Download Modeling Process Quality

ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Modeling Process Quality
Statistical methodology plays an important role in quality control and
quality improvement.
Descriptive statistics of a sample of data display the variation in a
quality characteristic.
Probability distributions are our main tool for modeling the variation
in a quality characteristic.
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Modeling Process Quality
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Describing Variation
Characteristics of the quality of goods and services always show
variability.
Examples:
Net content of a can of a soft drink, when measured precisely;
Time to complete an on-line banking transaction.
We shall use both graphical and numerical tools to describe this
variability.
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Stem-and-Leaf Plot
The stem-and-leaf display is a useful pencil-and-paper technique.
Shows the distribution of data, like a histogram.
Makes it easy to find the median, quartiles, and other quantiles
(percentiles).
Example 3.1
Table 3.1, Days to pay a health insurance claim
In R:
Table03p01 <- read.csv("Data/Table-03-01.csv");
stem(Table03p01$Days)
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Time Series Plot
The stem-and-leaf plot shows many aspects of the data, but not the
time order.
The time series plot is the same as a control chart. It can show:
Shift in the mean;
Trend;
Change in variability.
In R:
plot(Days ~ Claim, data = Table03p01)
# or simply plot(Table03p01$Days)
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Histogram
The number of leaves on each stem in a stem-and-leaf plot shows the
distribution of the data.
The histogram shows similar counts, with fewer restrictions on the
bins.
In R:
hist(Table03p01$Days)
hist(Table03p01$Days, right = FALSE) # same counts as in stem
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Example 3.2
Table 3.2, thickness of vapor deposition layers on silicon
In R:
Table03p02 <- read.csv("Data/Table-03-02.csv");
hist(Table03p02$Thickness)
# To look like Figure 3.3:
hist(Table03p02$Thickness, right = FALSE,
col = "brown", border = "white")
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Numerical Summary of Data
Graphical summaries like the stem-and-leaf plot and the histogram
give a visual impression of the distribution of a set of data.
Two key properties are:
Central tendency;
Dispersion.
We also need numerical summaries of these properties.
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
For a set of observations x1 , x2 , . . . , xn , numerical summaries of
central tendency include:
The sample average
Pn
xi
x1 + x2 + · · · + xn
x̄ =
= i=1 ;
n
n
The sample median;
Other trimmed means.
In R:
mean(Table03p01$Days)
# 33.25
median(Table03p02$Thickness)
# 450
mean(Table03p02$Thickness, trim = 0.25) # 449.94
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
The most common numerical
√ summary of dispersion is the sample
standard deviation s = s 2 , where
n
X
s2 =
(xi − x̄)2
i−1
n−1
is the sample variance.
In R:
sd(Table03p01$Days)
# 9.374679
sd(Table03p02$Thickness) # 13.42732
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Box Plot
The box plot is a graphical display that shows 5 useful numerical
summaries:
The median;
The two quartiles;
The two extremes.
It often also shows possible outliers.
In R:
# no outliers, whiskers extend to extremes:
boxplot(Table03p01$Days)
# possible outliers, whiskers extend only to non-outliers:
boxplot(Table03p02$Thickness)
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Comparison Box Plots
Box plots are also useful for comparing multiple sets of data.
In R:
batteries <- read.csv("Data/batteries.csv");
boxplot(Life ~ Temperature, data = batteries)
When a control chart is based on samples of characteristics, a
comparison box plot is a good way to look for shifts in center or
dispersion, possible outliers, and so on.
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Probability Distributions
We shall be working with two kinds of distribution:
Discrete distributions, characterized by a probability mass
function (pmf), like the binomial distribution; the pmf is
p(xi ) = P(X = xi ) for each possible value xi .
Continuous distributions, characterized by a probability density
function (pdf), like the normal distribution; the pdf f (x) is
interpreted through integrals:
Z b
P(a ≤ X ≤ b) =
f (x)dx.
a
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Mean and Variance
The mean µ and variance σ 2 of a probability distribution are defined
similarly to those of a sample:
P
 ∞ xi p(xi ) discrete X
i=1
µ = R∞

xf (x)dx continuous X
−∞
P
 ∞ (xi − µ)2 p(xi ) discrete X
i=1
2
σ = R∞

(x − µ)2 f (x)dx continuous X
−∞
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Examples
# binomial pmf:
plot(0:10, dbinom(0:10, size = 10, prob = 0.4), type = "h");
points(0:10, dbinom(0:10, size = 10, prob = 0.4))
# normal pdf:
curve(dnorm(x), from = -3, to = 3)
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Modeling Process Quality
Describing Variation
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Some Discrete Distributions
Hypergeometric Distribution
Sampling without replacement from a finite population.
Population size N, of which D have some characteristic of
interest.
Sample size n. The random variable X is the number in the
sample that are found to have that characteristic.
Then the probability mass function is
D
N −D
x
n−x
p(x) ==
, max(0, n + D − N) ≤ x ≤ min(n, D)
N
n
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Modeling Process Quality
Important Discrete Distributions
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
The mean and variance are
E(X ) =
and
nD
Var(X ) =
N
nD
N
D
N −n
1−
N
N −1
Application
Acceptance sampling: N is lot size, D is number defective in the lot.
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Modeling Process Quality
Important Discrete Distributions
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Binomial Distribution
Sampling with replacement from a finite population, or sampling
from an infinite population.
A sequence of n independent trials, each of which results in
“success” or “failure”.
The probability of success in each trial is p.
The random variable X is the number of successes found in the
n trials.
Then the probability mass function is
n x
p(x) = P(X = x) =
p (1 − p)n−x , 0 ≤ x ≤ n
x
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Modeling Process Quality
Important Discrete Distributions
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
The mean and variance are
E(X ) = np
and
Var(X ) = np(1 − p)
Sample fraction
The quantity p̂ =
X
n
is the fraction of trials that result in success.
p̂ is the sample analog of the population probability p, and is the
natural estimator of it.
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Modeling Process Quality
Important Discrete Distributions
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Poisson Distribution
Simplest model for random counts with no upper limit.
For example, the number of blemishes in a new car’s paint.
The probability mass function is
p(x) = P(X = x) =
e −λ λx
,x ≥ 0
x!
where λ is a positive parameter.
The mean and variance are
E(X ) = Var(X ) = λ
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Modeling Process Quality
Important Discrete Distributions
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
Negative Binomial and Geometric Distribution
Recall the sequence of independent trials, each with success
probability p.
For some r > 0, the random variable X is the number of the trial
when the r th success is seen.
Then the probability mass function is
x −1 r
p(x) = P(X = x) =
p (1 − p)x−r , x ≥ r
r −1
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Modeling Process Quality
Important Discrete Distributions
ST 435/535
Statistical Methods for Quality and Productivity Improvement / Statistical Process Control
The mean and variance are
E(X ) =
and
Var(X ) =
r
p
r (1 − p)
p2
In the special case r = 1, that is waiting for the first success, the pmf
simplifies to
p(x) = p(1 − p)x−1 , x ≥ 1,
the Geometric Distribution.
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Modeling Process Quality
Important Discrete Distributions