1. SIX SIGMA AND CALCULATION OF PROCESS CAPABILITY INDICES: SOME RECOMMENDATIONS
P.B. Dhanish
Department of Mechanical Engineering

2. Overview What is Six Sigma? The sample size problem
The distribution problem
The control problem
Conclusion

3. What is Six Sigma? Disciplined quality improvement
AIM: Near elimination of defects
NUMERICALLY: 3.4 DPMO!

4. Defect levels at various sigmas

5. What is sigma? Then, six sigma?
Statistics:
The process standard deviation
Then, six sigma?
Specification limits should be at
+/- six sigma

6. Then, how many defects? Assuming Normal distribution
The area under the normal curve beyond +/-six sigma
:Fraction non-conforming
Multiply by 1,000,000 to get DPMO
If the process is centred on target, DPMO

8. Such perfect centering
Not possible in practice
Allow +/-1.5 sigma shift
Then the defect level will be 3.4DPMO

10. The sample size problem
In practice, true sigma is unknowable
Sample standard deviation
From a finite number of samples
Sampling error in sigma level
Single value not meaningful
Hence: Give Confidence Limits

11. If x is normally distributed
an upper 100(1-α)% confidence limit for σ is
and an upper 100(1- α)% confidence limit for μ is

12. Assuming that mean and sigma are independent
to calculate an upper 100(1-α)% confidence limit for the proportion nonconforming p,
construct 100(1-α)1/2 % confidence limits for each parameter separately
using these two values, determine p

13. For example, If n=25, USL=6, LSL=-6, =0, and s=1,
a 97.47% upper confidence limit for σ is
and a 97.47% upper confidence limit for μ is

14. Then the proportion nonconforming:
= ppm

15. Alternative: Determine sample size for the required confidence level

16. Give confidence limits
Recommendation 1
Do NOT give a single value for the sigma level of your process
Instead,
Give confidence limits OR
the confidence level

17. The distribution problem
Above calculations utilise:
the tail end of the normal distribution
Does any process in nature match the values in the tail?

18. The values in the tail:

19. To check the correctness,
We need millions of samples!
No shift in the process during this production!
Hence impossible to verify
the exact values of the defect levels, say 3.4 or 5 DPMO, may not have practical significance

20. Not normally distributed:
Surface Finish
Circularity
Runout
Hence, take care

21. Cramer (1945): Lippman: everybody believes in the law of errors,
the experimenters because they think it is a mathematical theorem,
the mathematicians because they think it is an experimental fact

22. Recommendation 2 Verify that the distribution is normal
Otherwise, utilise the appropriate distribution
Realise that the exact values of low defect levels are meaningless

23. The Control Problem To claim that future performance would be similar,
the process should be stable OR
in statistical control

24. Deming (1986):
One sees much wrong practice in connection with capability of the process. It is totally wrong to take any number of pieces such as 8, 20, 50 or 100, measure them with calipers or other instruments, and take 6 standard deviations of these measurements as the capability of the process

25. Consider thirty observations:
-1.6, -1.2, -1.9, -0.6, -1.6, -1.4,
-0.5, -0.9, -0.2, -0.7, 0.2, -0.5,
0.3, -0.4, 0.5, -0.3, 0.4, -0.2,
0.8, 0.6, 0, 1.2, 2, 0.5,
0.9, 0.8, 0.1, 1.4, 0.6, 1.7
Mean 0
Standard deviation 1

26. Histogram:

27. An excellent process? If specification limits are +/-3 Wait,
Just plot a run chart for the given measurements

28. The process is drifting!

29. No capability can be ascribed!
QS9000: Distinguishes between Cpk and Ppk
An undisturbed process: Shouldn’t it be in control?
Very very unlikely
An SPC program necessary

30. Nelson (2001):
Getting a process in statistical control is not improvement (though it may be thought of as improvement of the operation), getting a process in statistical control only reveals the process, and after a process is in statistical control, improving it can begin.

31. Another pitfall: Samples taken too infrequently
This way, any process can be made to appear in control!
A common cause for failure of SPC

32. Recommendation 3:
Ensure that the process is stable or in statistical control

33. Any questions?

34. Conclusion: Industries while claiming Six Sigma, should
1. reveal the confidence level of their sigma calculation
2. The process distribution utilised
3. How process stability was verified