Download Capability Metrics (Q and R) - Universal Instruments Corporation

UNIVERSAL INSTRUMENTS CORPORATION
Capability Metrics
Mean
The sample mean is the arithmetic average of a set of
measurements. The sample mean estimates the population mean, and
represents the expected value of the population mean under most
circumstances.

x
Q&R 04/29/98 1
n
i 1
n
xi

x 1  x 2  ... x n
,
n
where x i  x 1 , x 2 ,.... x n and n  sample size
UNIVERSAL INSTRUMENTS CORPORATION
Capability Metrics
(Continued)
Accuracy
A measure of the difference between the mean of the process and the
target value. Typically, the target value is centered between the upper
and lower specifications. (Note: for some customers, accuracy has a
different meaning.)
Mean
Target
4
3. 75
3. 5
3. 25
3
2. 75
2. 5
2. 25
2
1. 75
1. 5
1. 25
1
0. 5
0. 75
0. 25
-1.421E-14
-0.5
-0.25
-1
-0.75
-1.5
-1.25
-1.75
-2
-2.25
-2.5
-2.75
-3
-3.25
-3.5
Q&R 04/29/98 2
-3.75
-4
Accuracy
UNIVERSAL INSTRUMENTS CORPORATION
Capability Metrics
(Continued)
Standard Deviation
A measure of the spread or dispersion of the values of a random
variable. It is defined on a sample by the equation given here.
Alternatively, the standard deviation is the expected distance any
one observation falls from the mean.
 x
n
s 
i 1
i
 x
n 1
2

x
 x    x 2  x  ...  x n  x 
2
1
2
n 1
2
, where x i  x 1  x 2 ... x n and n  sample size
Repeatability
Repeatability is the ability to repeat the same placement by the same
process, at or near the same time. It is usually measured by the
standard deviation of the process. (Note: for some customers, it is
measured by three standard deviations.)
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UNIVERSAL INSTRUMENTS CORPORATION
Capability Metrics
(Continued)
Cpk
Process capability index, which is a measure of the process’ ability to
produce product within specification. If the process distribution is known,
then a ppm defective rate can be calculated for the process using the
Cpk.
Cpk 
min  x  Lower Spec Limit , Upper Spec Limit  x

3s

Distance between the mean and the nearest spec
3s
where s = the standard deviation of the sample
USL
LSL
Mean
4
3.5
3.75
3
3.25
2.5
2.75
2
2.25
1.5
1.75
1
1.25
0.5
0.75
0.25
-1.421E-14
-0.5
-0.25
-1
-0.75
-1.5
-1.25
-2
-1.75
-2.5
-2.25
-3
-2.75
-3.5
-3.25
-4
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-3.75
Cpk Numerator
UNIVERSAL INSTRUMENTS CORPORATION
Capability Metrics
(Continued)
Cp
The Cp is a capability index which compares the spread of the process
to the distance between the upper and lower specifications. If the mean
could be shifted to the midpoint between the specification limits (the
mean could be dialed out), then the Cpk value would increase to the
value of the Cp. In this way, the Cp is the highest value that the Cpk can
take without reducing the process standard deviation.
Cp 
UpperSpecLimit  LowerSpecLimit
6s
where s = the standard deviation of the sample
USL
LSL
Cp Numerator
4
3. 5
3. 75
3
3. 25
2. 5
2. 75
2
2. 25
1. 5
1. 75
1
1. 25
0. 5
0. 75
0. 25
-1.421E-14
-0.5
-0.25
-1
-0.75
-1.5
-1.25
-2
-1.75
-2.5
-2.25
-3
-2.75
-3.5
-3.25
-4
-3.75
Q&R 04/29/98 5
UNIVERSAL INSTRUMENTS CORPORATION
Capability Metrics
(Continued)
Sigma Level
A process capability index, which is a measure of the process’ ability to
produce product within specifications. If the process is known, then a
ppm defective rate can be calculated for the process using the sigma
level. (Note: Motorola uses a different method for obtaining the ppm
defect rate. To avoid confusion, it is better to speak in terms of Cpk.)
SigmaLevel 
UpperSpecLimit  LowerSpecLimit
2s
where s = the standard deviation of the sample
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