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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
© Anita Lee-Post
Control charts
• Use statistical limits to identify whether
a sample of data falls within a normal
range of variation or not
© Anita Lee-Post
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)
© Anita Lee-Post
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)
© Anita Lee-Post
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
© Anita Lee-Post
Mean (x-bar) charts continued
• Use the x-bar chart established to
monitor sample averages as the
process continues:
© 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 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:
© Anita Lee-Post
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
© Anita Lee-Post
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
© Anita Lee-Post
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
© Anita Lee-Post
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
© Anita Lee-Post
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
© Anita Lee-Post
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
© Anita Lee-Post
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
© Anita Lee-Post
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.
© Anita Lee-Post
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