Chapter 3. Modeling Process Quality

Stem-and-Leaf Displaynumbers abovenumbers above

numbers within

• Percentile• Sample median or fiftieth percentile• First quartile (Q1) third quartile (Q3)

numbers below

• First quartile (Q1), third quartile (Q3)• Interquartile range (Q3-Q1)

Plot of Data in Time OrderMarginal plot produced by MINITAB

Histograms – Useful for large data setsHistograms  Useful for large data sets

Group values of the variable into bins then count the•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

o10Units in A (=10 meters)−

Histogram for discrete

d tdata

Numerical Summary of DataNumerical Summary of Data

Samples: x1, x2, x3, … , xnSamples: x1, x2, x3, … , xn

Sample average:

Previous wafer thickness example:

Sample VarianceSample Variance:

Sample Standard Deviation:

Previous wafer thickness example:

The Box Plot( d hi k l )(or Box‐and‐Whisker Plot)

minimum maximumQ3Sample medianQ1minimum maximum

Box plots can identify potential outliers.

Comparative Box PlotsComparative Box Plots

Probability DistributionsProbability Distributions• Statistics: based on sample analysis involving measurements• Probability: based on mathematical description of abstract model• Random variable: (real) value associated with variable of interest• Probability distribution: probability of occurrence of variable

Sometimes called a Sometimes called a probability mass

functionprobability density

function

Error. Should be25 25!

!(25 )!x x x⎛ ⎞

=⎜ ⎟ −⎝ ⎠

Mean

Variance

N: populationD: class of interest within populationn: random samples chosen from Nn: random samples chosen from Nx: belongs to class of interest among random samples

Example: A lot containing 100 products. There are five defects. Choose 10 random samples Find the probability that one or fewer defectsChoose 10 random samples. Find the probability that one or fewer defectsare contained in the sample.

N = 100D 5D = 5n = 10x ≤ 1

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

Example: n = 15, p = 0.1:

Error: This should be p(x).

Sample fraction defectiveSample fraction defective

ˆ is an estimator of .p p

[ ] is the largest integer less than or equal to .na na

( )2ˆ

1ˆˆMean of . Variance of p = p

p pp p σ

−= =p pp p

n

As λ gets large, p(x) looks more symmetric, i.e., looks like binomial.Binomial is an approximation for limiting Poisson distributionBinomial is an approximation for limiting Poisson distribution.n→∞ and p →0 such that np = λ, binomial distribution approximates Poisson distribution with λ.

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

• When r = 1 the Pascal distribution is• When r = 1 the Pascal distribution is known as the geometric distribution.

• The geometric distribution has many useful applications in statistical quality control.

2 2( , ) : is normally distributed with mean and variance .x N xµ σ µ σ∼

where

and Φ(.) is a standard normal distribution with mean 0 and t d d d i ti 1standard deviation 1.

Original normalOriginal normal distribution

Standard normal distribution

( )2

1 1 2 2 3 3

Let , and for i 1, 2,..., are independent.

Then, ...i i i i

n n

x N x n

y a x a x a x a x

µ σ∼ =

= + + + +

( )2 2 2 2 2 2 2 21 1 2 2 3 3 1 1 2 2 3 3 ... , ...n n n nN a a a a a a a aµ µ µ µ σ σ σ σ∼ + + + + + + + +

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

• When r is an integer, the gamma distribution is the result f i i d d tl d id ti ll ti lof summing r independently and identically exponential

random variables each with parameter λ

• When β = 1, the Weibull distribution d t th ti l di t ib tireduces to the exponential distribution

Probability Plot

• Determining if a sample of data might reasonably be

Probability Plot

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

• Probability plots are available for various y pdistributions

• Easy to construct with computer software (MINITAB)y p

• Subjective interpretation

Normal Probability Plot

Other Probability PlotsOther Probability Plots

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

Approximations: H = hypergeometric, B = binomial, P = Poisson, N = normal