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Chapter 2
Modeling Process Quality
Introduction to Statistical Quality Control,
4th Edition
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
2-1.1 The Stem-and-Leaf Plot
• 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
17| 558
18| 357
19| 00445589
20| 1399
21| 00238
22| 005
23| 5678
24| 1555899
25| 158
Introduction to Statistical Quality Control,
4th Edition
2-1.2 The Frequency Distribution
and Histogram
• Frequency Distribution
– 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
2-1.2 The Frequency Distribution
and Histogram
• Histogram
7
6
Frequency
5
4
3
2
1
0
170
180
190
200
210
220
230
240
250
260
C1
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
2-1.3 Numerical Summary of Data
Important summary statistics for a distribution
of data can include:
• Sample mean, x
• Sample variance, s2
• Sample standard deviation, s
• Sample median, M
Introduction to Statistical Quality Control,
4th Edition
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
40 215.50 211.00 634.5 25.19
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
2-1.4 The Box Plot
120.35
120.1
120.6
120.9
121.3
Figure 2-5. Box plot for the aircraft wing leading edge diameter data in Table 2-4.
Introduction to Statistical Quality Control,
4th Edition
2-1.5 Sample Computer Output
Introduction to Statistical Quality Control,
4th Edition
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
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
2-2 Important Discrete Distributions
2-2.1
2-2.2
2-2.3
2-2.4
The Hypergeometric Distribution
The Binomial Distribution
The Poisson Distribution
The Pascal and Related Distributions
Introduction to Statistical Quality Control,
4th Edition
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”.
3. 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
2-2.2 The Binomial Distribution
The binomial distribution with parameters
n  0 and 0 < p < 1, is
 n x
p( x)    p (1  p)n x
 x
The mean and variance of the binomial
distribution are
  np
 2  np(1  p)
Introduction to Statistical Quality Control,
4th Edition
2-2.3 The Poisson Distribution
The Poisson distribution is
e   x
p( x ) 
,
x!
x  0,1,
Where the parameter  > 0. The mean and
variance of the Poisson distribution are

 
2
Introduction to Statistical Quality Control,
4th Edition
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
2-3 Important Continuous
Distributions
2-3.1
2-3.2
2-3.3
2-3.4
The Normal Distribution
The Exponential Distribution
The Gamma Distribution
The Weibull Distribution
Introduction to Statistical Quality Control,
4th Edition
2-3.1 The Normal Distribution
The normal distribution
is an important
continuous distribution.
• Symmetric, bellshaped
• Mean, 
• Standard deviation, 
-4
-3
-2
-1
Introduction to Statistical Quality Control,
4th Edition
0
x
1
2
3
4
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.
-4
-3
-2
Introduction to Statistical Quality Control,
4th Edition
-1
0
x
1
2
3
4
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 occuring or percentages of nonconformities, etc..
A standardized normal random variable is:
x 
Z

Introduction to Statistical Quality Control,
4th Edition
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
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.
a) What is the probability that the tensile strength of a
sample is less than 40 lbs/in2?
b) 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
2-3.3 The Exponential Distribution
•
•
The exponential distribution is widely used in
the field of reliability engineering.
The exponential distribution is
p(x)  e ,
x0
The mean and variance are
1


 
2
1
2
Introduction to Statistical Quality Control,
4th Edition
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