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Chapter 5. Control Charts for Variables

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Control Charts for andx R

1 2

: quantity of interest ( , )

, , : samples of n

x x N

x x x x

,x Nn

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Subgroup Data with Unknown and

1 2, , : ranges of samplesmR R R m

: grand average of , best estimate forx x

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Phase I Application of and RCharts

Equations 5-4 and 5-5 are trial control limits. Determined from m initial samples.

Typically 20-25 subgroups of size n between 3 and 5.

Any out-of-control points should be examined for assignablecauses.

If assignable causes are found, discard points from calculationsand revise the trial control limits.

Continue examination until all points plot in control.

Adopt resulting trial control limits for use.

If no assignable cause is found, there are two options.

1. Eliminate point as if an assignable cause were found and revise limits.2. Retain point and consider limits appropriate for control.

If there are many out-of-control points they should be examinedfor patterns that may identify underlying process problems.

x

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Example 5-1

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Assume spec tolerance is 1.5 +/- 0.5 micron.

Nonconformance probability:

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Cp: Process Capability Ration (PCR)

2

Note: 6 spread is the basic definition of process capability. 3 above mean and 3 below.

R If is unknown, we can use = . in the example is 0.1398.

d

P: % of specification band the process uses up. Pcan be estimated as:

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Phase II Operation of Charts

Use of control chart for monitoring future production, after a set ofreliable limits are established, is calledphase IIof control chart

usage (Figure 5-4).

A run chart showing individuals observations in each sample, called

a tolerance chart or tier diagram (Figure 5-5), may reveal patterns or

unusual observations in the data.

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Control vs. Specification Limits

Control limits are derived fromnatural process variability, orthe natural tolerance limits of aprocess.

Specification limits aredetermined externally, forexample by customers ordesigners.

There is no mathematical orstatistical relationship betweenthe control limits and thespecification limits.

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Guidelines for Control Chart Design

Control chart design requires specification of sample size, controllimit width, and sampling frequency.

Exact solution requires detailed information on statistical characteristicsas well as economic factors.

The problem of choosing sample size and sampling frequency is one of

allocating sampling effort. For chart, choose as small a sample size consistent with

magnitude of process shift one is trying to detect. For moderate tolarge shifts, relatively small samples are effective. For small shifts,larger samples are needed.

For small samples, Rchart is relatively insensitive to changes in

process standard deviation. For larger samples (n > 10 or 12), s ors2 charts are better choices.

NOTE: Skip Section on Changing Sample Size (pages 209-212)

x

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Charts Based on Standard Values

D1 = d2- 3d3D2= d2+ 3d3d2 : mean of distribution of relative range

d3 : standard deviation of distribution of relative range

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Interpretation of and Chartsx R

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An assumption in performance properties is that the underlyingdistribution of quality characteristic is normal.

If underlying distribution is not normal, sampling distributions can bederived and exact probability limits obtained.

Usual normal theory control limits are very robust to normalityassumption.

In most cases, samples of size 4 or 5 are sufficient to ensurereasonable robustness to normality assumption for chart.

Sampling distribution ofRis not symmetric, thus symmetric 3-sigma

limits are an approximation and

-risk is not 0.0027. Rchart is moresensitive to departures from normality than chart.

Assumptions of normality and independence are not a primaryconcern in Phase I.

x

x

Effect of Nonnormality on and Chartsx R

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Operating Characteristic (OC) Function

is known. In-control mean: 0 out of control mean: 1 = 0+ k

Probability of not detecting shift: -risk

L: number ofs

ForL = 3, n = 5, k = 2.

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Average run length (r): shift is detected in the rth sample.

In the example.

Expected number of samples for detecting shift = 4.

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Average Run Length for Chartx

For Shewhart control chart:

Average time to signal (ATS)

Average number of individual units sampled for detection (I)

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Use the and charts instead of the and charts when:x s x R

Control Charts for andx s

2 2

5 4 4 6 4 43 1 and 3 1B c c B c c

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th

Assume no standard is given for . Need to estimate.

preliminary samples, each of size .

: standard deviation for samplei

m n

s i

4

: unbiased estimator for

chart has the following parameters:

S

c

s

2 2

3 4 4 4

4 4

3 3Note: 1 1 and 1 1 Then:B c B cc c

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4

When is used to estimate , chart has the following parameters:S

xc

3

4

3Define . Then:Ac n

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Example 5-3

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For chart:x For chart:s

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and Control Charts with Variable Sample Sizex s

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Example 5-4

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For chart:x For chart:s

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2Control Charts

Sometimes it is desired to use s2chart overs chart.

The parameters fors2chart are:

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Shewhart Control Chart for Individual Measurements

What if there is only one observation for each sample.

Use the moving range between two successive samples for range.

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Example 5-5

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Use the d2, D3 and D4 values from n = 2row for individual measurements.

Then:

Ph II O i d I i f Ch

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Phase II Operation and Interpretation of Charts

shift