TOTAL QUALITY MANAGEMENT

CONTROL CHARTS FOR VARIABLES AND ATTRIBUTES

Continuous Variables:

Take on a continuum of values. Ex.: length, diameter, thickness

Modeled by the Normal Distribution

Attribute Variables:

Take on discrete valuesEx.: present/absent, conforming/non-conforming

Discrete Variables Classes• Defectives

– The presence of a non-conformity ruins the entire unit – the unit is defective

• Example – fuses with disconnects

• Defects– The presence of one or more non-conformities may lower the value of

the unit, but does not render the entire unit defective• Example – paneling with scratches

• 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-Chart• Sample Size• (n)

X- BAR AND R CHART

Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 51 10.68 10.689 10.776 10.798 10.7142 10.79 10.86 10.601 10.746 10.7793 10.78 10.667 10.838 10.785 10.7234 10.59 10.727 10.812 10.775 10.735 10.69 10.708 10.79 10.758 10.6716 10.75 10.714 10.738 10.719 10.6067 10.79 10.713 10.689 10.877 10.6038 10.74 10.779 10.11 10.737 10.759 10.77 10.773 10.641 10.644 10.72510 10.72 10.671 10.708 10.85 10.71211 10.79 10.821 10.764 10.658 10.70812 10.62 10.802 10.818 10.872 10.72713 10.66 10.822 10.893 10.544 10.7514 10.81 10.749 10.859 10.801 10.70115 10.66 10.681 10.644 10.747 10.728

Example of x-bar and R charts: Step 1. Calculate sample means, sample ranges, mean of means, and mean of ranges.Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 Avg Range

1 10.68 10.689 10.776 10.798 10.714 10.732 0.1162 10.79 10.86 10.601 10.746 10.779 10.755 0.2593 10.78 10.667 10.838 10.785 10.723 10.759 0.1714 10.59 10.727 10.812 10.775 10.73 10.727 0.2215 10.69 10.708 10.79 10.758 10.671 10.724 0.1196 10.75 10.714 10.738 10.719 10.606 10.705 0.1437 10.79 10.713 10.689 10.877 10.603 10.735 0.2748 10.74 10.779 10.11 10.737 10.75 10.624 0.6699 10.77 10.773 10.641 10.644 10.725 10.710 0.13210 10.72 10.671 10.708 10.85 10.712 10.732 0.17911 10.79 10.821 10.764 10.658 10.708 10.748 0.16312 10.62 10.802 10.818 10.872 10.727 10.768 0.25013 10.66 10.822 10.893 10.544 10.75 10.733 0.34914 10.81 10.749 10.859 10.801 10.701 10.783 0.15815 10.66 10.681 10.644 10.747 10.728 10.692 0.103

Averages 10.728 0.220400

Example of x-bar and R charts: Step 2. Determine Control Limit Formulas and Necessary Tabled Values

x Chart Control Limits

UCL = x + A R

LCL = x - A R2

2

R Chart Control Limits

UCL = D R

LCL = D R4

3

n A2 D3 D42 1.88 0 3.273 1.02 0 2.574 0.73 0 2.285 0.58 0 2.116 0.48 0 2.007 0.42 0.08 1.928 0.37 0.14 1.869 0.34 0.18 1.82

10 0.31 0.22 1.7811 0.29 0.26 1.74

Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart and Plot Values

60110220405872810

85610220405872810

2

2

.)=.(-..R - AxLCL =

.)=.(..R + AxUCL =

10.550

10.600

10.650

10.700

10.750

10.800

10.850

10.900

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Sample

Mea

ns SamplemeanUCL

LCL

grandmean of x

Example of x-bar and R charts: Steps 5&6: Calculate R-chart and Plot Values

0

0.46504

)2204.0)(0(R D= LCL

)2204.0)(11.2(R D= UCL

3

4

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Sample

RRangeUCLLCLR-bar

© Wiley 2010 11

P-Chart Example: A production manager for a tire company has inspected the number of defective tires in five random samples with 20 tires in each sample. The table below shows the number of defective tires in each sample of 20 tires. Calculate the control limits.

Sample

Number of

Defective Tires

Number of Tires in each

Sample

Proportion

Defective

1 3 20 .152 2 20 .103 1 20 .054 2 20 .105 2 20 .05

Total 9 100 .09

Solution:

0.1023(.064).09σzpLCL

.2823(.064).09σzpUCL

0.6420

(.09)(.91)n

)p(1pσ

.09100

9Inspected Total

Defectives#pCL

p

p

p

npppLCL

pCLnpppUCL

)1(3

)1(3

© Wiley 2010 12

P- Control Chart

)1(3

)1(3

ppnpnLCL

pnCL

ppnpnUCL

ANSWER:

Upper Control Limit = 20.22Lower Control Limit = 1.54

© Wiley 2010 16

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.

Week Number of Complaints

1 32 23 34 15 36 37 28 19 310 1

Total 22

Solution:

02.252.232.2ccLCL

6.652.232.2ccUCL

2.21022

samples of #complaints#CL

c

c

z

z

negativeisLCLif0orc3cLCL

cCL

c3cUCL

© Wiley 2010 17

C- Control Chart

For ‘u’ chart refer KV web portal.

THANK YOU…!!!