Modeling Process Quality Statistical methodology plays an important - - PowerPoint PPT Presentation

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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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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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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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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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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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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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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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 ¯ x = x1 + x2 + · · · + xn n = n

i=1 xi

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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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 = √ s2, where s2 =

n

• i−1

(xi − ¯ x)2 n − 1 is the sample variance. In R:

sd(Table03p01$Days) # 9.374679 sd(Table03p02$Thickness) # 13.42732

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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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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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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: P(a ≤ X ≤ b) = b

a

f (x)dx.

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Mean and Variance The mean µ and variance σ2 of a probability distribution are defined similarly to those of a sample: µ =    ∞

i=1 xip(xi)

discrete X ∞

−∞ xf (x)dx

continuous X σ2 =    ∞

i=1(xi − µ)2p(xi)

discrete X ∞

−∞(x − µ)2f (x)dx

continuous X

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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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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 p(x) == D x N − D n − x

• N

n

• , max(0, n + D − N) ≤ x ≤ min(n, D)

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

The mean and variance are E(X) = nD N and Var(X) = nD N

• 1 − D

N N − n N − 1

• Application

Acceptance sampling: N is lot size, D is number defective in the lot.

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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 p(x) = P(X = x) = n x

• px(1 − p)n−x, 0 ≤ x ≤ n

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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 ˆ p = X

n is the fraction of trials that result in success.

ˆ p is the sample analog of the population probability p, and is the natural estimator of it.

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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−λλx x! , x ≥ 0 where λ is a positive parameter. The mean and variance are E(X) = Var(X) = λ

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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 p(x) = P(X = x) = x − 1 r − 1

• pr(1 − p)x−r, x ≥ r

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

The mean and variance are E(X) = r p and Var(X) = 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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