14 statistical process control

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Lesson 14 – Statistical Process Control Copyright – Harland E. Hodges, Ph.D 14-1 14 - 1 purpose is to assure that processes are performing in an acceptable manner 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 UCL LCL Sample number Center Out of Control Lesson 14 Statistical Process Control 14 - 2 Least Progressive Most Progressive Acceptance Sampling Inspection Before/After Production Process Control Corrective Action During Production Continuous Improvement Quality Built Into The Process Quality Assurance – An Evolutionary Process Developing an effective quality assurance program is an evolutionary process beginning with the quality assurance of incoming materials … evolving to total quality management. 14 - 3 Quality Control Concepts The manufacturing manager of the Hiney Winery is responsible for filling the Tiny Hiney wine bottles with 16 oz of wine. If the wine bottles are too full the Hiney Winery looses money and if they are do not contain 16 ounces their customers get upset. The steps involved in filling the Heiney Wine bottles is called a process. The process involves people, machinery, bottles, corks, etc. Do you think all Tiny Hiney’s will contain exactly 16 ounces of wine? Why or Why Not? 16 oz Tiny Hiney 16 oz Tiny Hiney 16 oz Tiny Hiney 16 oz Tiny Hiney

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Page 1: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-1

14 - 1

purpose is to assure that processes are performing in an acceptable manner

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

UCL

LCL

Sample number

Center

Out of Control

Lesson 14Statistical Process Control

14 - 2

Least Progressive Most Progressive

Acceptance Sampling

Inspection Before/After Production

Process Control

Corrective Action During

Production

Continuous Improvement

Quality Built Into The Process

Quality Assurance – An Evolutionary Process

Developing an effective quality assurance program is an evolutionary process beginning with the quality assurance of incoming materials …evolving to total quality management.

14 - 3

Quality Control Concepts

The manufacturing manager of the Hiney Winery is responsible for filling the Tiny Hiney wine bottles with 16 oz of wine. If the wine bottles are too full the Hiney Winery looses money and if they are do not contain 16 ounces their customers get upset.

The steps involved in filling the Heiney Wine bottles is called a process. The processinvolves people, machinery, bottles, corks, etc.

Do you think all Tiny Hiney’s will contain exactly 16 ounces of wine? Why or Why Not?

16 oz

Tiny Hiney

16 oz

Tiny Hiney

16 oz

Tiny Hiney

16 oz

Tiny Hiney

Page 2: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-2

14 - 4

Which sample has less variability?

16 oz.

Quality Control Concepts

16 oz.

Tiny Hiney

14 - 5

We measure process variability based on a sample statistic (mean, range, or proportion). The goal of statistical process control is to monitor and reduce process variability.

YY

TimeTime

16 oz.

All processes will vary over time!

Quality Control Concepts

14 - 6

Random or Natural Variation Random or Natural Variation (In Control)(In Control)

Special or Assignable Variation (Out of Control)

People

Machines

Materials

Methods

Measurement

Environment

What causes a process to vary?

Can you think of an example of random and assignable variation for each of the causes?

Can you think of a way to reduce it?

Page 3: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-3

14 - 7

Random or Assignable Variation? Why?

Quality Control Concepts

16 oz.

16 oz.

Tiny Hiney

14 - 8

Assignable Variation

Correctable problems

Not part of process design

Due to machine wear, unskilled workers, poor material, etc.

16 oz.

Can only be detected in a process which has stable or constant variation. The goal of process quality control is to identify assignable causes of variation and eliminate them. Random variation is much harder to identify; however, the goal remains to improve variability.

14 - 9

It is often impossible, impractical and cost prohibitive to inspect each and every element of production therefore statistical methods based on probability of failure are necessary to achieve a cost effective method of assuring quality.

. Inspection of inputs/outputs … Acceptance Sampling

. Inspection during the transformation process … ProcessControl

Acceptance Sampling

Process Control

Acceptance Sampling

Monitoring Production (Inspection)

Transformation OutputsOutputsInputs Transformation

process

Page 4: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-4

14 - 10

. How much? How Often?

. Where (at what points) should inspection occur?

. Centralized or On-site inspection?

The traditional view of the amount of inspection was that there was an optimum (cost) amount of inspection necessary to minimize risk of passing a defective. This was predicated on the facts:

. As the amount of inspection increases the defectives that “get through” will decrease (ie. The “cost of passing defectives go down”)

. As the amount of inspection increases the cost of inspection increases

The current view is that every effort involved in reducing the number of defectives will reduce costs.

Issues Involved In Inspection

14 - 11

Cost of inspection (goes up as amount increases)

Cost of passing defectives (goes down as amount increases)

Total Cost

Cost

Optimal Amount of Inspection

Traditional View – How Much To Inspect

14 - 12

How much to inspect and how often the inspection is done depends. On the item

.. Low cost, high volume items - little inspection (e.g. paper clips, pencils, glassware, etc)

.. High cost, low volume items with a high cost of a passing defective - more inspection (e.g. space shuttle)

. On the processes that produce the item.. Reliability of the equipment .. Reliability of the human element .. The stability of the process … stable processes

(those that infrequently go “out of control”) require less inspection than unstable processes

. On the lot (batch) size … large volumes of small lots will require more inspection than small volumes of large lots

Issues Involved In Inspection

Page 5: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-5

14 - 13

Where to inspect depends on. Raw materials and purchased parts. Finished products. Before a costly operation (don’t add costs to defective items). Before a covering process such as painting or plating. At points where there is a high variability in the output either as a result of mechanical or human variability

. Before an irreversible process

Issues Involved In Inspection

14 - 14

Centralized or On-site Inspection decisions are important. Centralized makes sense in retail environments, medical environments, when specialized testing equipment is required

. On-site makes sense when quicker decisions are necessary to ensure processes (mechanical/human) remain in control

Issues Involved In Inspection

Transformation Location 3Corporate Headquarters Transformation Location 4

Transformation Location 2Transformation Location 1

14 - 15

Type ofbusiness

Inspectionpoints

Characteristics

Fast Food CashierCounter areaEating areaBuildingKitchen

AccuracyAppearance, productivityCleanlinessAppearanceHealth regulations

Hotel/motel Parking lotAccountingBuildingMain desk

Safe, well lightedAccuracy, timelinessAppearance, safetyWaiting times

Supermarket CashiersDeliveries

Accuracy, courtesyQuality, quantity

Example Of Inspection – Service Business

Page 6: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-6

14 - 16

Statistical Process Control (SPC) is a methodology using statistical calculations to ensure a process is producing products which conform to quality standards (design based, manufacturing based, customer based).

Major activities. Monitor process variation. Diagnose causes of variation. Eliminate or reduce causes of variation

Designed to keep or bring a process into statistical control.

Statistical Process Control

14 - 17

Statistical Process Control involves statistical evaluation of a process throughout the whole production cycle. The control process requires the following steps:

. Define in sufficient detail the characteristics to be controlled, what “in” and “out” of control means (ie. What is the quality conformance standard?)

. Measure the characteristic (based on sample size data)

. Compare it to the standard

. Evaluate the results

. Take corrective action if necessary

. Evaluate the corrective action

Statistical Process Control

14 - 18

Produce Good or Service

Stop Process

YesYes

NoNo

Take Sample

Inspect Sample

Create/Update Control Chart

Start

Find Out Why

Special Causes

Statistical Process Control Steps

Page 7: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-7

14 - 19

All processes possess exhibit a natural (random) variability which are produced by a number of minor factors. If any one of these factors could be identified and eliminated, the change to the natural variability would be negligible.

An assignable variability is where a factor which contributes to the variability of a process can be identified and quantified. Usually this type of variation can be eliminated thus reducing the variability of the process.

When the measurements exhibit random patterns the process is considered to be in control. Non-random variation typically means that a process is out of control.

Process Variability

14 - 20

Process Variability can be measured and a probability distribution can be calculated … it might look something like this

Which process exhibits more stability (less variation)?

When the factors influencing process variability can be identified and eliminated the probability distribution will exhibit less variability

Statistical Process Control

14 - 21

Samplingdistribution

Processdistribution

Mean

The Central Limit Theorem states that (for large samples) the sampling distribution is approximately Normal regardless of the process distribution; therefore, the Normal Distribution can be used to evaluate measurement results and to determine whether or not a process is “in control”.

As the sample size increases, the variation in the sampling distribution decreases

The Normal Distribution and SPC

Page 8: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-8

14 - 22

Mean−3σ −2σ +2σ +3σ

95.5%

99.7%

σ = Standard deviation

The standard deviation (a measure of variability) of the sampling distribution is key to the statistical process control methods. 3standard deviations (99.7% confidence) are typically used to establish Control Limits (LCL, UCL) used to monitor process control.

The Normal Distribution and SPC

14 - 23

Samplingdistribution

Processdistribution

Mean

LCL UCL

In statistical process control, we are concerned with the samplingdistribution. Control Limits (LCL, UCL) are set using the standard deviation of the sampling distribution.

Note: When control limits are established some probability still exists in the tails.

Control Limits

14 - 24

MeanLCL UCL

α/2 α/2

α = Probabilityof Type I error Sample measurements beyond the LCL or UCL

suggest that a process is Out of Control

Sample measurements between the LCL and UCL usually indicate that a process is in control. The probability of concluding a process is out of control when it is actually is in control is called a Type I error. The amount of probability ( ) left in the tail is referred to as the probability of making a Type 1 error

In Control or Out of Control?

In Control

α

Page 9: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-9

14 - 25

Process Control Chart A Process Control Chart is a time series showing the sample statistics and their relationship to the Lower Control Limit (LCL) and Upper Control Limit (LCL). Once a control chart is established it can be used to monitor a process for future samples. The Process Control Chart may need to be re-evaluated if improvements are made.

YY

TimeTime

UCL

LCL

Center

14 - 26

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

UCL

LCL

Sample number

Center

We expect to see normal or random variation due to chance for measurements within the control limits

Measurements outside the control limits may indicate abnormal variation usually due to assignable sources

Using A Process Control Chart

14 - 27

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

UCL

LCL

Sample number

Center

We expect to see normal or random variation patterns within the control limits; however, this does not guarantee a process is in control. Consider the following example. What should a quality control manager do in this situation?

Using A Process Control Chart

Page 10: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-10

14 - 28

In Control or Out of ControlA process is deemed to be in control if the following two conditions are met.

1. Sample statistics are between the control limitsControl Charts are used to test this condition

2. There are no non-random patterns presentPattern analysis rules are used to test this condition

Otherwise, the process is deemed to be out of control.

14 - 29

Four commonly used control charts

. Control charts for variablesvariables (measured or continuous) . Range chart . Range chart - monitor variation (dispersion). Mean charts. Mean charts - monitor central tendency

. Control charts for attributesattributes (counted or discrete).. p.. p--chart chart - used to monitor the proportion of defectives

generated by a process.. c.. c--chart chart – used to monitor the number of defects per unit (e.g.

automobiles, hotel rooms, typed pages, rolls of carpet)

Control Charts

14 - 30

Used in conjunction with control charts to test for randomness of observational data. The presence of patterns in the data or trends indicates that non-random (assignable) variation is present.

A run is a sequence of observations with a certain characteristic.

Two useful run tests:. Runs above and below the centerline

data translated into A (above) and B (below). Runs up and down

data translated into U (up) and D (down)

Run Charts – Check For Randomness

Pattern Analysis

Page 11: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-11

14 - 31

Counting Above/Below Runs (7 runs)

Counting A/B Runs

B1

A2

A2

A4

A6

A6

B3

B5

B5

B5

B7

If a value is equal to the centerline, the A/B rating is different from the last A/B rating.

14 - 32

Counting Up/Down Runs (8 runs)

Note: the first value does not receive a notation

Counting U/D Runs

U1

D6

U7

U7

U5

U3

U1

D4

D2

D8

If two values are equal, the U/D rating is different from the last U/D rating.

14 - 33

Once the runs are counted they must be compared to the expected number of runs in a completely random series. The expected number of runs and the standard deviation of the expected number of runs are computed by the following formula:

E(r) = N2

+1 =N -1

4

E(r) =2N -1

3 =

16N - 2990

N = number of observations

a/b a / b

u/d u/d

σ

σ

Expected Number of Runs

Page 12: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-12

14 - 34

Next, we compare the observed number of runs to the expected number of runs by calculating the following Standard Normal Z-statistic:

Zr - E(r)

Z =r - E(r)

r = observed number of runs

a / ba / b

a / b

u /du / d

u /d

σ

Compare Observed Runs to Expected Runs

14 - 35

-3.00 3.00

For a degree of confidence of 99.7% (3 standard deviations), we compare the comparison to -3 standard deviations and + 3 standard deviations.

. If either the Zab or Zud is .. < lower limit then we have too few runs (Out of Control).. Between the lower limit and upper limit then we have an acceptable number

of runs (In Control).. > upper limit then we have too many runs (Out of Conrtrol)

Too Few RunsAcceptable number

Runs Too Many Runs

Run Test - In or Out of Control

14 - 36

Statistical Process Control

Range Chart Mean ChartsMean Chart P Charts C Charts

Qualitative Attribute Charts

Control Charts

Quantitative Variable Charts

Page 13: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-13

14 - 37

14 - 38

Statistical Process Control Menu

14 - 39

Goodman Tire & Rubber CompanyThe Goodman Tire & Rubber Company periodically tests its tires for tread wear under simulated road conditions. To monitor the manufacturing process 3 radial tires are chosen from each shift of production operation. The operations manager recently attended a statistical process control (SPC) seminar and decided to adopt SPC to control and study tread wear.

The manager believes the production process is under control and to establish control limits and monitor the process he took 3 randomly selected radial tires from 20 consecutive shifts. They were tested for tread wear and recorded in the table on the next slide.

Tread wear is recorded in millimeters which is a measured variable; therefore, a range and mean chart should be developed.

Let’s take a look at the steps the manager must use to establish andanalyze the control chart.

First of all, he notessample size = 3process variability is not known

Page 14: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-14

14 - 402.63.42.220

2.82.91.819

3.12.94.018

3.23.62.817

3.14.23.016

2.92.51.815

1.73.54.314

3.22.41.713

2.51.92.312

4.03.12.611

3.01.72.910

3.33.44.19

2.82.63.28

2.91.72.17

3.64.24.16

3.52.93.85

2.12.51.74

3.43.02.53

3.51.82.62

2.84.23.11

Tire 3Tire 2Tire 1Sample

2.73333333

2.5

3.33333333

3.2

3.43333333

2.4

3.16666667

2.43333333

2.23333333

3.23333333

2.53333333

3.6

2.86666667

2.23333333

3.96666667

3.4

2.1

2.96666667

2.63333333

3.36666667

Mean

1.2

1.1

1.1

0.8

1.2

1.1

2.6

1.5

0.6

1.4

1.3

0.8

0.6

1.2

0.6

0.9

0.8

0.9

1.7

1.4

Range The first step: calculate the range for each sample.

The next step: calculate the mean for each sample.

Then: calculate the grand (average) range for all samples.

1.14

Range

Grand

Then: calculate the grand (average) mean for all samples.

2.91666667

Average

Grand

14 - 41

Sample Size A2 D3 D4 d2 d32 1.880 0 3.267 1.128 0.8533 1.023 0 2.574 1.693 0.8884 0.729 0 2.282 2.059 0.8805 0.577 0 2.114 2.326 0.8646 0.483 0 2.004 2.534 0.8487 0.419 0.076 1.924 2.704 0.8338 0.373 0.136 1.864 2.847 0.8209 0.337 0.184 1.816 2.970 0.808

10 0.308 0.223 1.777 3.078 0.79711 0.285 0.256 1.744 3.173 0.78712 0.266 0.283 1.717 3.258 0.77813 0.249 0.307 1.693 3.336 0.77014 0.235 0.328 1.672 3.407 0.76315 0.223 0.347 1.653 3.472 0.75616 0.212 0.363 1.637 3.532 0.75017 0.203 0.378 1.622 3.588 0.74418 0.194 0.391 1.608 3.640 0.73919 0.187 0.403 1.597 3.689 0.73420 0.180 0.415 1.585 3.735 0.72921 0.173 0.425 1.575 3.778 0.72422 0.167 0.434 1.566 3.819 0.72023 0.162 0.443 1.557 3.858 0.71624 0.157 0.451 1.548 3.895 0.71225 0.153 0.459 1.541 3.931 0.708

3-Sigma Control Chart TableNext we must determine the center line, upper and lower control limits for the range and mean chart which are calculated by very simple formula.

There is only one formula for the rangechart.

However; there are two formulae for the mean chart. The one chosen is dependent up on whether the variability (standard deviation) of the process is

. known

. unknown

The 3-simga control chart table is used to calculate these formulae.

14 - 42

Range Chart (R Chart)The Range Chart measures the variability of a sample measurements in a process.The range chart must be interpreted first because the process variability must be in control before we can monitor whether a process is in or out of control.

The Range Chart Centerline, UCL and LCL are calculated by the following formula.

4

3

**

DRUCLDRLCL

RrangegrandCenterline

=

=

==

The Goodman Tire & Rubber Company range chart calculations are:

0.0000LCL

1.1400Centerline

2.9344UCL

3s Control Chart Settings

574.2*14.10*14.1

14.1

==

=

UCLLCLCenterline

Page 15: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-15

14 - 43

Control Chart

Centerline, 1.1400

LCL, 0.0000

UCL, 2.9344

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60 70

Range

Range Chart (R Chart)

1.2

1.1

1.1

0.8

1.2

1.1

2.6

1.5

0.6

1.4

1.3

0.8

0.6

1.2

0.6

0.9

0.8

0.9

1.7

1.4

Range

The sample range values are now plotted on the range control chart

14 - 44

Mean Chart ( Chart) - Known VariationThe Mean Chart measures the average of sample measurements in a process. Whenthe variation of the process is known the 3-sigma table is not used.

The Mean Chart Centerline, UCL and LCL (3 sigma limits) are calculated by the following when the variation of the process is known.

X

deviation standard process size sample :where

3

3

==

+=

−=

==

σ

σ

σ

nn

XUCL

nXLCL

XaveragegrandCenterline

14 - 45

Mean Chart ( Chart) - Unknown VariationThe Mean Chart measures the average of sample measurements in a process. Whenthe variation of the process is unknown the 3-sigma table is used.

The Mean Chart Centerline, UCL and LCL are calculated by the following formula when the variation of the process is unknown.

X

RAXUCL

RAXLCL

XaveragegrandCenterline

2

2

+=

−=

==

The Goodman Tire & Rubber Company mean chart calculations are:

1.7504LCL

2.9167Centerline

4.0829UCL

3s Control Chart Settings

023.1*14.19167.2023.1*14.19167.2

9167.2

+=−=

=

UCLLCLCenterline

Page 16: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-16

14 - 46

Control Chart

Centerline, 2.9167

LCL, 1.7504

UCL, 4.0829

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 10 20 30 40 50 60 70

Mean Unknown Variation

2.73333333

2.5

3.33333333

3.2

3.43333333

2.4

3.16666667

2.43333333

2.23333333

3.23333333

2.53333333

3.6

2.86666667

2.23333333

3.96666667

3.4

2.1

2.96666667

2.63333333

3.36666667

Mean

The sample mean values are now plotted on the mean control chart

Mean Chart ( Chart) - Unknown VariationX

14 - 47

1. Choose the RM (Range & Mean) menu option

14 - 48

Range & Mean Chart Worksheet

2. Enter Data

3. Select Known or Unknown

4. Return

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Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-17

14 - 49

5. Enter problem name then click OK

14 - 50

6. Choose the CC (Control Chart) menu item

14 - 51

Notice the range control chart shows the sample ranges

7. Choose the Range control chart

Pattern Analysis for AB and UD indicate the patterns are random (YES)

Therefore; the range chart indicates the variation of the process is in control

Page 18: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-18

14 - 52

8. Choose the Mean Unknown Variation control chart

Notice the mean control chart shows the sample averages

Pattern Analysis for AB and UD indicate the patterns are random (YES)

Therefore; the mean chart indicates the average of the process is in control

14 - 53

Both charts are necessary to effectively monitor a process.

The mean charts detect when a process mean is shifting

The range charts detect when the process variability is changing.

The range chart must be interpreted first. If the variability is not constant, the process is not in control.

Summary - Range & Mean Control Charts

14 - 54

UCL

LCLUCL

LCL

R-chart

x-Chart Detects shift in process mean

Does notdetect shift

(process mean is shifting upward)

SamplingDistribution

Summary - Range & Mean Control Charts

Page 19: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-19

14 - 55

UCL

LCL

LCL

R-chart Reveals increase

x-Chart

UCL

Does notreveal increase

(process variability is increasing)SamplingDistribution

Mean & Range Control Charts

14 - 56

Statistical Process Control

C Chart

Qualitative Attribute Charts

Control Charts

Quantitative Variable Charts

Range Chart Mean Chart P Chart

14 - 57

P-Chart (Proportion Chart)The P-Chart measures the proportions in a process.

The Proportion Chart Centerline, UCL and LCL are calculated by the following formula.

Centerline = p

LCL = p - 3p(1 - p)

n UCL = p + 3

p(1 - p)n

p =total num ber defective

n * (num ber of sam ples)n = sam ple size

$

$$ $

$$ $

$

Page 20: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-20

14 - 58

Automated mail sorting machines are used to sort mail by zip codes. These machines scan the zip codes and divert each letter to its proper carrier zone.

Even when a machine is properly operating some mail is improperly diverted (a defective).

Check the sorting machine here to see if it is in control.

Mail Sorting

SampleSample

Size Defective1 100 142 100 103 100 124 100 135 100 96 100 117 100 108 100 129 100 13

10 100 1011 100 812 100 1213 100 914 100 1015 100 1116 100 1017 100 818 100 1219 100 1020 100 16

14 - 59

Calculate the actual proportion defective per sample by dividing the observed defective by the sample size.

Mail Sorting

The average proportion is

1100.p̂ =

0.0161LCL0.1100Centerline0.2039UCL

3σ Control Chart Settings

SampleSample

Size DefectiveProportionDefective

1 100 14 0.142 100 10 0.103 100 12 0.124 100 13 0.135 100 9 0.096 100 11 0.117 100 10 0.108 100 12 0.129 100 13 0.13

10 100 10 0.1011 100 8 0.0812 100 12 0.1213 100 9 0.0914 100 10 0.1015 100 11 0.1116 100 10 0.1017 100 8 0.0818 100 12 0.1219 100 10 0.1020 100 16 0.16

The sample proportion values are now plotted on the proportion control chart

14 - 60

P-chart Worksheet

Page 21: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-21

14 - 61

Notice the proportion control chart shows the sample proportions

Choose the Proportion control chart

Pattern Analysis for AB and UD indicate the patterns are random (YES)

Therefore; the proportion chart indicates the process is in control

14 - 62

Types of Control Charts

C Charts

Qualitative Attribute Charts

Control Charts

Quantitative Variable Charts

Range Charts Mean Charts P Charts

14 - 63

C-Chart (Defects Per Unit)The C-Chart is used to monitor the number of defects per unit (e.g. automobiles, hotel rooms, typed pages, rolls of carpet).

The C Chart Centerline, UCL and LCL are calculated by the following formula.

UCL = c + c

LCL = c - c wherec = process average number of defects per unit

$ $

$ $

$

3

3

In the event c is unknown, it can be estimated by sampling and computing the average defects observed. In this case the average can be substituted in the formula for . Also, note that since the formula is an approximation the LCL can be negative. In these cases the LCL is set to 0.

$c

Page 22: 14 Statistical Process Control

Lesson 14 – Statistical Process Control

Copyright – Harland E. Hodges, Ph.D 14-22

14 - 64

Example 5: Rolls of coiled wire are monitored using a c-chart. Eighteen rolls of wire have been examined and the number of defects per roll has been recorded in the data provided here. Is the process in control? Use 3 standard deviations for the control limits.

Coiled Wire

SampleNumber Defects

1 32 23 44 55 16 27 48 19 2

10 111 312 413 214 415 216 117 318 1

14 - 65

The average number of defects per roll of coiled wire is

5.2c =)

Coiled WireNumber Defects

324512412134242131

0.0000LCL2.5000Centerline7.2434UCL

3σ Control Chart Settings

The sample number defects values are now plotted on the defects per unit control chart

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C-chart Worksheet

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Choose the Defects per unit (C) control chart

Notice the unit defective control chart shows the sample proportions.

Pattern Analysis for AB and UD indicate the patterns are random (YES)

Therefore; the C chart indicates the process is in control

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Process variability can significantly impact quality. Some commonly used terms refer to the variability of a process output.

. . Tolerances - specifications for the range of acceptable values established by engineering or customer specifications

. . Control limits - statistical limits that reflect the extent to which sample statistics such as means and ranges can vary due to randomness

. . Process variability - reflects the natural or inherent (e.g.randomvariability in a process. It is measured in terms of the process standard deviation.

. Process capability - the inherent variability of process output relative to the variation allowed by the design specification.

3-sigma control is when the process performs within 3 standard deviations of the mean.

6-sigma control is when the process performs within 6 standard deviations of the mean.

Process Variability

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Process Mean

Product Lowerspecification

Product Upperspecification

Three Sigma and Six Sigma Quality

1.35 ppt 1.35 ppt

+/- 3 Sigma

ppt – parts per thousand

1.7 ppm 1.7 ppm

+/- 6 Sigma

ppm – parts per million

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LowerSpecification

UpperSpecification

Process variability matches specifications

LowerSpecification

UpperSpecification

Process variability well within specifications

LowerSpecification

UpperSpecification

Process variability exceeds specifications

Process Capability

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Example 7: A manager has the option of using any one of the following three machines for a job. The machines and their standard deviations are listed below. Determine which machines are capable of doing the job if the specifications are 1.00mm to 1.60mm.

Process Capability - Example

Std Dev Machine (mm)

A 0.10B 0.08C 0.13

The specifications indicate that the tolerance width is .6mm. (e.g. 1.60 – 1.00)

1.00mm - Lower specification

1.60 - Upper specification

.60mm - tolerance width

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Process capability for 3-sigma control is deemed to be within 3 standard deviations of the mean. Therefore for this example the process capability is 6 * (standard deviation) shown in the table below.

Std Dev Machine Machine (mm) Capability

A 0.10 0.60B 0.08 0.48C 0.13 0.78

Therefore, Machine A & B are capable of meeting the specifications, Machine C is not.

Process Capability - Example

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HomeworkRead and understand all material in the chapter.

Discussion and Review Questions

Recreate and understand all classroom examples

Exercises on chapter web page