SPC for Real-World Processes

William A. Levinson, P.E.

Intersil Corporation

Mountaintop, PA

What’s wrong with this picture?

Particle counts, semiconductor manufacturing process

Particles

frequency

0 20 40 60 800

10

20

30

40

50

60

• The normal distribution does not fit the histogram well.

• The normal curve suggests that there can be fewer than zero particles.

Particles

frequency

0 20 40 60 800

10

20

30

40

50

60

• Particle data, gamma distribution fit

(Cells at right were merged for the chi square test)

Frequency

34

00 64Particles

Practical Implications of Nonnormality

• Suppose the upper specification limit is 80 particles.• Normal Cpk = 1.916, ~4.5 ppb

out of specification• Actual: 844,000 ppb,

corresponds to 1.05 Cpk

• The Shewhart chart false alarm rate will be much too high.

Practical Implications: Central Limit Theorem

• Averages of large subgroups follow the normal distribution even if the underlying distribution is nonnormal.

• Problem 1: Individuals, not averages, are in or out of specification. Can’t use the CLT for process capability.

• Problem 2: Subgroups are not always available.

Nonnormal Applications

• Gamma distribution• Continuous analogue of the

Poisson, for “undesirable random arrivals”

• Semiconductor industry: particle counts

• Chemical impurity levels

• Weibull distribution• Transformable to extreme

value distribution; failure at the weakest point

Nonnormal Applications: Clues

• A one-sided specification• Upper limit for impurities or

particles• Lower limit for strength

• A physical boundary on the measurement, such as zero for impurities and particles• with a realistic chance that a

normal distribution's tail will exceed the boundary

Nonnormal Process Capability

• -1(p) =inverse cumulative standard normal distribution

• p = nonconforming population fraction

• CPL or CPU = --1(p)/3• The same capability index as a

normal distribution with the same nonconforming fraction

• There might be only one index (CPL or CPU, = Cpk)

Nonnormal SPC

• Lower and upper control limits: 0.00135 and 0.99865 quantiles of the distribution• Same false alarm risk as

Shewhart 3-sigma limits for a normal distribution

• Center line: median (50th percentile) of the distribution• Can use the Western Electric

zone C test for 8 consecutive points above or below

• Zones A and B also can be constructed.• A: 0.0227, 0.9773 quantiles• B: 0.1587, 0.8413 quantiles

0

80

1 21 41 61 81 101 121 141 161 181 201 221 241

Sample

Zone C

0.99865 quantile

Particle control chart

Chart Interpretation

• An out of control signal can be desirable if there was a deliberate effort to improve the process.

• Particle counts or impurities• Above UCL ==> assignable or

special cause• Below LCL (or below threshold

parameter) ==> improvement probably worked.

• No LCL? Use zone tests.

Fitting Nonnormal Distributions to Data

• The gamma and Weibull distributions have three parameters:• shape parameter• scale parameter• threshold parameter

• The threshold parameter must be selected; then the shape and scale parameters are calculated.

Threshold Parameter

• The threshold parameter is the minimum possible measurement.• Guarantee time (minimum life)

in reliability testing• Lowest realistic impurity or

particle count• Lowest possible material

strength

• Measurement <: assignable or special cause

Threshold parameter: selection

• Select the threshold parameter that maximizes the likelihood function.• Selection can be from a graph

of the likelihood function (or its log) versus threshold parameters.

• The threshold parameter cannot be more than the smallest measurement in the data set.

Threshold parameter selection, contd.

• For a range of threshold parameters:• Compute the shape and scale

parameter for each• Calculate the likelihood

function, or its natural log

n

i i

i

n

i

xfL

xfL

1

1

ln,,ln

,,

f(x) is the probability density functionthat uses the indicated shape, scale,and threshold parameters.

Likelihood plot for the particle data: zero is the best threshold parameter.

Threshold Parameter

Natural Log, Likelihood Function

Another application (breaking strength data), with nonzero threshold parameter

Natural Log, Likelihood Function

Threshold Parameter

Best

Shape and Scale Parameters

• Use the shape and scale parameters that go with the threshold parameter that yields the highest likelihood function.

• Test the fitted distribution for fit to the data set.

Goodness of Fit Tests

• ALWAYS test the fitted distribution for goodness of fit to the data!

• Quantile-Quantile (Q-Q) plot• Similar in principle to the

normal probability plot, Weibull plot

• Chi square test for goodness of fit

Quantile-Quantile Plot

• Order the raw data from smallest to largest:• x(1), x(2), …, x(n)

• Plot x(k) against the (k-0.5)/n quantile of the fitted distribution.• Slope should be about 1.• Intercept should be about 0.• Correlation should be good.• Points should scatter randomly

around the regression line.

Q-Q plot for the particle count data

Quantile-Quantile Plot and Outliers

• A point that is far from the quantile-quantile plot’s regression line, especially at its ends, is a likely outlier.

Chi Square Test for Goodness of Fit

• Compares the fitted distribution’s histogram to that of the data.

• For k cells,• Ei = expected count in each

cell (from the fitted distribution)

• Oi = observed count in each cell (from the raw data)

• k-3 degrees of freedom

k

ii

ii

EEO

1

22

Chi square test, particle data

14.23, 12 degrees of freedom, can’t reject gamma fit

Frequency

34

00 64Particles

Chi Square Test: Caveats

• The results of this test depend on cell selection.

• For n data, k=n0.5 cells is a good starting point.

• The expected count for each cell must be 5 or greater.• Combine cells if necessary.

Summary

• Many real-world processes do not follow the normal distribution.• Treating them as normal can

yield bad estimates for capability indices, and control charts with excessive false alarm rates.

Summary, contd.

• Clues to the presence of a nonnormal distribution include:• One-sided specification limit• A significant fraction of the bell

curve exceeds practical limits for the measurement.

• e.g. zero for impurities, particles

• The shape of the data histogram

Summary, contd.

• Computerized algorithms can fit nonnormal distributions to data.• These were not available on a

desktop basis until fairly recently.

• Always check the selected distribution for goodness of fit.

Summary, contd.

• Once the nonnormal distribution has been selected, the user can:• Set control chart limits that

give the same false alarm rates as a traditional Shewhart chart.

• Compute normal-equivalent process capability indices.