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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
260250240230220210200190180170
7
6
5
4
3
2
1
0
C1
Freq
uenc
y
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 distributionof data can include:• Sample mean,
• Sample variance, s2
• Sample standard deviation, s
• Sample median, M
x
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
120.6120.35 120.9
120.1 121.3
Figure 2-5. Box plot for the aircraft wing leading edge diameter data in Table 2-4.
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 The Hypergeometric Distribution
2-2.2 The Binomial Distribution
2-2.3 The Poisson Distribution
2-2.4 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
The mean and variance of the binomial distribution are
p xn
xp px n x( ) ( )
1
np np p2 1( )
Introduction to Statistical Quality Control, 4th Edition
2-2.3 The Poisson Distribution
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
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 The Normal Distribution
2-3.2 The Exponential Distribution
2-3.3 The Gamma Distribution
2-3.4 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, bell-
shaped• Mean, • Standard deviation,
43210-1-2-3-4
x
Introduction to Statistical Quality Control, 4th Edition
2-3.1 The Normal Distribution
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
Introduction to Statistical Quality Control, 4th Edition
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:
Zx
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
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?
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
The mean and variance are
0x,e)x(p
22 11
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