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.
Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4
1
2
3
4
5
Special Metal Screw
Control ChartsControl Chartsfor Variablesfor Variables
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.
Control ChartsControl Chartsfor Variablesfor Variables
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
Control ChartsControl Chartsfor Variablesfor Variables
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
Control ChartsControl Chartsfor Variablesfor Variables
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
Control ChartsControl Chartsfor Variablesfor Variables
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
Control ChartsControl Chartsfor Variablesfor Variables
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
Control ChartsControl Chartsfor Variablesfor Variables
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
Control ChartsControl Chartsfor Variablesfor Variables
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 ChartsControl Chartsfor Variablesfor Variables
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 ChartsControl Chartsfor Variablesfor Variables
Control Charts - Special Metal Screw
R - Charts R = 0.0020 D4 = 2.2080
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 ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
R-Charts R = 0.0021 D4 = 2.282D3 = 0
UCLR = D4RLCLR = D3R
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
R-Charts R = 0.0021 D4 = 2.282D3 = 0
UCLR = 2.282 (0.0021) = 0.00479 in.
UCLR = D4RLCLR = D3R
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
R-Charts R = 0.0021 D4 = 2.282D3 = 0
UCLR = 2.282 (0.0021) = 0.00479 in.LCLR = 0 (0.0021) = 0 in.
UCLR = D4RLCLR = D3R
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
R-Charts R = 0.0021 D4 = 2.282D3 = 0
UCLR = 2.282 (0.0021) = 0.00479 in.LCLR = 0 (0.0021) = 0 in.
UCLR = D4RLCLR = D3R
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
Control ChartsControl Chartsfor Variablesfor Variables
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
Control ChartsControl Chartsfor Variablesfor Variables
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
_
Control ChartsControl Chartsfor Variablesfor Variables
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
_
Control ChartsControl Chartsfor Variablesfor Variables
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
_
Control ChartsControl Chartsfor Variablesfor Variables
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
_
Control ChartsControl Chartsfor Variablesfor Variables
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
_
Control ChartsControl Chartsfor Variablesfor Variables
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
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
X-Charts
UCLx = x + A2RLCLx = x - A2R
==
R = 0.0021x = 0.5027=
Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts - Special Metal Screw
R = 0.0020x = 0.5025
x - Charts
UCLx = x + A2RLCLx = x - A2R
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 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
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
x- Charts
UCLx = x + A2RLCLx = x - A2R
==
R = 0.0021 A2 = 0.729x = 0.5027=
Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
x-Charts
UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in.
UCLx = x + A2RLCLx = x - A2R
==
R = 0.0021 A2 = 0.729x = 0.5027=
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
x-Charts
UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in.LCLx = 0.5027 – 0.729 (0.0021) = 0.5012 in.
UCLx = x + A2RLCLx = x - A2R
==
R = 0.0021 A2 = 0.729x = 0.5027=
xx-Chart Special Metal -Chart Special Metal ScrewScrew
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.
Hometown BankHometown Bank
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
Hometown BankHometown Bank
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
147
12(2500)p =
n = 2500
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
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
p = 0.0049
n = 2500
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
UCLUCLpp = = pp + + zzpp
LCLLCLpp = = pp – – zzpp
pp = = pp(1 – (1 – pp))//nn
n = 2500 p = 0.0049
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
UCLUCLpp = = pp + + zzpp
LCLLCLpp = = pp – – zzpp
pp = 0.0049(1 – 0.0049)/2500 = 0.0049(1 – 0.0049)/2500
n = 2500 p = 0.0049
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
UCLUCLpp = = pp + + zzpp
LCLLCLpp = = pp – – zzpp
pp = 0.0014 = 0.0014
n = 2500 p = 0.0049
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
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)
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
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
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
pp = 0.0014 = 0.0014
n = 2500 p = 0.0049
p-ChartWrong Account Numbers
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.
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.
X-bar and R charts
Which control chart is Which control chart is appropriate?appropriate?
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.
Which control chart is Which control chart is appropriate?appropriate?
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