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An Introduction to ControlCharts

Somesh

Ravi

Manish

Basant


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Basic ConceptionsWhen to use a control chart? Controlling ongoing processes by finding and correcting

problems as they occur.

Predicting the expected range of outcomes from aprocess.

Determining whether a process is stable (in statisticalcontrol).

Analyzing patterns of process variation from specialcauses (non-routine events) or common causes (built intothe process).

Determining whether the quality improvement projectshould aim to prevent specific problems or to make

fundamental changes to the process.


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Basic Principles Basic components of control charts

A centerline, usually the mathematical

average of all the samples plotted; Lower and upper control limits defining

the constraints of common cause

variations; Performance data plotted over time.


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Process Control ChartsControl Chartsshow sample data plotted on a graph with CenterLine (CL), Upper Control Limit (UCL), and Lower Control Limit(LCL).


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Types of Control Charts Control chart for variablesare used to

monitor characteristics that can be measured,

e.g. length, weight, diameter, time, etc. Control charts for attributesare used to

monitor characteristics that have discretevalues and can be counted, e.g. % defective,

number of flaws in a shirt, number of brokeneggs in a box, etc.


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Control Charts for Variables Mean (x-bar) charts

Tracks the central tendency (the average

value observed) over time

Range (R) charts:

Tracks the spread of the distribution over

time (estimates the observed variation)


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x-bar and R charts

monitor different parameters!


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Constructing a X-bar Chart:A quality control inspector at the Cocoa Fizz soft drink company hastaken three samples withfour observationseach of the volumeof bottles filled. If the standard deviationof the bottling operation

is .2 ounces, use the data below to develop control charts withlimits of3standard deviations for the 16 oz. bottling operation.

Time 1 Time 2 Time 3

Observation 1 15.8 16.1 16.0

Observation 2 16.0 16.0 15.9

Observation 3 15.8 15.8 15.9

Observation 4 15.9 15.9 15.8


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Step 1:

Calculate the Mean of Each Sample

Time 1 Time 2 Time 3

Observation 1 15.8 16.1 16.0

Observation 2 16.0 16.0 15.9

Observation 3 15.8 15.8 15.9

Observation 4 15.9 15.9 15.8

Sample means(X-bar)

15.875 15.975 15.9


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Step 2: Calculate the Standard

Deviation of the Sample Mean

x .2 .1

n 4


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Step 3: Calculate CL, UCL, LCL Center line (x-double bar):

Control limits for 3limits (z = 3):

15.875 15.975 15.9

x 15.923

x x

x x

UCL x z 15.92 3 .1 16.22

LCL x z 15.92 3 .1 15.62


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Step 4: Draw the Chart


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An Alternative Method for the X-bar

Chart Using R-bar and the A2

Factor

Use this method whensigma for the processdistribution is notknown. Use factor A2from Table 6.1

Factor for x-Chart

A2 D3 D4

2 1.88 0.00 3.27

3 1.02 0.00 2.57

4

0.73

0.00

2.28

5 0.58 0.00 2.11

6 0.48 0.00 2.00

7 0.42 0.08 1.92

8 0.37 0.14 1.86

9 0.34 0.18 1.82

10

0.31

0.22

1.78

11 0.29 0.26 1.74

12 0.27 0.28 1.72

13 0.25 0.31 1.69

14 0.24 0.33 1.67

15 0.22 0.35 1.65

Factors for R-ChartSample Size

(n)


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Step 1: Calculate the Range of

Each Sample and Average RangeTime 1 Time 2 Time 3

Observation 1 15.8 16.1 16.0

Observation 2 16.0 16.0 15.9Observation 3 15.8 15.8 15.9

Observation 4 15.9 15.9 15.8

Sample ranges(R)

0.2 0.3 0.2

0.2 0.3 0.2R .233

3


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Step 2: Calculate CL, UCL, LCL Center line:

Control limits for 3limits:

2x

2x

15.875 15.975 15.9CL x 15.923

UCL x A R 15.92 0.73 .233 16.09

LCL x A R 15.92 0.73 .233 15.75


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Control Chart for Range (R-Chart)

Center Line and Control Limitcalculations:

4

3

0.2 0.3 0.2CL R .2333

UCL D R 2.28(.233) .53

LCL D R 0.0(.233) 0.0

Factor for x-Chart

A2 D3 D4

2 1.88 0.00 3.27

3 1.02 0.00 2.57

4

0.73

0.00

2.28

5 0.58 0.00 2.11

6 0.48 0.00 2.00

7 0.42 0.08 1.92

8 0.37 0.14 1.86

9 0.34 0.18 1.82

10

0.31

0.22

1.78

11 0.29 0.26 1.74

12 0.27 0.28 1.72

13 0.25 0.31 1.69

14 0.24 0.33 1.67

15 0.22 0.35 1.65

Factors for R-ChartSample Size

(n)


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R-Bar Control Chart


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Control Charts for Attributes P-Charts & C-Charts

Use P-Charts for quality characteristics thatare discrete and involve yes/no or good/baddecisions Percent of leaking caulking tubes in a box of 48 Percent of broken eggs in a carton

Use C-Charts for discrete defects when therecan be more than one defect per unit Number of flaws or stains in a carpet sample cut from a

production run

Number of complaints per customer at a hotel


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Constructing a P-Chart:

A Production manager for a tire company has inspected thenumber of defective tires in five random samples with 20tires in each sample. The table below shows the number ofdefective tires in each sample of 20 tires.

Sample SampleSize (n)

NumberDefective

1 20 3

2 20 2

3 20 1

4 20 2

5 20 1


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Step 1:Calculate the Percent defective of Each Sampleand the Overall Percent Defective (P-Bar)

Sample NumberDefective

SampleSize

PercentDefective

1 3 20 .15

2 2 20 .10

3 1 20 .05

4 2 20 .105 1 20 .05

Total 9 100 .09


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Step 2: Calculate the StandardDeviation of P.

p

p(1-p) (.09)(.91) = = =0.064

n 20


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Step 3: Calculate CL, UCL, LCL

CL p .09

Center line (p bar):


p

p

UCL p z .09 3(.064) .282

LCL p z .09 3(.064) .102 0


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Step 4: Draw the Chart


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Constructing a C-Chart:

The number ofweekly customercomplaints aremonitored in alarge hotel.Develop a threesigma control limits

For a C-Chart usingthe data table Onthe right.

Week Number ofComplaints

1 3

2 2

3 3

4 1

5 3

6 3

7 2

8 1

9 3

10 1

Total 22


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Calculate CL, UCL, LCL

Center line (c bar):


UCL c c 2.2 3 2.2 6.65

LCL c c 2.2 3 2.2 2.25 0

z

z

#complaints 22

CL 2.2# of samples 10

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