraction defective
| | | | | | | | | |
UCLp = p + zsp = (.02) = .10 ^
LCLp = p - zsp = (.02) = 0
UCLp = 0.10
LCLp = 0.00
p = 0.04
Possible assignable causes present
There is always a focus on finding and eliminating problems. But control charts find any process changed, good or bad. The clever company will be looking at Operator 3 and 19 as they reported no errors during this period. The company should find out why (find the assignable cause) and see if there are skills or processes that can be applied to the other operators.

33. Control Limits for c-Charts
Population will be a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics
UCLc = c + 3 c
LCLc = c - 3 c
Instructors may wish to point out the calculation of the standard deviation reflects the Poisson distribution of the population where the standard deviation equals the square root of the mean
where c = mean number defective in the sample

34. c-Chart for Cab Company
c = 54 complaints/9 days = 6 complaints/day
UCLc = c + 3 c =
= 13.35 | 1 2 3 4 5 6 7 8 9
Day
Number defective
14 –
12 –
10 –
8 –
6 –
4 –
2 –
0 –
UCLc = 13.35
LCLc = 0
c = 6
LCLc = c - 3 c =
= 0

35. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Normal behavior. Process is “in control.”
Figure S6.7

36. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
One plot out above (or below). Investigate for cause. Process is “out of control.”
Figure S6.7

37. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Trends in either direction, 5 plots. Investigate for cause of progressive change.
Figure S6.7

38. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Two plots very near lower (or upper) control. Investigate for cause.
Figure S6.7

39. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Run of 5 above (or below) central line. Investigate for cause.
Figure S6.7

40. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Erratic behavior. Investigate.
Figure S6.7

41. Which Control Chart to Use
Variables Data
Using an x-chart and R-chart:
Observations are variables
Collect samples of n = 4, or n = 5, or more, each from a stable process and compute the mean for the x-chart and range for the R-chart
Track samples of n observations each

42. Which Control Chart to Use
Attribute Data
Using the p-chart:
Observations are attributes that can be categorized in two states
We deal with fraction, proportion, or percent defectives
Have several samples, each with many observations

43. Which Control Chart to Use
Attribute Data
Using a c-Chart:
Observations are attributes whose defects per unit of output can be counted
The number counted is often a small part of the possible occurrences
Defects such as number of blemishes on a desk, number of typos in a page of text, flaws in a bolt of cloth