statistical process control. overview variation variation control charts control charts r charts r...

Statistical Statistical Process ControlProcess Control

OverviewOverview

VariationVariation

Control chartsControl charts

R chartsR charts

X-bar X-bar charts charts

P chartsP charts

)(x

Measures performance of a Measures performance of a processprocess Primary tool - statisticsPrimary tool - statistics Involves collecting, organizing, & Involves collecting, organizing, &

interpreting data interpreting data Used to: Used to:

Control the process as products are Control the process as products are producedproduced

Inspect samples of finished productsInspect samples of finished products

Statistical Quality Control Statistical Quality Control (SPC)(SPC)

Bottling CompanyBottling Company

Machine automatically fills a 20 oz bottle.Machine automatically fills a 20 oz bottle. Problem with filling too much? Problems Problem with filling too much? Problems

with filling to little?with filling to little? So Monday the average is 20.2 ounces.So Monday the average is 20.2 ounces. Tuesday the average is 19.6 ounces.Tuesday the average is 19.6 ounces. Is this normal? Do we need to be Is this normal? Do we need to be

concerned?concerned? Wed is 19.4 ounces.Wed is 19.4 ounces.

Natural Natural VariationVariation

Machine can not fill Machine can not fill every bottle exactly every bottle exactly the same amount – the same amount – close but not exactly.close but not exactly.

Bottle Amount1 19.92 20.23 20.14 20.05 19.9

Natural variation

19.820.020.220.420.620.821.021.2

1 2 3 4 5

Bottle

Ou

nc

es

Bottle Amount1 20.92 21.03 21.04 20.85 20.9

Assignable variationAssignable variation

A cause for part A cause for part of the variationof the variation

Assignable variation

19.820.020.220.420.620.821.021.2

1 2 3 4 5

Bottle

Ou

nce

s

SPCSPC

Objective: provide statistical signal Objective: provide statistical signal

when assignable causes of variation when assignable causes of variation

are presentare present

ControlCharts

RChart

VariablesCharts

AttributesCharts

XChart

PChart

CChart

Continuous Numerical Data

Categorical or Discrete Numerical Data

Control Chart TypesControl Chart Types

Characteristics for Characteristics for which you focus on which you focus on defectsdefects

Classify products as Classify products as either ‘good’ or either ‘good’ or ‘bad’, or count # ‘bad’, or count # defectsdefects e.g., radio works or e.g., radio works or

notnot Categorical or Categorical or

discrete random discrete random variablesvariables

AttributesAttributesVariablesVariables

Measuring qualityMeasuring quality

Characteristics Characteristics that you that you measure, e.g., measure, e.g., weight, lengthweight, length

May be in whole May be in whole or in fractional or in fractional numbersnumbers

Continuous Continuous random variablesrandom variables

Show changes in data patternShow changes in data pattern e.g., trendse.g., trends

Make corrections Make corrections beforebefore process is out of process is out of controlcontrol

Show causes of changes in dataShow causes of changes in data Assignable causesAssignable causes

Data outside control limits or trend in dataData outside control limits or trend in data Natural causesNatural causes

Random variations around averageRandom variations around average

Control Chart PurposesControl Chart Purposes

Figure S6.7Figure S6.7

Steps to Follow When Using Steps to Follow When Using Control ChartsControl Charts

TO SET CONTROL CHART LIMITS

1. Collect 20-25 samples of n=4 or n=5 a stable

process

compute the mean of each sample.

2. Calculate control limits

Compute the overall means

Calculate the upper and lower control limits.

Steps to Follow When Using Steps to Follow When Using Control Charts - continuedControl Charts - continued

TO MONITOR PROCESS USING THE CONTROL CHARTS:TO MONITOR PROCESS USING THE CONTROL CHARTS:

1.1. Collect and graph dataCollect and graph data

Graph the sample means and ranges on their respective Graph the sample means and ranges on their respective

control chartscontrol charts

Determine whether they fall outside the acceptable limits.Determine whether they fall outside the acceptable limits.

2.2. Investigate points or patterns that indicate the process is out of Investigate points or patterns that indicate the process is out of

control. Assign causes for the variations.control. Assign causes for the variations.

3.3. Collect additional samples and revalidate the control limits.Collect additional samples and revalidate the control limits.

Monitors variability in process Monitors variability in process

Variables control chartVariables control chart

Interval or ratio scaled numerical dataInterval or ratio scaled numerical data

Shows sample ranges over timeShows sample ranges over time

Difference between smallest & largest Difference between smallest & largest

values in inspection samplevalues in inspection sample

RR Chart Chart

Sample Range at Time i

# Samples

From Table S6.1

RR Chart Chart Control LimitsControl Limits

s

R R

RD LCL

RD UCL

i

s

1i

3R

4R

Control ChartsControl Chartsfor Variablesfor Variables

West Allis IndustriesWest Allis Industries

The management of West Allis Industries is concerned about the production of a special metal screw ordered by several of their largest customers. The diameter of the screw is critical.


Sample Sample

Number 1 2 3 4

1

2

3

4

5

Special Metal Screw


Sample Sample

Number 1 2 3 4

1 0.5014 0.5022 0.5009 0.5027

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5047

Special Metal Screw

Should be at least 20 samples of size 4 to calculate the control limits.


Sample Sample

Number 1 2 3 4 R

1 0.5014 0.5022 0.5009 0.5027

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

Special Metal Screw


Sample Sample

Number 1 2 3 4 R

1 0.5014 0.5022 0.5009 0.5027

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

0.5027 – 0.50090.5027 – 0.5009 == 0.00180.0018

Special Metal Screw


Sample Sample

Number 1 2 3 4 R

1 0.5014 0.5022 0.5009 0.5027 0.0018

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

0.5027 – 0.50090.5027 – 0.5009 == 0.00180.0018

Special Metal Screw


Sample Sample

Number 1 2 3 4 R

1 0.5014 0.5022 0.5009 0.5027 0.0018

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

0.5027 – 0.50090.5027 – 0.5009 == 0.00180.0018

0.5041 - 0.5020 0.5041 - 0.5020 = = 0.00210.0021

Special Metal Screw


Sample Sample

Number 1 2 3 4 R

1 0.5014 0.5022 0.5009 0.5027 0.0018

2 0.5021 0.5041 0.5024 0.5020 0.0021

3 0.5018 0.5026 0.5035 0.5023 0.0017

4 0.5008 0.5034 0.5024 0.5015 0.0026

5 0.5041 0.5056 0.5034 0.5047 0.0022

Special Metal Screw


Sample Sample

Number 1 2 3 4 R

1 0.5014 0.5022 0.5009 0.5027 0.0018

2 0.5021 0.5041 0.5024 0.5020 0.0021

3 0.5018 0.5026 0.5035 0.5023 0.0017

4 0.5008 0.5034 0.5024 0.5015 0.0026

5 0.5041 0.5056 0.5034 0.5047 0.0022

R = 0.0021

Special Metal Screw


Control Charts – Special Metal Screw

R-Charts R = 0.0021

UCLR = D4RLCLR = D3R

Control ChartsControl Chartsfor Variablesfor Variables Control Chart FactorsControl Chart Factors

Factor for UCLFactor for UCL Factor forFactor for FactorFactorSize ofSize of and LCL forand LCL for LCL forLCL for UCL forUCL forSampleSample xx-Charts-Charts RR-Charts-Charts RR-Charts-Charts

((nn)) ((AA22)) ((DD33)) ((DD44))

22 1.8801.880 0 0 3.2673.26733 1.0231.023 0 0 2.5752.57544 0.7290.729 0 0 2.2822.28255 0.5770.577 0 0 2.1152.11566 0.4830.483 0 0 2.0042.00477 0.4190.419 0.076 0.076 1.9241.924


Control Charts - Special Metal Screw

R - Charts R = 0.0020 D4 = 2.2080

Control Chart FactorsControl Chart Factors


((nn)) ((AA22)) ((DD33)) ((DD44))

22 1.8801.880 0 0 3.2673.26733 1.0231.023 0 0 2.5752.57544 0.7290.729 0 0 2.2822.28255 0.5770.577 0 0 2.1152.11566 0.4830.483 0 0 2.0042.00477 0.4190.419 0.076 0.076 1.9241.924


Control Charts—Special Metal Screw

R-Charts R = 0.0021 D4 = 2.282D3 = 0




R-Charts R = 0.0021 D4 = 2.282D3 = 0

UCLR = 2.282 (0.0021) = 0.00479 in.




R-Charts R = 0.0021 D4 = 2.282D3 = 0

UCLR = 2.282 (0.0021) = 0.00479 in.LCLR = 0 (0.0021) = 0 in.


Range Chart - Range Chart - Special Metal Special Metal

ScrewScrew

Monitors process average Monitors process average

Variables control chartVariables control chart

Interval or ratio scaled numerical dataInterval or ratio scaled numerical data

Shows sample means over timeShows sample means over time

XX Chart Chart

XX Chart Chart Control LimitsControl Limits

Sample Range at

Time i

# Samples

Sample Mean at Time i

From Table S6.1

RAxxLCL

RAxxUCL

s

R R

i

s

1i

s

x 1i

s

ix


Sample Sample

Number 1 2 3 4

1 0.5014 0.5022 0.5009 0.5027

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5047

Special Metal Screw


Sample Sample

Number 1 2 3 4 R x

1 0.5014 0.5022 0.5009 0.5027

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

Special Metal Screw

_


Sample Sample

Number 1 2 3 4 R x

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039(0.5014 + 0.5022 + 0.5009 + 0.5027)/4(0.5014 + 0.5022 + 0.5009 + 0.5027)/4 = 0.5018= 0.5018

Special Metal Screw

_


Sample Sample

Number 1 2 3 4 R x

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

(0.5021 + 0.5041 + 0.5024 + 0.5020)/4(0.5021 + 0.5041 + 0.5024 + 0.5020)/4 == 0.50270.5027

Special Metal Screw

_


Sample Sample

Number 1 2 3 4 R x

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018

2 0.5021 0.5041 0.5024 0.5020 0.0021 0.5027

3 0.5018 0.5026 0.5035 0.5023 0.0017 0.5026

4 0.5008 0.5034 0.5024 0.5015 0.0026 0.5020

5 0.5041 0.5056 0.5034 0.5047 0.0022 0.5045

Special Metal Screw

_


Sample Sample

Number 1 2 3 4 R x

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018

2 0.5021 0.5041 0.5024 0.5020 0.0021 0.5027

3 0.5018 0.5026 0.5035 0.5023 0.0017 0.5026

4 0.5008 0.5034 0.5024 0.5015 0.0026 0.5020

5 0.5041 0.5056 0.5034 0.5047 0.0022 0.5045

R = 0.0021

x = 0.5027

Special Metal Screw

=

_

Example 7.1



X-Charts

UCLx = x + A2RLCLx = x - A2R

==

R = 0.0021x = 0.5027=

Example 7.1


Control Charts - Special Metal Screw

R = 0.0020x = 0.5025

x - Charts


Control Chart FactorsControl Chart Factors


((nn)) ((AA22)) ((DD33)) ((DD44))

22 1.8801.880 00 3.2673.26733 1.0231.023 00 2.5752.57544 0.7290.729 00 2.2822.28255 0.5770.577 00 2.1152.11566 0.4830.483 00 2.0042.00477 0.4190.419 0.0760.076 1.9241.924



x- Charts


==

R = 0.0021 A2 = 0.729x = 0.5027=

Example 7.1



x-Charts

UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in.


==

R = 0.0021 A2 = 0.729x = 0.5027=



x-Charts

UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in.LCLx = 0.5027 – 0.729 (0.0021) = 0.5012 in.


==

R = 0.0021 A2 = 0.729x = 0.5027=

xx-Chart Special Metal -Chart Special Metal ScrewScrew

xx-Chart Special Metal -Chart Special Metal ScrewScrew

Investigate Cause

Shows % of nonconforming Shows % of nonconforming

itemsitems

Attributes control chartAttributes control chart

Nominally scaled categorical dataNominally scaled categorical data

e.g., good-bade.g., good-bad

pp Chart Chart

pp Chart Control Limits Chart Control Limits

# Defective Items in Sample i

Size of sample i

z = 2 for 95.5% limits;

z = 3 for 99.7% limits

s

ii

p

p

n

n

ppzpLCL

n

ppzpUCL

1

i

s

1ix

p

)1(

)1(

HOMETOWN BANK

Hometown BankHometown Bank

The operations manager of the booking services department of Hometown Bank is concerned about the number of wrong customer account numbers recorded by Hometown personnel. Each week a random sample of 2,500 deposits is taken, and the number of incorrect account numbers is recorded. The records for the past 12 weeks are shown in the following table. Is the process out of control? Use 3-sigma control limits.


UCLUCLpp = = pp + + zzpp

LCLLCLpp = = pp - - zzpp

pp = = pp(1 - (1 - pp))//nn

Sample WrongNumber Account

Number

1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 1010 1711 1512 3

Total 147

Total defectives

Total observationsp =

n = 2500

Control Charts for AttributesControl Charts for Attributes

Control ChartsControl Chartsfor Attributesfor Attributes




pp = = pp(1 - (1 - pp))//nn

Sample WrongNumber Account Number

1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 1010 1711 1512 3

Total 147

147

12(2500)p =

n = 2500





pp = = pp(1 - (1 - pp))//nn

Sample WrongNumber Account Number

1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 1010 1711 1512 3

Total 147

p = 0.0049

n = 2500




LCLLCLpp = = pp – – zzpp

pp = = pp(1 – (1 – pp))//nn

n = 2500 p = 0.0049





pp = 0.0049(1 – 0.0049)/2500 = 0.0049(1 – 0.0049)/2500

n = 2500 p = 0.0049





pp = 0.0014 = 0.0014

n = 2500 p = 0.0049



pp = 0.0014 = 0.0014

n = 2500 p = 0.0049

UCLUCLpp = 0.0049 + 3(0.0014) = 0.0049 + 3(0.0014)

LCLLCLpp = 0.0049 – 3(0.0014) = 0.0049 – 3(0.0014)



pp = 0.0014 = 0.0014

n = 2500 p = 0.0049

UCLUCLpp = 0.0049 + 3(0.0014) = 0.0049 + 3(0.0014)

LCLLCLpp = 0.0049 – 3(0.0014) = 0.0049 – 3(0.0014)Why 3?

3-sigma limits

Also to within 99.7%

UCLUCLpp = 0.0091 = 0.0091

LCLLCLpp = 0.0007 = 0.0007



pp = 0.0014 = 0.0014

n = 2500 p = 0.0049

p-ChartWrong Account Numbers

p-ChartWrong Account Numbers

Investigate Cause

Figure S6.7Figure S6.7

Which control chart is Which control chart is appropriate?appropriate?

Webster Chemical Company produces Webster Chemical Company produces mastics and caulking for the mastics and caulking for the construction industry. The product is construction industry. The product is blended in large mixers and then blended in large mixers and then pumped into tubes and capped.pumped into tubes and capped.

Webster is concerned whether the Webster is concerned whether the filling process for tubes of caulking is in filling process for tubes of caulking is in statistical control. The process should statistical control. The process should be centered on 8 ounces per tube. be centered on 8 ounces per tube. Several samples of eight tubes are Several samples of eight tubes are taken and each tube is weighed in taken and each tube is weighed in ounces. ounces.


Webster Chemical Company produces Webster Chemical Company produces mastics and caulking for the mastics and caulking for the construction industry. The product is construction industry. The product is blended in large mixers and then blended in large mixers and then pumped into tubes and capped.pumped into tubes and capped.

Webster is concerned whether the Webster is concerned whether the filling process for tubes of caulking is in filling process for tubes of caulking is in statistical control. The process should statistical control. The process should be centered on 8 ounces per tube. be centered on 8 ounces per tube. Several samples of eight tubes are Several samples of eight tubes are taken and each tube is weighed in taken and each tube is weighed in ounces. ounces.

X-bar and R charts


A sticky scale brings Webster’s A sticky scale brings Webster’s attention to whether caulking attention to whether caulking tubes are being properly capped. tubes are being properly capped. If a significant proportion of the If a significant proportion of the tubes aren’t being sealed, Webster tubes aren’t being sealed, Webster is placing their customers in a is placing their customers in a messy situation. Tubes are messy situation. Tubes are packaged in large boxes of 144. packaged in large boxes of 144. Several boxes are inspected. The Several boxes are inspected. The number of leaking tubes in each number of leaking tubes in each box is recorded. box is recorded.


A sticky scale brings Webster’s A sticky scale brings Webster’s attention to whether caulking attention to whether caulking tubes are being properly capped. tubes are being properly capped. If a significant proportion of the If a significant proportion of the tubes aren’t being sealed, Webster tubes aren’t being sealed, Webster is placing their customers in a is placing their customers in a messy situation. Tubes are messy situation. Tubes are packaged in large boxes of 144. packaged in large boxes of 144. Several boxes are inspected. The Several boxes are inspected. The number of leaking tubes in each number of leaking tubes in each box is recorded. box is recorded.

P charts

statistical process control. overview variation variation control charts control charts r charts r...

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