8. Control Charts for Variables

Control Charts for Variables

Variables due to Variation

• Concept of variation– No 2 things are alike

• Variation exists– Even if variation small and appears same, precision

instruments show differences• Ability to measure variation is necessary before can control


Variation Area

Basically 3 categories of variation in piece part production (e.g. Light bulbs, washer, nuts, etc.) Within piece - surface roughness Piece to piece - dimensions Time to time - different outcomes e.g. morning and

afternoon, tool wear, workers’ tiredness

Control Charts for VariablesSources of Variation• Equipment

- tool wear, electrical fluctuations for welding• Material

- tensile strength, moisture content (e.g. raw material) • Environment

- temperature, light, humidity etc.• Operator

- method, SOP followed, motivation level, training• Inspection

- inspector, inspection equipment, environment

Control Charts for VariablesX-bar chart

In this chart the sample means are plotted in order to control the mean value of a variable (e.g., size of piston rings, strength of materials, etc.).

R chart In this chart, the sample ranges are plotted in order to control the variability of a variable.

S chart In this chart, the sample standard deviations are plotted in order to control the variability of a variable.

S2 chart In this chart, the sample variances are plotted in order to control the variability of a variable.


Regardless of the distribution of population, the distribution of sample means drawn from the population will tend to follow a normal curve

1. The mean of the sampling distribution (x-double bar) will be the same as the population mean m

x = m

s nsx =

2. The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n

Central Limit Theorem

Control Charts for VariablesStatistical Basis• Usually µ and are unknown, so they are estimated from

preliminary samples and subgroups.• 20-25 samples are taken usually each of which contains 4-6

observations• If m samples are taken and n observations are made in each

sample then the best estimator of process mean µ is the grand average.

• Where are the average of each sample

Control Charts for VariablesStatistical Basis (contd.)

• If x1, x2, . . . , xn is the observations for sample size n, then the range of the sample is the difference between the largest and smallest observations; that is,

• Let R1, R2, . . . , Rm be the ranges of the m samples. The average range is


In case of 3-sigma R chart, the control limits can be written as follows

Where is the standard deviation of range values of m samples

In case of 3-sigma x chart, the control limits can be written as follows

Where is the standard deviation of range values of m samples


Control Charts for VariablesFactor 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)


Example


Example

1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 1718 19 20 21 22-0.02

-0.01

-1.73472347597681E-17

0.00999999999999998

0.02

0.03

0.04

0.05

0.06

0.07

RUCLCLLCL

R- chart


Example

X- chart

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2210.7

10.705

10.71

10.715

10.72

10.725

10.73

10.735

10.74

10.745

UCLCLLCL


R-chart(R-chart does not detect change in mean)

UCL

LCL

x-chart(x-chart detects shift in central tendency)

UCL

LCL

(Sampling mean is shifting upward but range is consistent)


R-chart(R-chart detects increase in dispersion)

UCL

LCL

(Sampling mean is constant but dispersion is increasing)

x-chart(x-chart does not detect the increase in dispersion)

UCL

LCL

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8. Control Charts for Variables