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
Underlying DistributionsUnderlying Distributions
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
Underlying DistributionsUnderlying Distributions
The standard deviation among the sample means is smaller by a factor of
Therefore,
n
1
Underlying DistributionsUnderlying Distributions
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
Control Limits for X-bar ChartControl Limits for X-bar Chart
UCLLCL
Xn
X
3n
X
3
Distribution of X
nXUCL
nXLCL
ˆ3
ˆ3
RAXUCL
RAXLCL
2
2
Control Limits for X-bar ChartControl Limits for X-bar Chart
Control Limits for X-bar ChartControl Limits for X-bar Chart
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)
Example of X-bar ChartExample of X-bar Chart
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:
Example of X-bar ChartExample of X-bar Chart
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?
Example of X-bar ChartExample of X-bar Chart
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
X-bar Chart Control LimitsX-bar Chart Control Limits
0.24
)5.8(02.133.15
66.6
)5.8(02.133.15
2
2
RAXUCL
RAXLCL
X-bar Chart Control LimitsX-bar Chart Control Limits
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.
Interpretation of X-bar ChartInterpretation of X-bar Chart
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.
Interpretation of X-bar ChartInterpretation of X-bar Chart
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.
Interpretation of X-bar ChartInterpretation of X-bar Chart
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
Underlying DistributionsUnderlying Distributions
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.