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 Also called common causes
Affect virtually all production processes
Expected amount of variation
Output measures follow a probability distribution
For any distribution there is a measure of central tendency and dispersion
If the distribution of outputs falls within acceptable limits, the process is said to be “in control”

3. Assignable Variations
Also called special causes of variation
Generally this is some change in the process
Variations that can be traced to a specific reason
The objective is to discover when assignable causes are present
Eliminate the bad causes
Incorporate the good causes

4. Samples
To measure the process, we take samples and analyze the sample statistics following these steps
(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

5. Samples
To measure the process, we take samples and analyze the sample statistics following these steps
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

6. 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.

7. Process Control
(a) In statistical control and capable of producing within control limits
Frequency
Lower control limit
Upper control limit
(b) In statistical control but not capable of producing within control limits
This slide helps introduce different process outputs.
It can also be used to illustrate natural and assignable variation.
(c) Out of control
(weight, length, speed, etc.)
Size
Figure S6.2

8. 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.

9. Central Limit Theorem
Regardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve
The mean of the sampling distribution (x) will be the same as the population mean m
x = m
This slide introduces the difference between “natural” and “assignable” causes.
The next several slides expand the discussion and introduce some of the statistical issues.
The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n s n
sx =

10. Population and Sampling Distributions
Three population distributions
Beta
Normal
Uniform
Distribution of sample means
Standard deviation of the sample means
= sx = s n
Mean of sample means = x
| | | | | | |
-3sx -2sx -1sx x +1sx +2sx +3sx
99.73% of all x
fall within ± 3sx
95.45% fall within ± 2sx
Figure S6.3

11. Sampling Distribution
Sampling distribution of means
Process distribution of means
x = m
(mean)
It may be useful to spend some time explicitly discussing the difference between the sampling distribution of the means and the mean of the process population.
Figure S6.4

12. 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
These two charts must be used together

13. 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

14. 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

15. 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 = 12 ounces
Average range R = .25
Sample size n = 5

19. Setting Control Limits
Process average x = 12 ounces
Average range R = .25
Sample size n = 5
UCLx = x + A2R
= 12 + (.577)(.25) =
= ounces
From Table S6.1

20. Setting Control Limits
Process average x = 12 ounces
Average range R = .25
Sample size n = 5
UCL =
Mean = 12
LCL =
UCLx = x + A2R
= 12 + (.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

23. 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

24. 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

25. 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

26. Steps In Creating Control Charts
Take samples from the population and compute the appropriate sample statistic
Use the sample statistic to calculate control limits and draw the control chart
Plot sample results on the control chart and determine the state of the process (in or out of control)
Investigate possible assignable causes and take any indicated actions
Continue sampling from the process and reset the control limits when necessary

27. 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)

28. 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 = ^
LCLp = p - zsp ^
Instructors may wish to point out the calculation of the standard deviation reflects the binomial distribution of the population
where p = mean fraction defective in the sample
z = number of standard deviations
sp = standard deviation of the sampling distribution
n = sample size ^

29. 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 ^

30. 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

31. Possible assignable causes present
p-Chart for Data Entry
UCLp = p + zsp = (.02) = .10 ^
Possible assignable causes present
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
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.

32. Managerial Issues and Control Charts
Three major management decisions:
Select points in the processes that need SPC
Determine the appropriate charting technique
Set clear policies and procedures

33. 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

34. 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

35. 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 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

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.
Normal behavior. Process is “in 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.
One plot out above (or below). Investigate for cause. Process is “out of control.”
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.
Trends in either direction, 5 plots. Investigate for cause of progressive change.
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.
Two plots very near lower (or upper) control. 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.
Run of 5 above (or below) central line. Investigate for cause.
Figure S6.7

41. 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

42. Process Capability
The natural variation of a process should be small enough to produce products that meet the standards required
A process in statistical control does not necessarily meet the design specifications
Process capability is a measure of the relationship between the natural variation of the process and the design specifications

43. Process Capability Ratio
Cp =
Upper Specification - Lower Specification 6s
A capable process must have a Cp of at least 1.0
Does not look at how well the process is centered in the specification range
Often a target value of Cp = 1.33 is used to allow for off-center processes
Six Sigma quality requires a Cp = 2.0

44. Process Capability Ratio
Insurance claims process
Process mean x = minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
Cp =
Upper Specification - Lower Specification 6s

45. Process Capability Ratio
Insurance claims process
Process mean x = minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
Cp =
Upper Specification - Lower Specification 6s
= = 1.938
6(.516)

46. Process Capability Ratio
Insurance claims process
Process mean x = minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
Cp =
Upper Specification - Lower Specification 6s
= = 1.938
6(.516)
Process is capable

47. Process Capability Index
Cpk = minimum of ,
Upper Specification - x Limit 3s
Lower x - Specification Limit
A capable process must have a Cpk of at least 1.0
A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes

48. Process Capability Index
New Cutting Machine
New process mean x = .250 inches
Process standard deviation s = inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches

49. Process Capability Index
New Cutting Machine
New process mean x = .250 inches
Process standard deviation s = inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches
Cpk = minimum of ,
(.251)
(3).0005

50. Process Capability Index
New Cutting Machine
New process mean x = .250 inches
Process standard deviation s = inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches
Cpk = minimum of ,
(.251)
(3).0005
(.249)
Both calculations result in
New machine is NOT capable
Cpk = = 0.67
.001
.0015

51. Interpreting Cpk Cpk = negative number Cpk = zero
Cpk = between 0 and 1
Cpk = 1
Cpk > 1
Figure S6.8

52. SPC and Process Variability
Lower specification limit
Upper specification limit
Process mean, m
(a) Acceptance sampling (Some bad units accepted)
(b) Statistical process control (Keep the process in control)
This may be a good time to stress that an overall goal of statistical process control is to “do it better,” i.e., improve over time.
(c) Cpk >1 (Design a process that is in control)
Figure S6.10