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Session IX “Control Charts for Variables”

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Page 1: “Control Charts for Variables” - WordPress.com ·  · 2016-07-092016-07-09 · Control Charts for Variables ... Within-subgroup variation is used to determine the control limits

Session IX

“Control Charts for Variables”

Page 2: “Control Charts for Variables” - WordPress.com ·  · 2016-07-092016-07-09 · Control Charts for Variables ... Within-subgroup variation is used to determine the control limits

Variation

The variation concept is a law of nature in that no

two natural items are the same.

The variation may be quite large and easily

noticeable

The variation may be very small. It may appear that

items are identical; however, precision instruments

will show difference

The ability to measure variation is necessary before

it can be controlled

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Variation

There are three categories of variation in piece part

production:

1. Within-piece variation “Surface”

2. Piece-to-piece variation “Among pieces

produced at the same time”

3. Time-to-time variation “Difference in product

produced at different times of the day”

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Materials

Tools

Operators MethodsMeasurement

Instruments

HumanInspectionPerformance

EnvironmentMachines

INPUTS PROCESS OUTPUTS

Sources of Variation in production processes:

Variation

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Sources of Variation1. Equipment:Tool wearMachine vibrationElectrical fluctuations etc.

2. MaterialTensile strengthDuctilityThicknessPorosity etc.

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Sources of Variation

3. Environment Temperature Light Radiation Humidity etc.

3. Operator Personal problem Physical problem etc.

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Sources of Variation

There is also a reported variation which is due to the

inspection activity which should be one tenth of the

four other sources of variation.

Variation may be due to chance causes (random

causes) or assignable causes.

When only chance causes are present, then the

process is said to be in a state of statistical control.

The process is stable and predictable.

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Control Charts

Variable data

x-bar and R-charts

x-bar and s-charts

Charts for individuals (x-charts)

Attribute data

For “defectives” (p-chart, np-chart)

For “defects” (c-chart, u-chart)

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ControlCharts

RChart

VariablesCharts

AttributesCharts

XChart

PChart

CChart

Continuous

Numerical DataCategorical or Discrete

Numerical Data

Control Charts

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Control Charts for Variables

The control chart for variables is a mean of

visualizing the variations that occur in the central

tendency and the mean of a set of observations.

It shows whether or not a process is in a stable

state.

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Control Charts

Example of a control chart

Page 12: “Control Charts for Variables” - WordPress.com ·  · 2016-07-092016-07-09 · Control Charts for Variables ... Within-subgroup variation is used to determine the control limits

Control Charts

Example of a method of reporting inspection

results

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Control Charts for Variables

The objectives of the variable control charts are:

1. For quality improvement

2. To determine the process capability

3. For decisions regarding product specifications

4. For current decisions on the production process

5. For current decisions on recently produced items

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Control Charts for Variables

Procedure for establishing a pair of control charts for

the average X-bar and the range R:

1. Select the quality characteristic

2. Choose the rational subgroup

3. Collect the data

4. Determine the trial center line and control limits

5. Establish the revised central line and control limits

6. Achieve the objective

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The Quality characteristic must be measurable.

It can expressed in terms of the seven basic units:

1. Length

2. Mass

3. Time

4. Electrical current

5. Temperature

6. Luminosity

Quality Characteristic

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Rational Subgroup

A rational subgroup is one in which the variation

within a group is due only to chance causes.

Within-subgroup variation is used to determine the

control limits.

Variation between subgroups is used to evaluate

long-term stability.

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Rational Subgroup

There are two schemes for selecting the subgroup

samples:

1. Select subgroup samples from product or service

produced at one instant of time or as close to

that instant as possible

2. Select from product or service produced over a

period of time that is representative of all the

products or services

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Rational Subgroup

The first scheme will have a minimum variation within a subgroup.

The second scheme will have a minimum variation among subgroups.

The first scheme is the most commonly used since it provides a particular time reference for determining assignable causes.

The second scheme provides better overall results and will provide a more accurate picture of the quality.

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Subgroup Size

As the subgroup size increases, the control

limits become closer to the central value, which

make the control chart more sensitive to small

variations in the process average

As the subgroup size increases, the inspection

cost per subgroup increases

When destructive testing is used and the item

is expensive, a small subgroup size is required

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Subgroup Size

From a statistical basis, a distribution of

subgroup averages are nearly normal for groups

of 4 or more even when samples are taken from

a non-normal distribution

When a subgroup size of 10 or more is used, the

s chart should be used instead of the R chart.

See Table 6-1 for sample sizes

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Data Collection

• Data collection can be accomplished using the

type of figure shown in Figure 6-2. It can also be

collected using the method in Table 6-2.

• It is necessary to collect a minimum of 25

subgroups of data.

• A run chart can be used to analyze the data in the

development stage of a product or prior to a state

of statistical control

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Run Chart

Run Chart for data of Table 6-2

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1 1

g g

i i

i i

i

i

X R

X and Rg g

where

X average of subgroup averages

X average of the ith subgroup

g number of subgroups

R average of subgroup ranges

R range of the ith subgroup

Trial Central Lines

Central Lines are obtained using:

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3 3

3 3

R RX X

R RX X

X

R

UCL X UCL R

LCL X LCL R

where

UCL=upper control limit

LCL=lower control limit

population standard deviation of the subgroup averages

population standard deviation of the range

Trial Control Limits

Trial control limits are established at ±3 standard

deviations from the central value

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2 4

2 3

RX

RX

UCL X A R UCL D R

LCL X A R LCL D R

Trial Control Limits

In practice calculations are simplified by using the

following equations where A2,D3 and D4 are factors

that vary with the subgroup size and are found in

Table B of the Appendix.

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Trial Control Limits

X-bar and R chart for preliminary data with trial control limits

Page 27: “Control Charts for Variables” - WordPress.com ·  · 2016-07-092016-07-09 · Control Charts for Variables ... Within-subgroup variation is used to determine the control limits

d dnew new

d d

d

d

d

X X R RX and R

g g g g

where

X discarded subgroup averages

g number of discarded subgroups

R discarded subgroup ranges

Revised Central Lines

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Standard Values

00 0 0

2

new new

RX X R R and

d

0 0 2 0

0 0 1 0

RX

RX

UCL X A UCL D

LCL X A LCL D

28

Page 29: “Control Charts for Variables” - WordPress.com ·  · 2016-07-092016-07-09 · Control Charts for Variables ... Within-subgroup variation is used to determine the control limits

Trial control limits and revised control limits for X-bar and R charts

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Achieve the Objective

Continuing use of control charts, showing improved quality

Page 31: “Control Charts for Variables” - WordPress.com ·  · 2016-07-092016-07-09 · Control Charts for Variables ... Within-subgroup variation is used to determine the control limits

d dnew new

d d

d

d

d

X X R RX and R

g g g g

where

X discarded subgroup averages

g number of discarded subgroups

R discarded subgroup ranges

Revised Central Lines

Page 32: “Control Charts for Variables” - WordPress.com ·  · 2016-07-092016-07-09 · Control Charts for Variables ... Within-subgroup variation is used to determine the control limits

1 1

3 4

3 3

g g

i ii i

sX

sX

s Xs X

g g

UCL X A s UCL B s

LCL X A s LCL B s

Sample Standard Deviation Control Chart

For subgroup sizes >=10, a s chart is more accurate

than an R Chart. Trial control limits are given by:

Page 33: “Control Charts for Variables” - WordPress.com ·  · 2016-07-092016-07-09 · Control Charts for Variables ... Within-subgroup variation is used to determine the control limits

0

00 0

4

0 0 6 0

0 0 5 0

4 5 6, , ,

dnew

d

d

new

d

sX

sX

d

X XX X

g g

s s ss s

g g c

UCL X A UCL B

LCL X A LCL B

where

s discarded subgroup averages

c A B B factors found in Table B

Revised Limits for s chart