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Control Charts for Variables

CHAPTER-3

Variation

There is no two natural items in any category are the same.

Variation may be quite large or very small.

If variation very small, it may appear that items are identical, but precision instruments will show differences.

3 Categories of variation

Within-piece variation One portion of surface is rougher than another

portion.

Apiece-to-piece variation Variation among pieces produced at the same time.

Time-to-time variation Service given early would be different from that given

later in the day.

Source of variation

Equipment Tool wear, machine vibration, …

Material Raw material quality

Environment Temperature, pressure, humadity

Operator Operator performs- physical & emotional

Control Chart Viewpoint

Variation due to

Common or chance causes

Assignable causes

Control chart may be used to discover

“assignable causes”

Some Terms

Run chart - without any upper/lower

limits

Specification/tolerance limits - not

statistical

Control limits - statistical

Control chart functions

Control charts are powerful aids to

understanding the performance of a process

over time.

PROCESS

Input Output

What’s causing variability?

Control charts identify variation

Chance causes - “common cause”

inherent to the process or random and not

controllable

if only common cause present, the process is

considered stable or “in control”

Assignable causes - “special cause”

variation due to outside influences

if present, the process is “out of control”

Control charts help us learn more about

processes

Separate common and special causes of

variation

Determine whether a process is in a state of

statistical control or out-of-control

Estimate the process parameters (mean,

variation) and assess the performance of a

process or its capability

Control charts to monitor processes

To monitor output, we use a control chart

we check things like the mean, range, standard

deviation

To monitor a process, we typically use two

control charts

mean (or some other central tendency measure)

variation (typically using range or standard

deviation)

Types of Data

Variable data

Product characteristic that can be measured

Length, size, weight, height, time, velocity

Attribute dataProduct characteristic evaluated with a discrete

choice

• Good/bad, yes/no

Control chart for variables

Variables are the measurablecharacteristics of a product or service.

Measurement data is taken and arrayed on charts.

Control charts for variables X-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.

X-bar and R charts

The X- bar chart is developed from the

average of each subgroup data.

used to detect changes in the mean between

subgroups.

The R- chart is developed from the ranges of

each subgroup data

used to detect changes in variation within

subgroups

Control chart components

Centerline

shows where the process average is centered or

the central tendency of the data

Upper control limit (UCL) and Lower control

limit (LCL)

describes the process spread

The Control Chart Method

R Control Chart:

UCL = D4 x Rmean

LCL = D3 x Rmean

CL = Rmean

Capability Study:

PCR = (UCL - LCL)/(6S); where S = Rmean /d2

X bar Control Chart:UCL = XDmean + A2 x Rmean LCL = XDmean - A2 x Rmean CL = XDmean

Control Chart Examples

Nominal

UCL

LCL

Sample number

Vari

ati

on

s

How to develop a control chart?

Define the problem

Use other quality tools to help determine the general problem that’s occurring and the process that’s suspected of causing it.

Select a quality characteristic to be measured

Identify a characteristic to study - for example, part length or any other variable affecting performance.

Choose a subgroup size to be sampled

Choose homogeneous subgroups

Homogeneous subgroups are produced under the

same conditions, by the same machine, the same

operator, the same mold, at approximately the

same time.

Try to maximize chance to detect differences

between subgroups, while minimizing chance

for difference with a group.

Collect the data

Generally, collect 20-25 subgroups (100 total

samples) before calculating the control limits.

Each time a subgroup of sample size n is

taken, an average is calculated for the

subgroup and plotted on the control chart.

Determine trial centerline

The centerline should be the population

mean,

Since it is unknown, we use X Double bar, or

the grand average of the subgroup averages.

m

m

i

i

1

X

X

Determine trial control limits - Xbar

chart

The normal curve displays the distribution of

the sample averages.

A control chart is a time-dependent pictorial

representation of a normal curve.

Processes that are considered under control

will have 99.73% of their graphed averages

fall within 6 .

UCL & LCL calculation

deviation standard

3XLCL

3XUCL

Determining an alternative value for

the standard deviation

m

m

i

i

1

R

R

RAXUCL 2

RAXLCL 2

Determine trial control limits - R chart

The range chart shows the spread or

dispersion of the individual samples within

the subgroup.

If the product shows a wide spread, then the

individuals within the subgroup are not similar to

each other.

Equal averages can be deceiving.

Calculated similar to x-bar charts;

Use D3 and D4 (appendix 2)

Example: Control Charts for Variable Data

Slip Ring Diameter (cm)

Sample 1 2 3 4 5 X R

1 5.02 5.01 4.94 4.99 4.96 4.98 0.08

2 5.01 5.03 5.07 4.95 4.96 5.00 0.12

3 4.99 5.00 4.93 4.92 4.99 4.97 0.08

4 5.03 4.91 5.01 4.98 4.89 4.96 0.14

5 4.95 4.92 5.03 5.05 5.01 4.99 0.13

6 4.97 5.06 5.06 4.96 5.03 5.01 0.10

7 5.05 5.01 5.10 4.96 4.99 5.02 0.14

8 5.09 5.10 5.00 4.99 5.08 5.05 0.11

9 5.14 5.10 4.99 5.08 5.09 5.08 0.15

10 5.01 4.98 5.08 5.07 4.99 5.03 0.10

50.09 1.15

Calculation

From Table above:

Sigma X-bar = 50.09

Sigma R = 1.15

m = 10

Thus;

X-Double bar = 50.09/10 = 5.009 cm

R-bar = 1.15/10 = 0.115 cm

Note: The control limits are only preliminary with 10 samples.

It is desirable to have at least 25 samples.

Trial control limit

UCLx-bar = X-D bar + A2 R-bar = 5.009 + (0.577)(0.115) = 5.075 cm

LCLx-bar = X-D bar - A2 R-bar = 5.009 -(0.577)(0.115) = 4.943 cm

UCLR = D4R-bar = (2.114)(0.115) = 0.243 cm

LCLR = D3R-bar = (0)(0.115) = 0 cm

For A2, D3, D4: see Table B, Appendix n = 5

3-Sigma Control Chart Factors

Sample size X-chart R-chart

n A2 D3 D4

2 1.88 0 3.27

3 1.02 0 2.57

4 0.73 0 2.28

5 0.58 0 2.11

6 0.48 0 2.00

7 0.42 0.08 1.92

8 0.37 0.14 1.86

X-bar Chart

4.94

4.96

4.98

5.00

5.02

5.04

5.06

5.08

5.10

0 1 2 3 4 5 6 7 8 9 10 11

Subgroup

X b

ar

LCL

CL

UCL

R Chart

0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7 8 9 10 11

Subgroup

Range

LCL

CL

UCL

Run Chart

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20 25

Subgroup number

Ra

ng

e, R

6.30

6.35

6.40

6.45

6.50

6.55

6.60

6.65

6.70

0 5 10 15 20 25

Subgroup numberM

ean

, X

-bar

Another Example of X-bar & R chart

Subgro

up X1

X2

X3

X4

X-bar UCL-X-bar

X-Dbar LCL-X-bar

R UCL-R

R-bar LCL-R

1 6.35 6.4 6.32 6.37 6.36 6.47 6.41 6.35 0.08 0.20 0.0876 0

2 6.46 6.37 6.36 6.41 6.4 6.47 6.41 6.35 0.1 0.20 0.0876 0

3 6.34 6.4 6.34 6.36 6.36 6.47 6.41 6.35 0.06 0.20 0.0876 0

4 6.69 6.64 6.68 6.59 6.65 6.47 6.41 6.35 0.1 0.20 0.0876 0

5 6.38 6.34 6.44 6.4 6.39 6.47 6.41 6.35 0.1 0.20 0.0876 0

6 6.42 6.41 6.43 6.34 6.4 6.47 6.41 6.35 0.09 0.20 0.0876 0

7 6.44 6.41 6.41 6.46 6.43 6.47 6.41 6.35 0.05 0.20 0.0876 0

8 6.33 6.41 6.38 6.36 6.37 6.47 6.41 6.35 0.08 0.20 0.0876 0

9 6.48 6.44 6.47 6.45 6.46 6.47 6.41 6.35 0.04 0.20 0.0876 0

10 6.47 6.43 6.36 6.42 6.42 6.47 6.41 6.35 0.11 0.20 0.0876 0

11 6.38 6.41 6.39 6.38 6.39 6.47 6.41 6.35 0.03 0.20 0.0876 0

12 6.37 6.37 6.41 6.37 6.38 6.47 6.41 6.35 0.04 0.20 0.0876 0

13 6.4 6.38 6.47 6.35 6.4 6.47 6.41 6.35 0.12 0.20 0.0876 0

14 6.38 6.39 6.45 6.42 6.41 6.47 6.41 6.35 0.07 0.20 0.0876 0

15 6.5 6.42 6.43 6.45 6.45 6.47 6.41 6.35 0.08 0.20 0.0876 0

16 6.33 6.35 6.29 6.39 6.34 6.47 6.41 6.35 0.1 0.20 0.0876 0

17 6.41 6.4 6.29 6.34 6.36 6.47 6.41 6.35 0.12 0.20 0.0876 0

18 6.38 6.44 6.28 6.58 6.42 6.47 6.41 6.35 0.3 0.20 0.0876 0

19 6.35 6.41 6.37 6.38 6.38 6.47 6.41 6.35 0.06 0.20 0.0876 0

20 6.56 6.55 6.45 6.48 6.51 6.47 6.41 6.35 0.11 0.20 0.0876 0

21 6.38 6.4 6.45 6.37 6.4 6.47 6.41 6.35 0.08 0.20 0.0876 0

22 6.39 6.42 6.35 6.4 6.39 6.47 6.41 6.35 0.07 0.20 0.0876 0

23 6.42 6.39 6.39 6.36 6.39 6.47 6.41 6.35 0.06 0.20 0.0876 0

24 6.43 6.36 6.35 6.38 6.38 6.47 6.41 6.35 0.08 0.20 0.0876 0

25 6.39 6.38 6.43 6.44 6.41 6.47 6.41 6.35 0.06 0.20 0.0876 0

Given Data (Table 5.2)

Calculation

From Table 5.2:

Sigma X-bar = 160.25

Sigma R = 2.19

m = 25

Thus;

X-double bar = 160.25/29 = 6.41 mm

R-bar = 2.19/25 = 0.0876 mm

Trial control limit

UCLx-bar = X-double bar + A2R-bar = 6.41 + (0.729)(0.0876) = 6.47 mm

LCLx-bar = X-double bar - A2R-bar = 6.41 –(0.729)(0.0876) = 6.35 mm

UCLR = D4R-bar = (2.282)(0.0876) = 0.20 mm

LCLR = D3R-bar = (0)(0.0876) = 0 mm

For A2, D3, D4: see Table B Appendix, n = 4.

X-bar Chart

R Chart

Revised CL & Control Limits

Calculation based on discarding subgroup 4 & 20 (X-bar chart) and subgroup 18 for R chart:

= (160.25 - 6.65 - 6.51)/(25-2)

= 6.40 mm

= (2.19 - 0.30)/25 - 1

= 0.079 = 0.08 mmd

d

newmm

RRR

d

d

newmm

XXX

New Control Limits

New value:

Using standard value, CL & 3 control limit obtained using formula:

2

,,d

RRRXX O

onewonewo

oRoR

ooXooX

DLCLDUCL

AXLCLAXUCL

12 ,

,

From Table B:

A = 1.500 for a subgroup size of 4,

d2 = 2.059, D1 = 0, and D2 = 4.698

Calculation results:

mmXX newo 40.6 mmd

RRR o

onewo 038.0059.2

079.0,079.0

2

mmAXUCL ooX46.6)038.0)(500.1(40.6

mmAXLCL ooX34.6)038.0)(500.1(40.6

mmDUCL oR 18.0)038.0)(698.4(2

mmDLCL oR 0)038.0)(0(1

Trial Control Limits & Revised Control Limit

6.30

6.35

6.40

6.45

6.50

6.55

6.60

6.65

0 2 4 6 8

Subgroup

Mean

, X

-bar

0.00

0.05

0.10

0.15

0.20

0 2 4 6 8

Subgroup

Ran

ge, R

UCL = 6.46

CL = 6.40

LCL = 6.34

LCL = 0

CL = 0.08

UCL = 0.18

Revised control limits

Revise the charts

In certain cases, control limits are revised

because:

out-of-control points were included in the

calculation of the control limits.

the process is in-control but the within

subgroup variation significantly

improves.

Revising the charts

Interpret the original charts

Isolate the causes

Take corrective action

Revise the chart Only remove points for which you can determine an

assignable cause

Process in Control

When a process is in control, there occurs a

natural pattern of variation.

Natural pattern has:

About 34% of the plotted point in an imaginary

band between 1 on both side CL.

About 13.5% in an imaginary band between 1

and 2 on both side CL.

About 2.5% of the plotted point in an imaginary

band between 2 and 3 on both side CL.

The Normal

Distribution

-3 -2 -1 +1 +2 +3Mean

68.26%

95.44%

99.74%

= Standard deviation

LSL USL

-3 +3CL

34.13% of data lie between and 1 above the mean ( ).

34.13% between and 1 below the mean.

Approximately two-thirds (68.28 %) within 1 of the mean.

13.59% of the data lie between one and two standard deviations

Finally, almost all of the data (99.74%) are within 3 of the mean.

Define the 3-sigma limits for sample means as follows:

What is the probability that the sample means will lie

outside 3-sigma limits?

Note that the 3-sigma limits for sample means are

different from natural tolerances which are at

Normal Distribution Review

94345

0503015

3

07755

0503015

3

.).(

. Limit Lower

.).(

. Limit Upper

n

n

3

Common Causes

Process Out of Control

The term out of control is a change in the

process due to an assignable cause.

When a point (subgroup value) falls outside

its control limits, the process is out of control.

Assignable Causes

(a) Mean

Grams

Average

Assignable Causes

(b) Spread

Grams

Average

Assignable Causes

(c) Shape

Grams

Average

Control Charts

UCL

Nominal

LCL

Assignable

causes

likely

1 2 3

Samples

Control Chart Examples

Nominal

UCL

LCL

Sample number

Vari

ati

on

s

Control Limits and Errors

LCL

Process

average

UCL

(a) Three-sigma limitsType I error:

Probability of searching for

a cause when none exists

Control Limits and Errors

Type I error:

Probability of searching for

a cause when none exists

UCL

LCL

Process

average

(b) Two-sigma limits

Type II error:

Probability of concluding

that nothing has changed

Control Limits and Errors

UCL

Shift in process

average

LCL

Process

average

(a) Three-sigma limits

Type II error:

Probability of concluding

that nothing has changed

Control Limits and Errors

UCL

Shift in process

average

LCL

Process

average

(b) Two-sigma limits

Achieve the purpose

Our goal is to decrease the variation inherent

in a process over time.

As we improve the process, the spread of the

data will continue to decrease.

Quality improves!!

Improvement

Examine the process

A process is considered to be stable and

in a state of control, or under control,

when the performance of the process

falls within the statistically calculated

control limits and exhibits only chance, or

common causes.

Consequences of misinterpreting the

process Blaming people for problems that they cannot

control

Spending time and money looking for problems that

do not exist

Spending time and money on unnecessary process

adjustments

Taking action where no action is warranted

Asking for worker-related improvements when

process improvements are needed first

Process variation

When a system is subject to only

chance causes of variation, 99.74% of

the measurements will fall within 6

standard deviations

If 1000 subgroups are measured,

997 will fall within the six sigma

limits.

-3 -2 -1 +1 +2 +3Mean

68.26%

95.44%

99.74%

Chart zones

Based on our knowledge of the normal curve, a

control chart exhibits a state of control when:

♥ Two thirds of all points are near the center

value.

♥ The points appear to float back and forth

across the centerline.

♥ The points are balanced on both sides of the

centerline.

♥ No points beyond the control limits.

♥ No patterns or trends.