© Anita Lee-Post

Quality ControlPart 2

Quality ControlPart 2

ByBy

Anita Lee-PostAnita Lee-Post

ByBy

Anita Lee-PostAnita Lee-Post

© Anita Lee-Post

Statistical process control methodsStatistical process control methods

• Control charts for variables: process Control charts for variables: process characteristics are measured on a characteristics are measured on a continuous scale, e.g., weight, volume, continuous scale, e.g., weight, volume, widthwidth• Mean (X-bar) chartMean (X-bar) chart• Range (R) chartRange (R) chart

• Control charts for attributes: process Control charts for attributes: process characteristics are counted on a discrete characteristics are counted on a discrete scale, e.g., number of defects, number of scale, e.g., number of defects, number of scratchesscratches• Proportion (P) chart Proportion (P) chart • Count (C) chartCount (C) chart

• Process capability ratio and indexProcess capability ratio and index

• Control charts for variables: process Control charts for variables: process characteristics are measured on a characteristics are measured on a continuous scale, e.g., weight, volume, continuous scale, e.g., weight, volume, widthwidth• Mean (X-bar) chartMean (X-bar) chart• Range (R) chartRange (R) chart

• Control charts for attributes: process Control charts for attributes: process characteristics are counted on a discrete characteristics are counted on a discrete scale, e.g., number of defects, number of scale, e.g., number of defects, number of scratchesscratches• Proportion (P) chart Proportion (P) chart • Count (C) chartCount (C) chart

• Process capability ratio and indexProcess capability ratio and index

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Control chartsControl charts

• Use statistical limits to identify whether Use statistical limits to identify whether a sample of data falls within a normal a sample of data falls within a normal range of variation or notrange of variation or not

• Use statistical limits to identify whether Use statistical limits to identify whether a sample of data falls within a normal a sample of data falls within a normal range of variation or notrange of variation or not

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Setting Limits Requires Balancing RisksSetting Limits Requires Balancing Risks

• Control limits are based on a willingness to think Control limits are based on a willingness to think that something is wrong when it’s actually not that something is wrong when it’s actually not (Type I or alpha error), balanced against the (Type I or alpha error), balanced against the sensitivity of the tool - the ability to quickly reveal sensitivity of the tool - the ability to quickly reveal a problem (failure is Type II or beta error)a problem (failure is Type II or beta error)

• Control limits are based on a willingness to think Control limits are based on a willingness to think that something is wrong when it’s actually not that something is wrong when it’s actually not (Type I or alpha error), balanced against the (Type I or alpha error), balanced against the sensitivity of the tool - the ability to quickly reveal sensitivity of the tool - the ability to quickly reveal a problem (failure is Type II or beta error)a problem (failure is Type II or beta error)

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Control Charts for Variable DataControl Charts for Variable Data

• Mean (x-bar) chartsMean (x-bar) charts

• Tracks the central tendency (the average Tracks the central tendency (the average value observed) over timevalue observed) over time

• Range (R) charts:Range (R) charts:

• Tracks the spread of the distribution over Tracks the spread of the distribution over time (estimates the observed variation)time (estimates the observed variation)

• Mean (x-bar) chartsMean (x-bar) charts

• Tracks the central tendency (the average Tracks the central tendency (the average value observed) over timevalue observed) over time

• Range (R) charts:Range (R) charts:

• Tracks the spread of the distribution over Tracks the spread of the distribution over time (estimates the observed variation)time (estimates the observed variation)

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Mean (x-bar) chartsMean (x-bar) charts

pulationprocess/po the of deviation standard the : n

meanprocess the of deviation standard the :x

deviation standard normal of number :z

samples of number :k

size) (sample nsobservatio of number :n

i, nobservatio :ix ,...1

ix where

...21 meanprocess

nnxx

xzxLCL

xzxUCL

kkxxx

xCL

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Mean (x-bar) charts continuedMean (x-bar) charts continued

• Use the x-bar chart established to Use the x-bar chart established to monitor sample averages as the monitor sample averages as the process continues:process continues:

• Use the x-bar chart established to Use the x-bar chart established to monitor sample averages as the monitor sample averages as the process continues:process continues:

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An exampleAn example

The diameters of five C&A bagels are The diameters of five C&A bagels are sampled each hour during a 8-hour period. sampled each hour during a 8-hour period. The data collected are shown as follows:The data collected are shown as follows:

The diameters of five C&A bagels are The diameters of five C&A bagels are sampled each hour during a 8-hour period. sampled each hour during a 8-hour period. The data collected are shown as follows:The data collected are shown as follows:

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An example continuedAn example continued

a)a) Develop an x-bar chart with the control limits set Develop an x-bar chart with the control limits set to include 99.74% of the sample means and the to include 99.74% of the sample means and the standard deviation of the production process (standard deviation of the production process () ) is known to be 0.2 Inches.is known to be 0.2 Inches.

Step 1. Compute the sample mean x-bar:Step 1. Compute the sample mean x-bar:

a)a) Develop an x-bar chart with the control limits set Develop an x-bar chart with the control limits set to include 99.74% of the sample means and the to include 99.74% of the sample means and the standard deviation of the production process (standard deviation of the production process () ) is known to be 0.2 Inches.is known to be 0.2 Inches.

Step 1. Compute the sample mean x-bar:Step 1. Compute the sample mean x-bar:

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An example continuedAn example continued

Step 2. Compute the process mean or center line of Step 2. Compute the process mean or center line of the control chart:the control chart:

Step 2. Compute the process mean or center line of Step 2. Compute the process mean or center line of the control chart:the control chart:

04.48

34.32

8

03.404.408.403.401.418.494.303.4

xCL

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An example continuedAn example continued

Step 3. Compute the upper and lower control limits:Step 3. Compute the upper and lower control limits:

To include 99.74% of the sample means implies that the To include 99.74% of the sample means implies that the number of normal standard deviations is 3. i.e., z=3number of normal standard deviations is 3. i.e., z=3

Step 3. Compute the upper and lower control limits:Step 3. Compute the upper and lower control limits:

To include 99.74% of the sample means implies that the To include 99.74% of the sample means implies that the number of normal standard deviations is 3. i.e., z=3number of normal standard deviations is 3. i.e., z=3

8.35

2.0304.4

3.45

2.0304.4

xzxLCL

xzxUCL

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An example continuedAn example continued

b.b. C&A collects the process characteristics (i.e., C&A collects the process characteristics (i.e., diameter) of their bagels in days 2 through 10. Is diameter) of their bagels in days 2 through 10. Is the process in control?the process in control?

b.b. C&A collects the process characteristics (i.e., C&A collects the process characteristics (i.e., diameter) of their bagels in days 2 through 10. Is diameter) of their bagels in days 2 through 10. Is the process in control?the process in control?

Day 2 3 4 5 6 7 8 9 10Average of 5 observations 4.21 3.99 3.89 4.05 4.22 4.28 3.55 3.78 5.00

Diameter of Sample

The process is not in control because the means of recent sample averages fall outside the upper and lower control limits

x-bar Chart for Samples of 5 Bagels

3.50

3.75

4.00

4.25

4.50

4.75

5.00

5.25

2 3 4 5 6 7 8 9 10

Sample

Av

era

ge

Dia

me

ter

(in

in

ch

es

)

LCL = 3.83

UCL = 4.25

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Range (R) chartsRange (R) charts

LCL for factor chart-R the :3D

UCL for factor chart-R the :4D

i sample of nobservatio the :ix

)imin(x)i max(x i sample of range the :iR

samples of number :k

range sample the of average the :R where3

4

...21

RDLCL

RDUCLk

kRRRRCL

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An exampleAn example

The diameters of five C&A bagels are The diameters of five C&A bagels are sampled each hour during a 8-hour period. sampled each hour during a 8-hour period. The data collected are shown as follows:The data collected are shown as follows:

The diameters of five C&A bagels are The diameters of five C&A bagels are sampled each hour during a 8-hour period. sampled each hour during a 8-hour period. The data collected are shown as follows:The data collected are shown as follows:

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An example continuedAn example continued

a)a) Develop a range chart.Develop a range chart.

Step 1. Compute the average range or CL:Step 1. Compute the average range or CL:

a)a) Develop a range chart.Develop a range chart.

Step 1. Compute the average range or CL:Step 1. Compute the average range or CL:

0.27 8

13.021.013.021.012.007.110.018.0

RCL

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An example continuedAn example continued

Step 2. Compute the upper and lower control limits:Step 2. Compute the upper and lower control limits:Step 2. Compute the upper and lower control limits:Step 2. Compute the upper and lower control limits:

Control Limit Factors for Range Charts

Sample size, n D3 D4

2 0.00 3.27

3 0.00 2.57

4 0.00 2.28

5 0.00 2.11

6 0.00 2.00

7 0.08 1.92

8 0.14 1.86

0

27.00

57.0

27.011.2

3

4

RDLCL

RDUCL

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An example continuedAn example continued

b.b. C&A collects the process characteristics (i.e., diameter) C&A collects the process characteristics (i.e., diameter) of their bagels in days 2 through 10. Is the process in of their bagels in days 2 through 10. Is the process in control?control?

b.b. C&A collects the process characteristics (i.e., diameter) C&A collects the process characteristics (i.e., diameter) of their bagels in days 2 through 10. Is the process in of their bagels in days 2 through 10. Is the process in control?control?

Day 2 3 4 5 6 7 8 9 10Range of 5 observations 0.30 0.20 0.33 0.20 0.14 0.11 0.05 0.35 0.20

Diameter of Sample

The process is in control because the ranges of recent samples fall within the upper and lower control limits

Range Chart for Samples of 5 Bagels

0.00

0.10

0.20

0.30

0.40

0.50

0.60

2 3 4 5 6 7 8 9 10

Sample

Ran

ge

UCL = 0.57

LCL = 0

CL = 0.27

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Using both mean & range chartsUsing both mean & range charts

• Mean (x-bar) chart: measures the central tendency Mean (x-bar) chart: measures the central tendency of a processof a process

• Range (R) chart: measures the variance of a Range (R) chart: measures the variance of a processprocess

Case 1: a process showing a drift in its mean but Case 1: a process showing a drift in its mean but not its variance not its variance

can be detected only by a mean (x-bar) chartcan be detected only by a mean (x-bar) chart

• Mean (x-bar) chart: measures the central tendency Mean (x-bar) chart: measures the central tendency of a processof a process

• Range (R) chart: measures the variance of a Range (R) chart: measures the variance of a processprocess

Case 1: a process showing a drift in its mean but Case 1: a process showing a drift in its mean but not its variance not its variance

can be detected only by a mean (x-bar) chartcan be detected only by a mean (x-bar) chart

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Using both mean & range charts continuedUsing both mean & range charts continued

Case 2: a process showing a change in its variance Case 2: a process showing a change in its variance but not its mean but not its mean

can be detected only by a range (R) chartcan be detected only by a range (R) chart

Case 2: a process showing a change in its variance Case 2: a process showing a change in its variance but not its mean but not its mean

can be detected only by a range (R) chartcan be detected only by a range (R) chart

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Construct x-bar chart from sample rangeConstruct x-bar chart from sample range

chart bar-x for factor limit Control :2A

samples of number :k

i sample for nobservatio :ix

)min()i max(x:iR

...21RR

2

2

...1

ixk

kRRwhere

RAxLCL

RAxUCL

kkxx

xCL

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Control Charts for AttributesControl Charts for Attributes

• p-Charts:p-Charts:

• Track the proportion defective in a Track the proportion defective in a samplesample

• c-Charts:c-Charts:

• Track the average number of defects per Track the average number of defects per unit of outputunit of output

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Proportion (p) chartsProportion (p) charts

• Data requirements:Data requirements:• Sample size Sample size • Number of defectsNumber of defects• Sample size is large enough so that the Sample size is large enough so that the

attributes will be counted twice in each attributes will be counted twice in each sample, e.g., a defect rate of 1% will sample, e.g., a defect rate of 1% will require a sample size of 200 units.require a sample size of 200 units.

• Data requirements:Data requirements:• Sample size Sample size • Number of defectsNumber of defects• Sample size is large enough so that the Sample size is large enough so that the

attributes will be counted twice in each attributes will be counted twice in each sample, e.g., a defect rate of 1% will sample, e.g., a defect rate of 1% will require a sample size of 200 units.require a sample size of 200 units.

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Proportion (p) charts continuedProportion (p) charts continued

size sample :n

)1(

deviation standard sample the

deviation standard normal ofnumber :z where

size Sample samples ofNumber

samples all from defects ofNumber Total

p

n

pp

zpLCL

zpUCL

pCL

p

p

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Count (c) chartsCount (c) charts

• Data requirementsData requirements

• Number of defectsNumber of defects• Monitoring processes in which the items Monitoring processes in which the items

of interest (in this case, defects) are of interest (in this case, defects) are infrequent and/or occur in time or space, infrequent and/or occur in time or space, e.g., errors in newspaper, bad circuits in e.g., errors in newspaper, bad circuits in a microchip, complaints from customers.a microchip, complaints from customers.

• Data requirementsData requirements

• Number of defectsNumber of defects• Monitoring processes in which the items Monitoring processes in which the items

of interest (in this case, defects) are of interest (in this case, defects) are infrequent and/or occur in time or space, infrequent and/or occur in time or space, e.g., errors in newspaper, bad circuits in e.g., errors in newspaper, bad circuits in a microchip, complaints from customers.a microchip, complaints from customers.

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Count (c) charts continuedCount (c) charts continued

deviation standardof number :z where

cz c LCL

cz c UCL

/unitsdays/weeks of number :n where,n

x

defects of number average cCLn

1ii