1. IES 331 Quality Control Chapter 6 Control Charts for Attributes
Week 7-8
July 19-28, 2005

2. Attribute Data
If item does not conform to standard on one or more of these characteristics, it is classified as nonconforming
Conforming / Nonconforming units
Non-defective / Defecting units
Good / Bad
Pass / Fail
Nonconforming unit will contain at least on nonconformity
Nonconformities / Defects
Each specific point at which a specification is not satisfied
ex: scratch, chip, dirty spots, accident

3. 6-2 Control Chart for Fraction Nonconforming
Fraction of Nonconforming: Ratio of the number of nonconforming items in a population to the total number of items in that population
Fraction of nonconforming ~ Binomial Distribution
p Probability that any unit will not conform to specifications
n Random sample of n unit. Sample size
D The number of units of products that are nonconforming

4. The Control Chart for Fraction Nonconforming
Sample fraction nonconforming: ratio of the number of nonconforming units in the sample, D, to the sample size n
Mean and Variances

5. The Control Chart for Fraction Nonconforming
With specified standard value

6. The Control Chart for Fraction Nonconforming
When p is not known, it must be estimated from collected data
Average of these individual sample fractions nonconforming
Fraction Nonconforming control chart: No Standard Given
“Trial Control Limit”

7. Ex 1 (Exercise 6-1) Also see Example 6-1
Sample #
Number of Noncomforming Assemblies 1 7 2 4 3 5 6 8 10 9 11 12 15 13 14 16 17 18 19 20
The data that follow give the number of nonconforming bearing and seal assemblies in sample size of 100
Construct a fraction nonconforming control chart for these data.
If any points plot out of control, assume that assignable causes can be found and determine the revised control limits

8. The np Control Chart Alternative to p Control Chart
Based on the number nonconforming rather than the fraction nonconforming
If standard value p is not known, use the estimator
Revisit Ex 1

9. Variable Sample Size 3 approaches to deal with variable sample size
Variable-width control limits: to determine control limits for each individual sample that are based on specific sample size
Control limits based on average sample size: to obtain an approximate set of control limits (constant control limits)
The standardized control chart:
The points are plotted in standard deviation units
Center line at zero
Upper and lower control limits  +3 and - 3

10. Ex 2 (Exercise 6-3) Also see Example 6-2
Day
Units Inspected
Nonconforming units
Fraction Nonconforming 1 80 4
0.050 2
110 7
0.064 3 90 5
0.056 75 8
0.107
130 6
0.046
120 70
0.057
125
0.040 9
105
0.076 10 95
0.074
The following data represent the results of inspecting all units of a personal computer produced for the last 10 days.
Does the process appear to be in control?

11. The Operating-Characteristic Function
OC curve provides a measure of the sensitivity of the control chart
Ability to detect the shift in the process fraction nonconforming from the nominal value to some other value
Probability of incorrectly accepting the hypothesis of statistical control (i.e., type II or β-error)

13. Average Run Length (ARL)
ARL for any Shewhart control chart
If the process in control, ARL0
If the process is out of control, ARL1

14. 6-3 Control Charts for Nonconformities (Defects)
We can develop control charts for
Total number of nonconformities in a unit, or
Average number of nonconformities per unit
Defects or nonconformities ~ Poisson Distribution
Where x is the number of nonconformities and C > 0
c chart: same sample size

15. Ex 3 (Exercise 6-41) Also see Example 6-3
Sample Number
Number of Nonconformities 1 2 3 4 7 5 8 6 10 13 9 19 11 24 12 14 15 16 17 18 20 21 22
Ex 3 (Exercise 6-41) Also see Example 6-3
The data represent the number of nonconformities per 1000 meters in telephone cable.
From analysis of these data, would you conclude that the process is in control?

16. Choice of Sample Size: u chart
Average number of nonconformities per inspection unit
It may requires several inspection units in the sample
u chart: setting up the control chart based on the average number of nonconformities per inspection unit

17. Ex 4 (Exercise 6-44) Also see Example 6-4
Sample Number
Number of Nonconformities 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
An automobile manufacturer wishes to control the number of nonconformities in a subassembly area producing manual transmissions.
The inspection unit is defined as four transmissions, and data from 16 samples (each of size 4) are shown here.
Set up a control chart for nonconformities per unit

18. Control charts for nonconformities with variable sample size
c chart will be difficult to interpret – varying center line and control limits
Alternative approaches are:
u chart (nonconformities per unit)
Constant center line
Control limit will vary inversely with the n1/2
Use control limits based on an average sample size
Use a standardized control chart (preferred)

19. Demerit Systems for Defects

20. The Operating Characteristic Function
OC curves for c and u charts can be obtained from the Poisson distribution

21. Action taken to improve a process