1. Modeling Process Quality
Chapter 2
Modeling Process Quality
Introduction to Statistical Quality Control, 4th Edition

2. 2-1. Describing Variation
Graphical displays of data are important tools for investigating samples and populations.
Displays can include stem and leaf plots, histograms, box plots, and dot diagrams.
Graphical displays give an indication of the overall “distribution” of the data.
Introduction to Statistical Quality Control, 4th Edition

3. 2-1.1 The Stem-and-Leaf Plot
17| 558
18| 357
19|
20| 1399
21| 00238
22| 005
23| 5678
24|
25| 158
The numbers on the left are the “stems”
The values on the right are the “leaves”
The smallest number in this set of data is 175
The median is 211
Introduction to Statistical Quality Control, 4th Edition

4. 2-1.2 The Frequency Distribution and Histogram
Arrangement of data by magnitude
More compact than a stem-and-leaf display
Graphs of observed frequencies are called histograms.
Introduction to Statistical Quality Control, 4th Edition

5. 2-1.2 The Frequency Distribution and Histogram
Introduction to Statistical Quality Control, 4th Edition

6. Introduction to Statistical Quality Control, 4th Edition
Graphical Displays
What is the overall shape of the data?
Are there any unusual observations?
Where is the “center” or “average” of the data located?
What is the spread of the data? Is the data spread out or close to the center?
Introduction to Statistical Quality Control, 4th Edition

7. 2-1.3 Numerical Summary of Data
Important summary statistics for a distribution
of data can include:
Sample mean,
Sample variance, s2
Sample standard deviation, s
Sample median, M
Introduction to Statistical Quality Control, 4th Edition

8. 2-1.3 Numerical Summary of Data
For the data shown in the previous histogram and stem and leaf plot, the summary statistics are:
N Mean Median Var StDev
Introduction to Statistical Quality Control, 4th Edition

9. Introduction to Statistical Quality Control, 4th Edition
2-1.4 The Box Plot
The Box Plot is a graphical display that
provides important quantitative
information about a data set. Some of
this information is
Location or central tendency
Spread or variability
Departure from symmetry
Identification of “outliers”
Introduction to Statistical Quality Control, 4th Edition

10. Introduction to Statistical Quality Control, 4th Edition
The Box Plot
Introduction to Statistical Quality Control, 4th Edition

11. 2-1.5 Sample Computer Output
Introduction to Statistical Quality Control, 4th Edition

12. 2-1.6 Probability Distributions
Definitions
Sample A collection of measurements selected from some larger source or population.
Probability Distribution A mathematical model that relates the value of the variable with the probability of occurrence of that value in the population.
Random Variable variable that can take on different values in the population according to some “random” mechanism.
Introduction to Statistical Quality Control, 4th Edition

13. 2-1.6 Probability Distributions
Two Types of Probability Distributions
Continuous When a variable being measured is expressed on a continuous scale, its probability distribution is called a continuous distribution. The probability distribution of piston-ring diameter is continuous.
Discrete When the parameter being measured can only take on certain values, such as the integers 0, 1, 2, …, the probability distribution is called a discrete distribution. The distribution of the number of nonconformities would be a discrete distribution.
Introduction to Statistical Quality Control, 4th Edition

14. 2-2 Important Discrete Distributions
The Hypergeometric Distribution
The Binomial Distribution
The Poisson Distribution
The Pascal and Related Distributions
Introduction to Statistical Quality Control, 4th Edition

15. 2-2.2 The Binomial Distribution
A quality characteristic follows a binomial
distribution if:
1. All trials are independent.
2. Each outcome is either a “success” or “failure”.
The probability of success on any trial is given as p. The probability of a failure is 1- p.
4. The probability of a success is constant.
Introduction to Statistical Quality Control, 4th Edition

16. 2-2.2 The Binomial Distribution
The binomial distribution with parameters
n and 0 < p < 1, is
The mean and variance of the binomial distribution are
Introduction to Statistical Quality Control, 4th Edition

17. 2-2.3 The Poisson Distribution
The Poisson distribution is
Where the parameter  > 0. The mean and variance of the Poisson distribution are
Introduction to Statistical Quality Control, 4th Edition

18. 2-2.3 The Poisson Distribution
The Poisson distribution is useful in quality engineering
Typical model of the number of defects or nonconformities that occur in a unit of product.
Any random phenomenon that occurs on a “per unit” basis is often well approximated by the Poisson distribution.
Introduction to Statistical Quality Control, 4th Edition

19. 2-3 Important Continuous Distributions
The Normal Distribution
The Exponential Distribution
The Gamma Distribution
The Weibull Distribution
Introduction to Statistical Quality Control, 4th Edition

20. 2-3.1 The Normal Distribution
is an important
continuous distribution.
Symmetric, bell-shaped
Mean, 
Standard deviation, 
Introduction to Statistical Quality Control, 4th Edition

21. 2-3.1 The Normal Distribution
For a population that is
normally distributed:
approx. 68% of the data will lie within 1 standard deviation of the mean;
approx. 95% of the data will lie within 2 standard deviations of the mean, and
approx. 99.7% of the data will lie within 3 standard deviations of the mean.
Introduction to Statistical Quality Control, 4th Edition

22. 2-3.1 The Normal Distribution
Standard normal distribution
Many situations will involve data that is normally distributed. We will often want to find probabilities of events occurring or percentages of nonconformities, etc.. A standardized normal random variable is:
Introduction to Statistical Quality Control, 4th Edition

23. 2-3.1 The Normal Distribution
Standard normal distribution
Z is normally distributed with mean 0 and standard deviation, 1.
Use the standard normal distribution to find probabilities when the original population or sample of interest is normally distributed.
Tables, calculators are useful.
Introduction to Statistical Quality Control, 4th Edition

24. 2-3.2 The Normal Distribution
Example
The tensile strength of paper is modeled by a normal
distribution with a mean of 35 lbs/in2 and a standard
deviation of 2 lbs/in2.
What is the probability that the tensile strength of a sample is less than 40 lbs/in2?
If the specifications require the tensile strength to exceed 30 lbs/in2, what proportion of the samples is scrapped?
Introduction to Statistical Quality Control, 4th Edition

25. 2-3.3 The Exponential Distribution
The exponential distribution is widely used in the field of reliability engineering.
The exponential distribution is
The mean and variance are
Introduction to Statistical Quality Control, 4th Edition

26. 2-4 Some Useful Approximations
In certain quality control problems, it is sometimes useful to approximate one probability distribution with another. This is particularly useful if the original distribution is difficult to manipulate analytically.
Some approximations:
Binomial approximation to the hypergeometric
Poisson approximation to the binomial
Normal approximation to the binomial
Introduction to Statistical Quality Control, 4th Edition