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3.1 Describing Variation Stem-and-Leaf Display Easy to find percentiles of the data; see page 69 2
Plot of Data in Time Order Marginal plot produced by MINITAB Also called a run chart 3
Histograms Useful for large data sets 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 4
10 bins 5
Additional Minitab Graphs 15 bins 6
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Numerical Summary of Data Sample average: 8
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The Standard Deviation 10
The Box Plot (or Box-and-Whisker Plot) 120.1 1 120.2 2 120.3 3 120.4 4 120.5 5 120.5 6 120.7 7 120.8 8 120.9 9 120.9 10 121.1 11 121.3 12 11
Comparative Box Plots 12
Probability Distributions 13
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Sometimes called a probability mass function Sometimes called a probability density function Will see many examples in the text 15
1 2 3 4 5 6 7 8. 25 0.01 0.99 0.99 0.99 0.99 0.99 0.99 0.99. 0.99 16
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The mean is the point at which the distribution exactly balances. x P(x) 1 2 3 4 0.1 0.5 0.3 0.1 x p(x) xp(x) 1 0.1 0.1 2 0.5 1 3 0.3 0.9 4 0.2 0.8 SUM 2.8 18
The mean is not necessarily the 50 th 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) 19
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Discrete Distributions The Hypergeometric Distribution N (50 Marbles) D (20 RED) N-D (30 GREEN) Pick n marbles without replacement Looking for p(x) No. of red in n 21
3.2 Important Discrete Distributions The Hypergeometric Distribution 22
Discrete distributions are used frequently in designing acceptance sampling plans see Chapter 15 23
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The Binomial Distribution 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 26
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Binomial Distributions 28
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The Poisson Distribution Frequently used as a model for count data 30
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The Negative Binomial Distribution The random variable x is the number of Bernoulli trials upon which the rth success occurs 33
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 34
Geometric Distribution 35
3.3 Important Continuous Distributions The Normal Distribution 36
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The Central Limit Theorem Practical interpretation the sum of independent random variables is approximately normally distributed regardless of the distribution of each individual random variable in the sum 42
The Lognormal Distribution 43
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The Exponential Distribution 46
Relationship between the Poisson and exponential distributions 47
Lack-of-memory property 48
The Gamma Distribution 49
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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. 51
The Weibull Distribution Chapter 3 Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. 52 Copyright (c) 2012 John Wiley & Sons, Inc.
When β = 1, the Weibull reduces to the exponential 53
An Application of the Weibull Distribution 54
3.4 Probability Plots 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 55
Normal Probability Plot 56
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The Normal Probability Plot on Standard Graph Paper 58
Other Probability Plots What is a reasonable choice as a probability model for these data? 59
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3.5 Some Useful Approximations 61
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