Statistical Quality Control

N.Obeidi

Descriptive Statistics

• Descriptive Statistics include:Descriptive Statistics include:

– The Mean- measure of The Mean- measure of central tendencycentral tendency

– The Range- difference The Range- difference between largest/smallest between largest/smallest observations in a set of observations in a set of datadata

– Standard Deviation Standard Deviation measures the amount of measures the amount of data dispersion around data dispersion around meanmean

– Distribution of Data shapeDistribution of Data shape• Normal or bell shaped Normal or bell shaped

oror• SkewedSkewed

n

xx

n

1ii

1n

Xxσ

n

1i

2

i

Statistics – Statistics – ‘Mode‘Mode’’

Mode = most frequently occurring value

Find the mode of 4,6,7,9,4

The most popular, or mode is 4

Normal DistributionF

requ

ency

4.7’4.8’4.9’Mean5.1’5.2’5.3’

X

Normal Distribution

02468

10121416

# of

Obs

erva

tions

192 194 196 198 200 202 204 206 208 210 212

Serum glucose (mg/dL)

Mean

Distribution of DataDistribution of Data• Normal distributions • Skewed distribution

Setting Control Limits

• Percentage of values under normal curve

sampleeach w/in nsobservatio of# theis

(n) and means sample of # theis )( wheren

σσ ,

...xxxx x

n21

k

k

Constructing an X-bar Chart: A quality control inspector at the Cocoa Fizz soft drink company has taken three samples with four observations each of the volume of bottles filled. If the standard deviation of the bottling operation is .2 ounces, use the below data to develop control charts with limits of 3 standard deviations for the 16 oz. bottling operation.

Time 1

Time 2Time 3

Observation 1

15.816.116.0

Observation 2

16.016.015.9

Observation 3

15.815.815.9

Observation 4

15.915.915.8

Sample means (X-bar)

15.875

15.97515.9

Sample ranges (R)

0.20.30.2

Solution and Control Chart (x-bar)

• Center line (x-double bar):

15.923

15.915.97515.875x

Levey-Jennings Chart

12

Levey-Jennings Chart

Levey-Jennings Chart

Chapter 814

Introduction to Statistical Quality Control, 5th Edition by Douglas C. Montgomery.

Copyright (c) 2005 John Wiley & Sons, Inc.

Cusum ChartCusum Chart

C-Chart Example: The number of weekly customer complaints are monitored in a large hotel using a

c-chart. Develop three sigma control limits using the data table below.

WeekNumber of Complaints

13

22

33

41

53

63

72

81

93

101

Total22

Solution:

02.252.232.2ccLCL

6.652.232.2ccUCL

2.210

22

samples of #

complaints#CL

c

c

z

z

Interpreting patterns in control charts

Downward trend in R-chart…

Moving Range I-chart

0

2.458

8.031

0.000

1.000

2.000

3.000

4.000

5.000

6.000

7.000

8.000

9.000

0 5 10 15 20 25 30

Trend in the moving range indicates a process not in control

Levey-Jennings Chart -Record and Evaluate the Control Values

80

85

90

95

100

105

110

115

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Mean

Day

+1SD

+2SD

+3SD

-1SD

-2SD

-3SD

Westgard RulesWestgard Rules• ““Multirule Quality Control” Multirule Quality Control” • Uses a combination of Uses a combination of

decision criteria or control decision criteria or control rulesrules

• Allows determination of Allows determination of whether an analytical run is whether an analytical run is “in-control” or “out-of-“in-control” or “out-of-control”control”

Westgard RulesWestgard Rules ( (Generally used where 2 levels of Generally used where 2 levels of control material are analyzed per control material are analyzed per

run)run)

• 112S2S rule rule

• 113S3S rule rule

• 222S2S rule rule

• RR4S4S rule rule

• 441S1S rule rule

• 1010XX rule rule

Westgard – 1Westgard – 12S2S Rule Rule

• ““warning rule”warning rule”• One of two control results falls One of two control results falls

outside ±2SDoutside ±2SD• Alerts tech to possible problemsAlerts tech to possible problems• Not cause for rejecting a runNot cause for rejecting a run

• Must then evaluate the 1Must then evaluate the 13S 3S rulerule

12S Rule = A warning to trigger

careful inspection of the control data

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Mean

Day

+1SD

+2SD

+3SD

-1SD

-2SD

-3SD

12S rule violation

Westgard – 1Westgard – 13S3S Rule Rule

• If either of the two control If either of the two control results falls outside of results falls outside of ±3SD, rule is violated±3SD, rule is violated

• Run must be rejectedRun must be rejected

• If 1If 13S3S not violated, check not violated, check 222S2S

113S3S Rule Rule = Reject the run when a single control = Reject the run when a single control

measurement exceeds the +3SD or -3SD control measurement exceeds the +3SD or -3SD control limitlimit

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Mean

Day

+1SD

+2SD

+3SD

-1SD

-2SD

-3SD

13S rule violatio

n

Westgard – 2Westgard – 22S2S Rule Rule

• 2 consecutive control values for 2 consecutive control values for the same level fall outside of the same level fall outside of ±2SD in the same direction, or±2SD in the same direction, or

• Both controls in the same run Both controls in the same run exceed ±2SD exceed ±2SD

• Patient results cannot be Patient results cannot be reportedreported

• Requires corrective actionRequires corrective action

222S2S Rule Rule = Reject the run when 2 consecutive control = Reject the run when 2 consecutive control

measurements exceed the same +2SD or -2SD control measurements exceed the same +2SD or -2SD control limitlimit

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Mean

Day

+1SD

+2SD

+3SD

-1SD

-2SD

-3SD

22S rule violatio

n

Westgard – RWestgard – R4S4S Rule Rule

• One control exceeds the mean by One control exceeds the mean by –2SD, and the other control –2SD, and the other control exceeds the mean by +2SDexceeds the mean by +2SD

• The range between the two The range between the two results will therefore exceed 4 results will therefore exceed 4 SDSD

• Random error has occurred, test Random error has occurred, test run must be rejectedrun must be rejected

RR4S4S Rule Rule = Reject the run when 1 control = Reject the run when 1 control

measurement exceed the +2SD and the other exceeds measurement exceed the +2SD and the other exceeds the -2SD control limitthe -2SD control limit

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Mean

Day

+1SD

+2SD

+3SD

-1SD

-2SD

-3SD

R4S rule violatio

n

Westgard – 4Westgard – 41S1S Rule Rule

• Requires control data from Requires control data from previous runsprevious runs

• Four consecutive QC results for Four consecutive QC results for one level of control are outside one level of control are outside ±1SD, or±1SD, or

• Both levels of control have Both levels of control have consecutive results that are consecutive results that are outside ±1SDoutside ±1SD

Westgard – 10Westgard – 10XX Rule Rule• Requires control data from Requires control data from

previous runsprevious runs• Ten consecutive QC results for Ten consecutive QC results for

one level of control are on one one level of control are on one side of the mean, orside of the mean, or

• Both levels of control have five Both levels of control have five consecutive results that are on consecutive results that are on the same side of the meanthe same side of the mean

1010xx Rule Rule = Reject the run when 10 = Reject the run when 10

consecutive control measurements fall on consecutive control measurements fall on one side of the meanone side of the mean

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Mean

Day

+1SD

+2SD

+3SD

-1SD

-2SD

-3SD

10x rule violatio

n

Westgard Multirule QC