Its been a while since I have posted because I have been very busy living the Kaizen Kaikaku Life!
This post is very important in the area of engineering and can help businesses improve and sell there products by proving quality!
First I want to give you the background to be able perform a robust margin analysis then I will show you how to apply it to Kaizen which will ultimately lead to Kaikaku.
There are many ways to perform margin analysis but my favorite is using K-Factor because most things in life are natural and random and fall to the central limit theorem; thus, using the Normal Distribution (Guassian) would make sense.
Equation above: Gaussain Distribution
The K-Factor Equation is K=M/U or simply stated Margin over Uncertainty.
M=Mean - Lower Performance Requirements (LPR) and U = Sigma
If you arrive at K equaling 1 than you are 1 standard deviation (sigma) away from the mean or 68.3% of the numbers will fall within this range within the full distribution; if you obtain a K of 2 then you are two sigma away from mean or 95.4% and 3 sigma is 99.7 %.
Now here is the tricky part that most people have a hard time understanding when I explain this to them.
The higher the K is better!
I often hear but 3 standard deviations away from the mean is worse than 2. Which is false.
I understand where a person might say that but what we as engineers do here is put limits, tolerances, boundaries on numbers and those limits are the lower and upper performance requirements. Therefore, the more Sigma that fit within those customer imposed limits the better our margin.
Lets perform an example together by first looking at this histogram/bell curve below:
From the histogram we see there we have an initial population size of 150 units in which all were tested. We can see the mean is 4.065, a standard deviation of 0.726, and an observed 99th percentile of 5.59 (estimated by the average of the two largest values in the observed dataset). The performance characteristic has an upper limit of 7.1
Example Calculation:
First Step: Find a tolerance bound that contains 99% of the population with 95% confidence. We can do this by taking the upper performance requirements (7.1) and subtracting the approximate 99th percentile figure (5.6) to obtain our margin of M=1.5
Second Step: Using the K-Factor equation from above we get:
K= 1.5/.73 = 2.05
Note: We have a K-Factor of 2.05 which tells us 2.05 SD fit within the boundaries with 95% confidence.
Third Step: The K-Factor is considered to be "large enough" when its a greater number than that specified of the normal distribution i.e. .995. .995 is equal to a 2.576 (2.58) Z Score. You can look this up on the internet. For our problem we want a 95% confidence with an alpha tail of .05 would give us a z-score of 1.645.
Fourth Step: Is our K factor larger than 1.645 to say with 95% confidence we will meet requirements? Yes. However, if we used the 99.5% confidence we would be short.
Optional Fifth Step: If you didn't have a population (N) but a sample (n): If its a sample, this does not take into account the relative error rate which is a function of sample size. See the equation and subsequent chart derived from the equation for your K bounds that was derived from the CpK equation (capability index):
If our K- Factor is 2.05 we multiply by the error rate
2.05*.86 for Lower and 2.05*1.14 for Upper giving you 1.7663 and 2.337.
Which would still be higher that the 95% confidence rate you were after.
Conclusion: We can claim now that there is sufficient margin to conclude that 95% of the units will meet performance requirements.
Note: Judging by the analysis of the Sample step, if we had a sample size of 10 which would give 2.05*.53 giving a Lower Relative Error Rate K-factor of 1.08, we could not say with 95% that we will be within margins. Basically the more we sample the better within reason cost wise.
So how does this relate to the Kaizen Kaikaku Life?
By knowing your margins in meeting requirements, you can have a very good baseline for improvement. Remember what Taichi Ohno said, "Where there is not standard, there can be no Kaizen!"
Let me know if you have any questions or you want to get into some Margin Analysis that is less dependent on the Normal Distribution like the Tolerance Bound Approach.
Let me know in the comment section or e-mail me.
And as always stay improving my friends! Below I'm improving my skating skills!!
