Chapter 3 1Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Learning Objectives

Chapter 3 2Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 3Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Easy to find percentiles of the data; see page 69

Stem-and-Leaf Display3.1 Describing Variation

Chapter 3 4Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Also called a run chart

Plot of Data in Time OrderMarginal plot produced by MINITAB

Chapter 3 5Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Group values of the variable into bins, then count the number of observations that fall into each bin

Plot frequency (or relative frequency) versus the values of the variable

Histograms – Useful for large data sets

Chapter 3 6Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Additional Minitab Graphs

Chapter 3 7Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 8Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 9Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Numerical Summary of DataSample average:

Chapter 3 10Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The Standard Deviation

Chapter 3 11Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 12Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The Box Plot(or Box-and-Whisker Plot)

Chapter 3 13Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Comparative Box Plots

Chapter 3 14Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Probability Distributions

Chapter 3 15Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 16Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Will see many examples in the text

Sometimes called a probability mass function

Sometimes called a probability density function

Chapter 3 17Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 18Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 19Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The mean is the point at which the distribution exactly “balances”.

Chapter 3 20Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The mean is not necessarily the 50th percentile of the distribution (that’s the median)

The mean is not necessarily the most likely value of the random variable (that’s the mode)

Chapter 3 21Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 22Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The Hypergeometric Distribution

3.2 Important Discrete Distributions

Chapter 3 23Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Discrete distributions are used frequently in designing acceptance sampling plans – see Chapter 15

Chapter 3 24Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 25Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Basis is in Bernoulli trials

The random variable x is the number of successes out of n Bernoulli trials with constant probability of success p on each trial

The Binomial Distribution

Chapter 3 26Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Binomial Distributions

Chapter 3 27Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 28Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 29Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Frequently used as a model for count data

The Poisson Distribution

Chapter 3 30Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 31Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 32Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The random variable x is the number of Bernoulli trials upon which the rth success occurs

The Negative Binomial Distribution

Chapter 3 33Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

•The negative binomial distribution is also sometimes called the Pascal distribution

•When r = 1 the negative binomial distribution is known as the geometric distribution

• The geometric distribution has many useful applications in SQC

Geometric Distribution

Chapter 3 34Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 35Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The Normal Distribution3.3 Important Continuous Distributions

Chapter 3 36Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 37Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 38Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 39Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 40Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The Central Limit Theorem

Chapter 3 41Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Practical interpretation – the sum of independent random variables is approximately normally distributed regardless of the distribution of each individual random variable in the sum

Chapter 3 42Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The Lognormal Distribution

Chapter 3 43Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 44Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 45Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The Exponential Distribution

Chapter 3 46Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Relationship between the Poisson and exponential distributions

Lack-of-memory property

Chapter 3 47Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 48Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The Gamma Distribution

Chapter 3 49Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 50Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

When r is an integer, the gamma distribution is the result of summing r independently and identically exponential random variables each with parameter λ.

The gamma distribution has many applications in reliability engineering.

Chapter 3 51Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The Weibull Distribution

Chapter 3 52Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

When β = 1, the Weibull reduces to

the exponential

An Application of the Weibull Distribution

Chapter 3 53Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 54Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

• Determining if a sample of data might reasonably be assumed to come from a specific distribution

• Probability plots are available for various distributions

• Easy to construct with computer software (MINITAB)

• Subjective interpretation

3.4 Probability Plots

Normal Probability Plot

Chapter 3 55Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 56Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

The Normal Probability Plot on Standard Graph Paper

Chapter 3 57Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 58Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Other Probability Plots

• What is a reasonable choice as a probability model for these data?

Chapter 3 59Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 60Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

3.5 Some Useful Approximations

Chapter 3 61Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Chapter 3 62Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.

Learning Objectives