introduction to statistical quality control, 4th edition chapter 2 modeling process quality

Introduction to Statistical Quality Control, 4th Edition

Chapter 2

Modeling Process Quality


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.


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


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.


2-1.2 The Frequency Distribution and Histogram

• Histogram

260250240230220210200190180170

7

6

5

4

3

2

1

0

C1

Freq

uenc

y


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?


2-1.3 Numerical Summary of Data

Important summary statistics for a distributionof data can include:• Sample mean,

• Sample variance, s2

• Sample standard deviation, s

• Sample median, M

x


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


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”


2-1.4 The Box Plot

120.6120.35 120.9

120.1 121.3

Figure 2-5. Box plot for the aircraft wing leading edge diameter data in Table 2-4.


2-1.5 Sample Computer Output


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.


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.


2-2 Important Discrete Distributions

2-2.1 The Hypergeometric Distribution

2-2.2 The Binomial Distribution

2-2.3 The Poisson Distribution

2-2.4 The Pascal and Related Distributions



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.



The binomial distribution with parameters

n 0 and 0 < p < 1, is

The mean and variance of the binomial distribution are

p xn

xp px n x( ) ( )

1

np np p2 1( )



The Poisson distribution is

Where the parameter > 0. The mean and variance of the Poisson distribution are

,1,0x,!x

e)x(p

x

2



• 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.


2-3 Important Continuous Distributions

2-3.1 The Normal Distribution

2-3.2 The Exponential Distribution

2-3.3 The Gamma Distribution

2-3.4 The Weibull Distribution



The normal distribution

is an important

continuous distribution.• Symmetric, bell-

shaped• Mean, • Standard deviation,

43210-1-2-3-4

x



For a population that isnormally 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.

43210-1-2-3-4

x



• 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:

Zx



• 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.



ExampleThe 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?


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

0x,e)x(p

22 11


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 chapter 2 modeling process quality

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