Six Sigma Metrics
The point of most Six Sigma projects is to reduce costs. The use of the Six Sigma metrics provides a common way to measure all processes so both the baseline and post-project improvements can be established and validated.
First, some simple terms will be explained, followed by three examples of how Six Sigma metrics are measured.
A unit of product can be defective if it contains 1 or more defects. And, a unit of product can have more than 1 opportunity to have a defect. With Six Sigma, we measure the defects per opportunity so that both simple and complicated processes can have common metrics. To determine the number of opportunities,
1. Determine all the possible opportunities for problems,
then
2. Reduce the list by excluding the trivial and grouping
similar defect types , and then
3. Define the opportunities consistently across different
processes and locations.
The Proportion defective (p) is determined by the following formula:
p = no. of defective units/total no. of product units
Then the yield ( Y 1st-pass , or Y final ) can be determined:
Y = 1 - p
Rolled throughput yield , RTY, is the accumulation of i yields after multiple processes or process steps:
RTY = Y 1 x Y 2 x Y 3 x Y 4 x Y i
RTY Example
If 10 processes each have a yield of 99%, then
RTY = Y i = .99^10 = 90.4%
It doesn’t take very long for RTYs to drop significantly when there are multiple process steps and/or individual process yields that are not at very low defect rates.
Note: Yield can be converted to a sigma value using the Area
Under the Normal Curve (Z) tables, which are available in any statistics book. An example will follow later.
Some other relationships and metrics used in Six Sigma are defects per unit, defects per opportunity, and defects per million opportunities.
Defects per Unit (dpu, or u in SPC):
- dpu = no. of defects/total no. of product units
Defects per Opportunity (dpo):
- dpo = no. of defects/(no. of units x no. of defect
opportunities per unit)
Defects per Million Opportunities (dpmo, or ppm (parts per million)):
- dmpo = dpo x 1,000,000
Since processes that have been improved can reach very low defect rates, a measure that conveniently reflects rates below 100 defects per million opportunities is needed. The dpmo can be then converted to sigma & equivalent C P values (see third example and the table at end of article).
Calculating dpmo Example
If there are 9 defects among 150 invoices, and there are 8 opportunities for errors for every invoice, what is the dpmo?
dpu = 9/150 = .06 dpu
dpo = 9/(150 X 8) = .0075 dpo
dmpo = .0075 X 1,000,000 = 7,500 dpmo
The CP index, used as a measure of process capability in manufacturing, is a ratio of the product characteristic’s upper specification limit (USL) minus the lower specification limit (LSL), divided by six standard deviations ( s ). The specification limits are established by the design engineer, and the standard deviation is a calculated measure of the actual variation of the process.
CP = (USL – LSL)/6 s
CP is a ratio of the engineering spread over the process spread. The Six Sigma method of converting yield to sigma and CP metrics provides a common way to measure both manufacturing and transactional (business) processes.
Converting Yield to Sigma and CP Metrics Example
- Yield = 1 - p = .990
- Z Table value for .990 = 2.32 s (This means that 1%
of the area under the normal distribution curve lies more than 2.32 standard deviations, s, to the right of the mean).
- Estimate process capability by adding 1.5 s to reflect
the typical “real-world” shift in the process mean over
time:
2.32 s + 1.5 s = 3.82 s
- This s value can be converted to an equivalent CP
process capability index by dividing it by 3 s:
CP = 3.82/3 s = 1.27
The table below contains these conversions already computed. Note: CPK cannot be estimated by this method.
SIGMA TABLE
Yield |
dpmo |
Sigma |
CP Equiv. |
COPQ |
|
|
|
|
|
| 0.84 |
160,000 |
2.50 |
0.83 |
40% |
| 0.87 |
130,000 |
2.63 |
0.88 |
|
| 0.9 |
100,000 |
2.78 |
0.93 |
|
| 0.93 |
70,000 |
2.97 |
0.99 |
|
| 0.935 |
65,000 |
3.01 |
1.00 |
|
| 0.94 |
60,000 |
3.05 |
1.02 |
|
| 0.945 |
55,000 |
3.10 |
1.03 |
30% |
| 0.95 |
50,000 |
3.14 |
1.05 |
|
| 0.955 |
45,000 |
3.20 |
1.06 |
|
| 0.96 |
40,000 |
3.25 |
1.08 |
|
| 0.965 |
35,000 |
3.31 |
1.10 |
|
| 0.97 |
30,000 |
3.38 |
1.13 |
|
| 0.975 |
25,000 |
3.46 |
1.15 |
|
| 0.98 |
20,000 |
3.55 |
1.18 |
20% |
| 0.985 |
15,000 |
3.67 |
1.22 |
|
| 0.99 |
10,000 |
3.82 |
1.27 |
|
| 0.995 |
5,000 |
4.07 |
1.36 |
|
| 0.998 |
2,000 |
4.37 |
1.46 |
|
| 0.999 |
1,000 |
4.60 |
1.53 |
10% |
| 0.9995 |
500 |
4.79 |
1.60 |
|
| 0.99975 |
250 |
4.98 |
1.66 |
5% |
| 0.9999 |
100 |
5.22 |
1.74 |
|
| 0.99998 |
20 |
5.61 |
1.87 |
|
| .9999966 |
3.4 |
6.00 |
2.00 |
– The Six Sigma Level of Quality
|
This conversion table allows you to determine the Sigma and equivalent CP values for any type of process, without having to do the math. The last column, COPQ, shows the approximate Cost of Poor Quality as a percentage of sales dollars for several of the yield values. The COPQ would contain not only scrap, returns, and rework, but extra inspection, handling, missed schedules, firefighting, extra inventory, expediting, etc.
The use of the Six Sigma measurements provides a common way to measure all types of manufacturing and transactional processes. They are used both in the Define phase when the baseline is established and in the Improve and Control phases when the metrics are validated.
