Technical
Definition of Six Sigma
If you're into mathematics
you might wonder why Six Sigma is related to 3.4 defect per million
rather than the 2 defects per billion you get if you calculate
the defect rate for a perfectly centered six sigma process.
Motorola
created the definition of a Six Sigma Process which recognizes the
difficultly of keeping the center of the process output (or mean) on target over time. Noise
factors tend to move the process mean subtly. Once this shift is
significant enough to be detected, the process can be adjusted to move the
output back on target. The Motorola definition allows plus or minus 1.5
standard deviations of process shift in the mean of the process output. This
is (maybe coincidentally, or maybe not,) the amount of shift in the process mean which, when plotted on a
control chart with a sub group size n=4, would trigger an out-of-control
condition in just one sample point.
The animated chart
shows the distribution process output in red. The process target = 0,
the Lower Specification Limit = -6 and the Upper Specification
Limit = +6 (i.e. the limits of the chart are the acceptance boundary). m
= process mean, s = process standard deviation.
The blue curve shows a perfectly centered six sigma process and the
green curve a six sigma process at the maximum allowable positive shift,
for comparison.
The animation changes
through the following transitions.
| Step |
Transition |
Cp |
Cpk |
defects per million (dpm) |
| 1 |
Mean, m = 4.2, Sigma Capability 2.5
(=6/s) |
0.83 |
0.25 |
226,638 |
| 2 |
to Sigma Capability 4.5 |
1.5 |
0.45 |
88,508 |
| 3 |
to Mean, m = 2.0 (+ 1.5s
shift) |
1.5 |
1.0 |
1,350 |
| 4 |
to Mean, m = 0 |
1.5 |
1.5 |
6.8 |
| 5 |
to Sigma Capability 4.645 |
1.55 |
1.55 |
3.4 |
| 6 |
to Sigma Capability 6.0 |
2.0 |
2.0 |
0.002 |
| 7 |
to Mean, m
= -1.5 (-1.5s shift) |
2.0 |
1.5 |
3.4 |
The effect of the shift taking the defect rate from 2 per billion to 3.4
per million can be seen in steps 6 to 7.
At worst
a Six Sigma process exhibits no more defects than the number of
defects that would be observed in one of the tails of an absolutely
perfectly centered 4.5 sigma capable process. (Step 4)
(or At worst
a Six Sigma process exhibits no more
defects than the number of defects that would be observed overall (in both tails) of
an absolutely perfectly centered 4.645 sigma capable process) (Step 5)
The formal
definition of a Six Sigma rated process is:
Cp=2, Cpk=1.5,
dpm=3.4
Cp
and Cpk are traditional process performance measures relating
process quality to natural tolerance i.e. +/-3s
Cp =
Specification Width / (2 X 3s)
(Note: Cp takes no account of process mean position)
Cpk =
the distance from the process mean to the nearest spec limit / (3s)
Steps 6 and
7 (in green) in the table above
achieve (or better) the Motorola definition of Six Sigma.
Step 5
(in yellow)
achieves the required defects per million, but fails on one important criteria, Cp. Over
time the process is unlikely to sustain this defect rate. As soon as the
process mean moves slightly the number defects will rise considerably!
"Variation
is the enemy!"
|
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