Chapter 6 Part 3 X-bar and R Control Charts. Attribute Data Data that is discrete Discrete data is based on “counts.” Assumes integer values G Number.

Chapter 6 Part 3Chapter 6 Part 3

X-bar and R Control Charts

Attribute DataAttribute Data

Data that is discrete Discrete data is based on “counts.” Assumes integer values

Number of defective units Number of customers who are “very satisfied” Number of defects

Variables DataVariables Data

X-bar and R chart is used to monitor mean and variance of a process when quality characteristic is continuous.

Continuous values (variables data) can theoretically assume an infinite number of values in some interval. Time Weight Ounces Diameter

X-bar and R ChartX-bar and R Chart

X-bar chart monitors the process mean by using the means of small samples taken frequently

R chart monitors the process variation by using the sample ranges as the measure of variability Range = Maximum value – Minimum value

NotationNotation

samples ofnumber k

n

X

XX

bar)-(mean sample

size samplen

sticcharacteriquality X

NotationNotation

sXX theofmean

bar)-( ofmean sample XXX

mean overall the

or mean, process estimated the

"bar, double"

called also is

X

X

X ofdeviation standard estimatedˆ

NotationNotation

Xn

oferror standard estimatedˆ

k

RR

R

ranges ofMean

valueMinimum - valueMaximum

Range Sample

Example of NotationExample of Notation

A company monitors the time (in minutes) it takes to assemble a product.

The company decides to sample 3 units of the product at three different times tomorrow: 9 AM 12 Noon 3 PM

What is the sample size, n? What is k, the number of samples?

Suppose the following data are obtained.

How would you compute X-bar? R R-bar X double bar

Hour

Assembly Time (minutes)

X1 X2 X3

9:00 AM 5 12 4

12 Noon 6 8 10

3:00 PM 9 4 2

Example of NotationExample of Notation

Hour

Assembly Time (minutes)

Sample

Mean,

Sample Range, R

X1 X2 X3

9:00 AM 5 12 4 21/3 = 7 12 – 4 = 8

12 Noon 6 8 10 24/3 = 8 10 – 6 = 4

3:00 PM 9 4 2 15/3 = 5 9 – 2 = 7

= 20/3 = 6.7 =19/3 =6.3

X

X R

73

21

3

4125

n

XX

n

XX

Sample MeansSample Means

First Sample

Sample MeansSample Means

Second Sample

83

24

3

1086

n

XX

53

15

3

249

n

XX

Third Sample

Estimated Process Mean Estimated Process Mean

7.6

3

20

3

587

k

XX

Sample RangesSample Ranges

72-9

:Sample Third

46-10

:Sample Second

8412

:SampleFirst

Value Minimum - Value Maximum

R

R

R

R

3.6

3

19

3

748

k

RR

Mean of Mean of RR

Underlying DistributionsUnderlying Distributions

When constructing an X-bar chart, we actually have two distributions to consider:

The distribution of the sample means , and

The process distribution, the distribution of the quality characteristic itself, X.

The distribution of is a distribution of averages.

The distribution of X is a distribution of ???

X

X


These distributions have the same mean

Their variances (or standard deviations) are different.

Which distribution has the bigger variance? Would you expect more variability among

averages or among individual values? The variability among the individual values

is ???

XX ofMean ofMean

n

ˆ1ˆ of Std

of Std

nX

X


The standard deviation among the sample means is smaller by a factor of

Therefore,

n

1


X

Samplingdistribution of Distribution

of X

X

Distribution of Distribution of XX

UCLLCL M

M

m

M

M

m

M

M

m

X X

The distribution of X is assumed to be normal. This assumption needs to be tested in practice.

Distribution of Distribution of XX-bar-bar

UCLLCL M

M

m

M

M

m

M

M

m

n

X X

If the distribution of X is normal, the distribution of X-bar will be normal for any sample size.

Control Limits for X-bar ChartControl Limits for X-bar Chart

Since we are plotting sample means on the X-bar chart, the control limits are based on the distribution of the sample means.

The control limits are therefore

nXUCL

3

nXLCL

3


UCLLCL

Xn

X

3n

X

3

Distribution of X

nXUCL

nXLCL

ˆ3

ˆ3

RAXUCL

RAXLCL

2

2



RAn

2

ˆ3

A2 is a factor that depends on the n, the sample size,

and will be given in a table.

Example of Example of XX-bar Chart-bar Chart

A company that makes soft drinks wants to monitor the sugar content of its drinks.

The sugar content (X) is normally distributed, but the means and variances are unknown.

The target sugar level for one of its drinks is 15 grams.

The lower spec limit is 10 grams. The upper spec limit is 20 grams.

Example of X-bar ChartExample of X-bar Chart

The company wants to know how much sugar on average is being put into this soft drink and how much variability there is in the sugar content in each bottle.

The company also wants to know if the mean sugar content is on target.

Lastly, the company wants to know the percentage of drinks that are too sweet and the percentage that are not sweet enough. (Next section)


To obtain this information, the company decides to sample 3 bottles of the soft drink at 3 different time each day: 10 A.M, 1:00 P.M. and 4:00 P.M.

The company will use this data to construct an X-bar and R chart. (In practice, you need 25-30 samples to construct the control limits.)

For the past two days, the following data were collected:


Day Hour X1 X2 X3

1 10 am 17 13 6

1 pm 15 12 24

4 pm 12 21 15

2 10 am 13 12 17

1 pm 18 21 15

4 pm 10 18 17

What is n?

What is the k?

What is the next step?


Day Hour X1 X2 X3 R

1 10 am 17 13 6 36/3 =12 11

1 pm 15 12 24 51/3 =17 12

4 pm 12 21 15 48/3 =16 9

2 10 am 13 12 17 42/3 =14 5

1 pm 18 21 15 54/3 =18 6

4 pm 10 18 17 45/3 =15 8

= 92/6

= 15.33

= 51/6

= 8.5 X R

X

X-bar Chart Control LimitsX-bar Chart Control Limits

RAXUCL

RAXLCL

2

2

Table A: X-bar Chart Factor, Table A: X-bar Chart Factor, AA22

n A2

2 1.88

3 1.02

4 0.73

5 0.58

02.1

3

textof 182 p. 1,-6 Tableor notesin A Table From

5.8

33.15

2

A

n

R

X


0.24

)5.8(02.133.15

66.6

)5.8(02.133.15

2

2

RAXUCL

RAXLCL


X-bar Chart for Sugar Content

0.00

5.00

10.00

15.00

20.00

25.00

30.00

10 1 4 10 1 4

Hour Hour

1 2

Day

Interpretation of X-bar ChartInterpretation of X-bar Chart

The X-bar chart is in control because ???? This means that the only source of

variation among the sample mean is due to random causes.

The process mean is therefore stable and predictable and, consequently, we can estimate it.

Interpretation of X-bar Chart Interpretation of X-bar Chart

Our best estimate of the mean is the center line on the control chart, which is the overall mean (X-double bar) of 15.33 grams.

If the process remains in control, the company can predict that all bottles of this soft drink produced in the future will have a sugar content of, on average, 15.33 grams.

Interpretation of X-bar Chart Interpretation of X-bar Chart

This prediction, however, indicates that there is a problem with the location of the mean.

The process mean is off target by 0.33 grams (15.33 -15.00).

The process mean, although stable and predictable, is at the wrong level and should be corrected to the target.


Since the process mean is in control, there are no special causes of variation that may be responsible for the mean being off target.

Since the operators are responsible for correcting problems due to special causes and management is responsible for correcting problems due to random causes of variation, management action is required to fix this problem.


The reason is that, because the process is in control, the filling machines require more than a simple adjustments (typically due to special causes) which can be made by the operators.

The machines may require new parts, a complete overhaul, or they may simply not be capable of operating on target, in which case a new machine is required.


Expecting the operators to adjust the mean to the target when the process is in control is analogous to requiring that you produce zero heads (head = defective unit) if you are hired to toss a fair coin 100 times each day.

Why?

R ChartR Chart

Monitors the process variability (the variability of X)

Tells us when the process variability has changed or is about to change.

R chart must be in control before we can use the X-bar chart.

R ChartR Chart

Rules for detecting changes in variance: If at least one sample range falls above the upper

control limit, or there is an upward trend within the control limits, process variability has increased.

If at least one sample range falls on or below the lower control limit, or there is a downward trend within the control limits, process variability has decreased.

RDUCL

RDLCL

4

3

R Chart Control LimitsR Chart Control Limits

n D3 D4

2 0 3.27

3 0 2.57

4 0 2.28

5 0 2.11

Table B: Factors for R ChartTable B: Factors for R Chart

85.21

)5.8(57.2

0

)5.8(0

UCL

LCL

R Chart Control LimitsR Chart Control Limits

57.2

0

3

4

3

D

D

n

R Chart for Weight

05

1015

2025

10 1 4 10 1 4

Hour Hour

1 2

Day

R

LCL

UCL

R-bar

Interpretation of R ChartInterpretation of R Chart

Since all of the sample ranges fall within the control limits, the R chart is in control.

The standard deviation is stable and predictable and can be estimated—done in next section.

This does not necessarily mean that the amount of variation in the process is acceptable.

Interpretation of R ChartInterpretation of R Chart

Continuous improvement means the company should continuously reduce the variance.

Since the process variation is in control, management action is required to reduce the variation.

X-bar Chart

UCL

LCL

Expected Pattern in a Stable ProcessExpected Pattern in a Stable Process

Time

Expected pattern is a normal distribution

x-Chart

UCL

Does notreveal increase

How Non-Random Patterns Show UpHow Non-Random Patterns Show Up

UCL

LCL

LCL

R-chart Reveals increase

(process variability is increasing)SamplingDistribution

How Non-Random Patterns Show UpHow Non-Random Patterns Show Up

UCL

LCL

UCL

LCL

R-chart

x-Chart Detects shift

Does notdetect shift

(process mean is shifting upward)

SamplingDistribution

Is a Stable Process a Good Process?Is a Stable Process a Good Process?

“In control” indicates that the process mean is stable and hence predictable.

A stable process, however, is not necessary a “good” (defect free) process.

The process mean, although stable, may be far off target, resulting in the production of defective product.

In this case, we have, as Deming puts it, “A stable process for the production of defective product.”

Control Limits vs. Spec. LimitsControl Limits vs. Spec. Limits

Control limits apply to sample means, not individual values.

Mean diameter of sample of 5 parts, X-bar

Spec limits apply to individual values

Diameter of an individual part, X

Control Limits vs. Spec. LimitsControl Limits vs. Spec. Limits

Samplingdistribution, X-bar

Processdistribution, X

Mean=Target

Lowercontrol

limit

Uppercontrol

limit

LSLUSL


X Distribution

of X

X USLLSL

LCL UCL

Control limits are put on distribution of X-barSpec limits apply to the distribution of X

X

Samplingdistribution of

Responsibility for Corrective ActionResponsibility for Corrective Action

Special Causes

(Process out of control)

Random Variation

(Process in Control)

Operators

(workers)

Management

Benefits of Control ChartsBenefits of Control Charts Control charts prevent unnecessary

adjustments. If process is in control, do not adjust it. Adjustments will increase the variance. Management action is required to improve process.

Adjustments should be made only when special causes occur.

Benefits of Control ChartsBenefits of Control Charts

Control charts assign responsibility for corrective action.

Control charts are the only statistical valid way to estimate the mean and variance of a process or product.

Control charts make it possible to predict future performance of a process and thereby take early corrective action.

Chapter 6 Part 3 X-bar and R Control Charts. Attribute Data Data that is discrete Discrete data is based on “counts.” Assumes integer values G Number.

Documents

x slide

sample slide

xbar chart slide

distribution of x

notation slide

sample ranges slide

x bar chart

pm942 slide