1. Statistical Quality Control
MBA 8452 Systems and Operations Management
Statistical Quality Control

2. Objective: Quality Analysis
Process Variation
Be able to explain Taguchi’s View of the cost of variation.
Statistical Process Control Charts and Process Capability
Be able to develop and interpret SPC charts.
Be able to calculate and interpret Cp and Cpk
Be able to explain the difference between process control and process capability
Sample Size
Be able to explain the importance of sample size

3. Statistical Quality Control Approaches
Statistical Process Control (SPC)
Sampling to determine if the process is within acceptable limits (under control)
Acceptance Sampling
Inspects a random sample of a product
to determine if the lot is acceptable

4. Line graph shows plot of data and variation from the average
Process
target or
average 1 2 3 4 5 6 7 8 9 10
Sample number

5. Why Statistical Quality Control?
Variations in Manufacturing/Service Processes
Any process has some variations: common and/or special
Variations are causes for quality problems
If a process is stable (no special variation), it is able to produce product/service consistently
As variation is reduced, quality is improved
Statistics is the only science that is dedicated to dealing with variations.
Measure, monitor, and reduce variations in the process

6. Types of Variation Natural (common) Assignable (special)
Inherent to process
Random
Cannot be controlled
Cannot be prevented
Examples
weather
accuracy of measurements
capability of machine
Exogenous to process
Not random
Controllable
Preventable
Examples
tool wear
human factors (fatigue)
poor maintenance

7. Cost of Variation: Traditional vs. Taguchi’s View
Incremental
Cost of
Variability
High
Zero
Lower
Spec
Target
Upper
Taguchi’s View
Incremental
Cost of
Variability
High
Zero
Lower
Spec
Target
Upper
Traditional View

8. Statistical Process Control
On-line quality control tool used when the product/service is being produced
Purpose: prevent systematic quality problems
Procedure
Take periodic random samples from a process
Plot the sample statistics on control chart(s)
Determine if the process is under control
If the process is under control, do nothing
If the process is out of control, investigate and fix the cause

9. Statistical Process Control Types Of Data
Attribute data (discrete values)
Quality characteristic evaluated about whether it meets the required specifications
Good/bad, yes/no
Variable data (continuous values)
Quality characteristic that can be measured
Length, size, weight, height, time, velocity
Attribute data describe product or service characteristics that can be counted or classified, such as number of defective items, number of customer complaints, or percent of students who passed a test.
Variable data describe product or service characteristics that can be measured, such as weight, length, or time.

10. Statistical Process Control Control Charts
Charts for attributes
p-chart (for proportions)
c-chart (for counts)
Charts for variables
R-chart (for ranges)
-chart (for means)
A p-chart measures the number of defectives as a percentage of the total number of possible defectives.
A c-chart measures the actual number of defectives in a sample when the total number of possible defectives is not known. For example, the number of blemishes in a paint job, the number of bubbles in a sheet of glass, or the number of picks in a bolt of cloth would be graphed on a c-chart.

11. Control Chart General Structure
Upper
control
limit (UCL)
Process
target or
average
Control charts are graphs that visually show if sample results are within statistical control limits.
Lower
control
limit (LCL) 1 2 3 4 5 6 7 8 9 10
Sample number

12. A Process Is In Control If ...
No sample points outside control limits
Most points near the process average
About an equal # points above & below the centerline
Points appear randomly distributed

13. Common Out-of-control Signs
One observation outside the limits
Sample observations consistently below or above the average
Sample observations consistently decrease or increase

14. Issues In Building Control Charts
Number of samples: around 25
Size of each sample: large (100) for attributes and small (25) for variables
Frequency of sampling: depends
Control limits: typically 3-sigma away from the process mean

15. Control Limits: The Normal Distribution
99.74 %
95 % m
m+1s
m+2s
m+3s
m-1s
m-2s
m-3s
Figure 4.2 X
If we establish control limits at +/- 3 standard deviations (s), then we would expect 99.74% of observations (X) to fall within these limits.

16. Control Limits: General Formulas
UCL = mean + z (stand dev)
LCL = mean – z (stand dev)
z is the # of standard deviations
z = 3.00 is the most commonly used value with 99.7% confidence level
Other z values can be used (e.g. z=2 for 95% confidence and z=2.58 for 99% confidence)

17. Control Charts for Attributes p-charts

18. . . . . . . p-Chart Example total defectives p = 200 = 0.10 =
20 Samples of 100 pairs of jeans each were randomly selected from the Western Jean Company’s production line.
total defectives
total sample observations
200
20 (100)
p = =
= 0.10
n=100 jeans in each sample
Proportion
Sample Defect Defective
Total

19. p-Chart Example p 0.2 UCL 0.18 0.16 0.14 0.12 0.1 Proportion defective
0.08
0.06
0.04
0.02
LCL 2 4 6 8 10 12 14 16 18 20 . .
Sample number

20. Control Charts For Variables X-bar charts and R-charts
Where X = average of sample means = Xi / m
R = average of sample ranges = Ri / m
Xi = mean of sample i, i = 1,2,…,m
Ri = range of sample i, i = 1,2,…,m
m = total number of samples
A2, D3, and D4 are constants from Exhibit TN7.7

21. Example
If a company makes jeans, there are a specifications that must be met.
The back pockets of the jeans can’t be too small or too large.
The control chart can be established to monitor the measurements of the back pocket
Given 15 samples with 5 observations each, we can determine the Upper and Lower control limits for the range and x-bar charts.

22. X-bar and R Charts Example
(Xi)
(Ri) X R

23. X-bar and R Charts Example
Exhibit TN7.7
Since n=5, from Exhibit TN7.7 (also right table), we find
A2=0.58
D3=0
D4=2.11

24. X-bar and R Charts: Example
UCL
LCL R

25. X-bar and R Charts Example
X-bar chart
UCL
LCL X

26. Process Capability
The ability of a process to meet product design/technical specifications
Design specifications for products (Tolerances)
upper and lower specification limits (USL, LSL)
Process variability in production process
natural variation in process (3 from the mean)
Process may not be capable of meeting specifications if natural variation in a process exceeds allowable variation (tolerances)

27. Process Capability Illustrations
specification
natural variation
(a)
(b)
(c)
(d)

28. Process Capability Further Illustrations
LSL
Target
USL
LSL
Target
USL
Process variation
Tolerance variation
Capable process
Highly capable process
Process not capable
Process not capable

29. Specification Limits  Control Limits
Specification limits are pre-established for products before production
Control limits are used to monitor the actual production process performance
It is possible that a process is under control, but not capable to meet specifications
It is also possible that a process that is within specifications is out-of-control

30. Control Limits Vs. Specification Limits Illustrations
USL
LCL
UCL
LSL
(2) In control but exceeds specifications
USL
LSL
LCL
UCL
(1) In control and within specifications
USL
LSL
UCL
LCL
(3) Out-of-control and within specifications

31. Process Capability Index: Cp -- Measure of Potential Capability
LSL
USL
Cp = 1
Cp < 1

32. Process Capability Index: Cpk -- Measure of Actual Capability
is the standard deviation of the production process
Cpk considers both process variation () and process location (X)

33. Process Capability Index Example
A manufacturing process produces a certain part with a mean diameter of 2 inches and a standard deviation of 0.03 inches. The lower and upper engineering specification limits are 1.90 inches and 2.05 inches.
Therefore, the process is not capable (the variation is too much and the process mean is not on the target)

34. Impact of Process Location on Process Capability
Cp = 2.0
Cpk = 2.0
Cp = 2.0
Cpk = 1.5
Cp = 2.0
Cpk = 1.0
Cp = 2.0
Cpk = 0

35. Acceptance Sampling
Determines whether to accept or reject an entire lot of goods based on sample results
Measures quality in percent defective
Usually applied to incoming raw materials or outgoing finished goods

36. Sampling Plan Guidelines for accepting or rejecting a lot
Single sampling plan
N = lot size
n = sample size
c = max acceptance number of defects
d = number of defective items in sample
If d <= c, accept lot; else reject
Sampling plan is developed based on the tradeoff between producer’s risk and consumer’s risk

37. Producer’s & Consumer’s Risk
Producer’s Risks
reject a good lot (TYPE I ERROR)
a = producer’s risk = P(reject good lot)
5% is common
Consumer’s Risks
accept a bad lot (TYPE II ERROR)
b = consumer’s risk = P(accept bad lot)
10% is typical

38. Quality Definitions Acceptable quality level (AQL)
Acceptable proportion of defects on average
“good lot” = the proportion of defects of the lot is less than or equal to AQL
Lot tolerance percent defective (LTPD)
Maximum proportion of defects in a lot
“bad lot” = the proportion of defects of the lot is greater than LTPD

39. Operating Characteristic Curve
AQL
LTPD
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 2 3 4 5 6 7 8 9 10 11 12
Probability of acceptance 
=.10
(consumer’s risk) a
= .05 (producer’s risk)
Percent defective in a lot