Transcript qc2
Quality Control
Part 2
By
Anita Lee-Post
© Anita Lee-Post
Statistical process control methods
• Control charts for variables: process
characteristics are measured on a
continuous scale, e.g., weight, volume,
width
•
•
Mean (X-bar) chart
Range (R) chart
• Control charts for attributes: process
characteristics are counted on a discrete
scale, e.g., number of defects, number of
scratches
•
•
Proportion (P) chart
Count (C) chart
• Process capability ratio and index
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Control charts
• Use statistical limits to identify whether
a sample of data falls within a normal
range of variation or not
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Setting Limits Requires Balancing Risks
• Control limits are based on a willingness to think
that something is wrong when it’s actually not
(Type I or alpha error), balanced against the
sensitivity of the tool - the ability to quickly reveal
a problem (failure is Type II or beta error)
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Control Charts for Variable Data
• Mean (x-bar) charts
•
Tracks the central tendency (the average
value observed) over time
• Range (R) charts:
•
Tracks the spread of the distribution over
time (estimates the observed variation)
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Mean (x-bar) charts
CL process
mean x
x1 x2 ... xk
k
UCL x z x
LCL x z x
x ...xn
where xi 1
, xi : observation i,
n
n : number of observations (sample size)
k : number of samples
z : number of normal standard deviation
x : the standard deviation of the process mean
n
: the standard deviation of the process/population
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Mean (x-bar) charts continued
• Use the x-bar chart established to
monitor sample averages as the
process continues:
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An example
The diameters of five C&A bagels are
sampled each hour during a 8-hour period.
The data collected are shown as follows:
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An example continued
a)
Develop an x-bar chart with the control limits set
to include 99.74% of the sample means and the
standard deviation of the production process ()
is known to be 0.2 Inches.
Step 1. Compute the sample mean x-bar:
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An example continued
Step 2. Compute the process mean or center line of
the control chart:
4.03 3.94 4.18 4.01 4.03 4.08 4.04 4.03
CL x
8
32.34
4.04
8
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An example continued
Step 3. Compute the upper and lower control limits:
To include 99.74% of the sample means implies that the
number of normal standard deviations is 3. i.e., z=3
UCL x z x
0. 2
4.3
4.04 3
5
LCL x z x
0.2
3.8
4.04 3
5
© Anita Lee-Post
An example continued
b.
C&A collects the process characteristics (i.e.,
diameter) of their bagels in days 2 through 10. Is
the process in control?
Diameter of Sample
Day
2
3
4
5
6
7
8
9
10
Average of 5 observations 4.21 3.99 3.89 4.05 4.22 4.28 3.55 3.78 5.00
x-bar Chart for Samples of 5 Bagels
Average Diameter (in
inches)
5.25
5.00
4.75
4.50
UCL = 4.25
4.25
4.00
3.75
LCL = 3.83
3.50
2
3
4
5
6
Sample
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7
8
9
10
The process is
not in control
because the
means of recent
sample averages
fall outside the
upper and lower
control limits
Range (R) charts
CL R
R1 R2 ... Rk
k
UCL D4R
LCL D3R
where R : the average of the sample range
k : number of samples
Ri : the range of sample i max(xi) min(xi)
xi : the observatio n of sample i
D4 : the R - chart factor for UCL
D3 : the R - chart factor for LCL
© Anita Lee-Post
An example
The diameters of five C&A bagels are
sampled each hour during a 8-hour period.
The data collected are shown as follows:
© Anita Lee-Post
An example continued
a)
Develop a range chart.
Step 1. Compute the average range
or CL:
0.18 0.10 1.07 0.12 0.21 0.13 0.21 0.13
8
0.27
CL R
© Anita Lee-Post
An example continued
Step 2. Compute the upper and lower control limits:
Control Limit Factors for Range Charts
Sample size, n
D3
D4
2
0.00
3.27
3
0.00
2.57
4
0.00
2.28
5
0.00
2.11
6
0.00
2.00
7
0.08
1.92
8
0.14
1.86
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UCL D4 R 2.11 0.27
0.57
LCL D3 R 0 0.27
0
An example continued
b.
C&A collects the process characteristics (i.e., diameter)
of their bagels in days 2 through 10. Is the process in
control?
Day
Range of 5 observations
Diameter of Sample
2
3
4
5
6
7
8
9
10
0.30 0.20 0.33 0.20 0.14 0.11 0.05 0.35 0.20
Range Chart for Samples of 5 Bagels
UCL = 0.57
0.60
Range
0.50
0.40
CL = 0.27
0.30
0.20
0.10
LCL = 0
0.00
2
3
4
5
6
Sample
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7
8
9
10
The process is in
control because
the ranges of
recent samples
fall within the
upper and lower
control limits
Using both mean & range charts
• Mean (x-bar) chart: measures the central tendency
of a process
• Range (R) chart: measures the variance of a
process
Case 1: a process showing a drift in its mean but
not its variance
can be detected only by a mean (x-bar) chart
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Using both mean & range charts continued
Case 2: a process showing a change in its variance
but not its mean
can be detected only by a range (R) chart
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Construct x-bar chart from sample range
CL x
x1 ... xk
k
UCL x A2R
LCL x A2R
R R2 ... Rk
where R 1
k
Ri : max(xi) min(xi )
xi : observation for sample i
k : number of samples
A 2 : Control limit factor for x - bar chart
© Anita Lee-Post
Control Charts for Attributes
• p-Charts:
•
Track the proportion defective in a
sample
• c-Charts:
•
Track the average number of defects per
unit of output
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Proportion (p) charts
• Data requirements:
•
Sample size
• Number of defects
• Sample size is large enough so that the
attributes will be counted twice in each
sample, e.g., a defect rate of 1% will
require a sample size of 200 units.
© Anita Lee-Post
Proportion (p) charts continued
Total Number of defects from all samples
CL p
Number of samples Sample size
UCL p z p
LCL p z p
where z : number of normal standard deviation
p the sample standard deviation
p (1 p )
n
n : sample size
© Anita Lee-Post
Count (c) charts
• Data requirements
•
Number of defects
• Monitoring processes in which the items
of interest (in this case, defects) are
infrequent and/or occur in time or space,
e.g., errors in newspaper, bad circuits in
a microchip, complaints from customers.
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Count (c) charts continued
CL c average number of defects
n
xi
i 1 , where n : number of days/weeks/units
n
UCL c z c
LCL c z c
where z : number of standard deviation
© Anita Lee-Post