1. Quality Control
Agenda
- What is quality?
- Approaches in quality control
- Accept/Reject testing
- Sampling (statistical QC)
- Control Charts
- Robust design methods

2. What is ‘Quality’
Performance:
- A product that ‘performs better’ than others at same function
Example:
Sound quality of Apple iPod vs. iRiver…
- Number of features, user interface
Examples:
Tri-Band mobile phone vs. Dual-Band mobile phone
Notebook cursor control (IBM joystick vs. touchpad)

3. - A product that needs frequent repair has ‘poor quality’
What is ‘Quality’
Reliability:
- A product that needs frequent repair has ‘poor quality’
Example:
Consumer Reports surveyed the owners of > 1 million vehicles. To calculate predicted reliability for 2006 model-year vehicles, the magazine averaged overall reliability scores for the last three model years (two years for newer models)
Best predicted reliability: Sporty cars/Convertibles Coupes
Honda S2000
Mazda MX-5 Miata (2005)
Lexus SC430
Chevrolet Monte Carlo (2005)

4. - A product that has longer expected service life
What is ‘Quality’
Durability:
- A product that has longer expected service life
Adidas Barricade 3 Men's Shoe
(6-Month outsole warranty)
Nike Air Resolve Plus Mid Men’s Shoe
(no warranty)

5. What is ‘Quality’
Aesthetics:
- A product that is ‘better looking’ or ‘more appealing’
Examples? or ?

6. Defining quality for producers..
Example: [Montgomery]
- Real case study performed in ~1980 for a US car manufacturer
- Two suppliers of transmissions (gear-box) for same car model
Supplier 1: Japanese; Supplier 2: USA
- USA transmissions has 4x service/repair costs than Japan transmissions
Lower variability 
Lower failure rate
Distribution of critical dimensions from transmissions

7. Definitions
Quality is inversely proportional to variability
Quality improvement is the reduction in variability
of products/services.
How to reduce in variability of products/services ?

8. QC Approaches
(1) Accept/Reject testing
(2) Sampling (statistical QC)
(3) Statistical Process Control [Shewhart]
(4) Robust design methods (Design Of Experiments) [Taguchi]

9. Accept/Reject testing
- Find the ‘characteristic’ that defines quality
- Find a reliable, accurate method to measure it
- Measure each item
- All items outside the acceptance limits are scrapped
Lower Specified Limit
Upper Specified Limit
target
Measured characteristic

10. Problem with Accept/Reject testing
(1) May not be possible to measure all data
Examples:
Performance of Air-conditioning system, measure temperature of room
Pressure in soda can at 10°
(2) May be too expensive to measure each sample
Service time for customers at McDonalds
Defective surface on small metal screw-heads

11. Problems with Accept/Reject testing
Solution: only measure a subset of all samples
This approach is called: Statistical Quality Control
What is statistics?

12. Background: Statistics
Average value (mean) and spread (standard deviation)
Given a list of n numbers, e.g.: 19, 21, 18, 20, 20, 21, 20, 20.
Mean = m = S ai / n = ( ) / 8 =
The variance s2 = ≈
The standard deviation = s =
= √( s2) ≈

13. Background: Statistics..
Example. Air-conditioning system cools the living room and bedroom to 20;
Suppose now I want to know the average temperature in a room:
- Measure the temperature at 5 different locations in each room.
Living Room: 18, 19, 20, 21, 22.
Bedroom: 19, 20, 20, 20, 19.
What is the average temperature in the living room?
m = S ai / n = ( ) / 5 = 20.
BUT: is m = m ?

14. Background: Statistics...
Example (continued)
m = S ai / n = ( ) / 5 = 20.
BUT: is m = m ?
If: sample points are selected randomly,
thermometer is accurate, …
then m is an unbiased estimator of m.
- take many samples of 5 data points,
- the mean of the set of m-values will approach m
- how good is the estimate?

15. Background: Statistics....
Example. Air-conditioning system cools the living room and bedroom to 20;
Suppose now I want to know the variation of temperature in a room:
- Measure the temperature at 5 different locations in each room.
Living Room: 18, 19, 20, 21, 22.
sn = ≈
BUT: is sn = s ? No!
The unbiased estimator of stdev of a sample = s =

16. Sampling: Example
Soda can production:
Design spec: pressure of a sealed can 50PSI at 10C
Testing: sample few randomly selected cans each hour
Questions:
How many should we test?
Which cans should we select?
To Answer:
We need to know the distribution of pressure among all cans
Problem:
How can we know the distribution of pressure among all cans?

17. Sampling: Example..
How can we know the distribution of pressure among all cans?
Plot a histogram showing %-cans with pressure in different ranges

18. Limit (as histogram step-size)  0: probability density function
Sampling: Example…
Limit (as histogram step-size)  0: probability density function 50 55 60 65 70 45 40 35 30
pressure (psi)
pdf is (almost) the familiar bell-shaped Gaussian curve!
why?
True Gaussian curve: [-∞ , ∞]; pressure: [0, 95psi]

19. Why is everything normal?
pdf of many natural random variables ~ normal distribution
WHY ?
Central Limit Theorem
Let X random variable, any pdf, mean, m, and variance, s2
Let Sn = sum of n randomly selected values of X;
As n  ∞ Sn approaches normal distribution
with mean = nSn, and variance = ns2.

20. Central limit theorem.. Example X1 = -1, with probability 1/31 S1
p(S1)
Example
X X X1 + X2
X1 + X2 =
-2, with probability 1/9
-1, with probability 2/9
0, with probability 3/9
1, with probability 2/9
2, with probability 1/9 -1 1 -2 2 S2
p(S2)
X1 + X2 + X3 =
-3, with probability 1/27
-2, with probability 3/27
-1, with probability 6/27
0, with probability 7/27
1, with probability 6/27
2, with probability 3/27
3, with probability 1/27 -1 1 -2 2 -3 3 S3
p(S3)
Gaussian curve
Curve joining p(S3)

21. (Weaker) Central Limit Theorem...
Let Sn = X1 + X2 + … + Xn
Different pdf, same m and s
normalized Sn is ~ normally distributed
Another Weak CLT:
Under some constraints, even if Xi are from different pdf’s,
with different m and s, the normalized sum is nearly normal!

22. Central Limit Therem....
Observation: For many physical processes/objects
variation is f( many independent factors)
effect of each individual factor is relatively small
Observation + CLT 
The variation of parameter(s) measuring the
physical phenomenon will follow Gaussian pdf

23. Sampling for QC
Soda Can Problem, recalled:
How can we know the distribution of pressure among all cans?
Answer:
We can assume it is normally distributed
Problem:
But what is the m, s ?
Answer:
We will estimate these values

24. Outline