Let me illustrate the idea of the capability score on the following slide.
Just as a reminder, this is the definition of the capability score.
The upper specification limit minus the lower specification limit, divided by six
times the standard deviation. Now, imagine two distributions,
One of them has a density function that has a little larger variance.
And you notice here, how you can go three standard deviations from the mean of the
distribution before you hit the specification limit.
Now in the lower case, you see a distribution that has a lower variance,
and you need to go six standard deviations before you going to incur a defect.
Now clearly defects are less likely in the lower case.
There is simply less probability mess at the tails here.
So, this suggest that we can compute, or we can translate the capability scores of
a distribution into the probability of defects.
Let me illustrate this calculation by going back into our spreadsheet.
So how likely is it going to be that we can encounter a bag of M and M's that are
heavier than 52 grams? So what's the probability that the bag is
too heavy? I can get to this by using the normal
distribution function in Excel, and looking at the 52 grams, relative.
To a distribution with 50 as a mean, and 1.1 as a standard deviation.
That probability is 96% that it stays below this.
Or, the probability that this is too heavy is simply one minus it, which is 3.4%.
Now next, ask ourselves, what's the probability that this is too small of a
bag, or that this is too light. Well for that, I have to look at the
normal distribution. This time with the lowest specification
limit. 50.
And 1.1 is the standard deviation. And this is equal to 3.3%.
Now for a defect, I either need to have the bag.
Be too heavy or too light and so the sum of those two is simply the probability of
a defect. Now, I can take this number, and I can
multiply this with, say, a million units. In the production run to get a number that
is known as the ppm, the parts per million.
So we have 67,818 parts defective per million parts.
So you notice that the capability score for around 0.6 as we just saw, saw or, in
the M and M example. This equating to a defect probability of
around 0.0. 67 percent.
Or, put differently, 67,000 defects in a million parts.
Now, in this table, we show the relationship between the capability score
and defect probabilities. For example, at a capability score of one,
you can go three sigmas from the mean to either side of the specification levels.
And we have a defect probability of 0.027. Put differently, you're gonna have 2,700
defects per million parts. Now, where do these numbers come from?
Now, let's first look at the three sigma process.
What's the defect probability? Well, I have to go three standard
deviations before I hit the specification limit.
Let's assume the underlying distribution is a standard normal distribution which
is, having a mean of zero and a standard deviation of one.
So for a defect I have to hit the number three.
So I'm looking at the normal distribution, mean zero, standardization of one, which
is a value of 99.865%. So one minus that probability, tells me
the probability of the part being too large.
This is also the probability of the part being too small, so assuming the symmetry,
I can just double this number and that gives you the 0.027 that you all saw on
the earlier table. Now lets move this further and look at
just six sigma process. With six sigma, we have to go six standard
deviations, and you notice that this number here becomes ridiculously small.
So a little hotter in temperature, so let not look at it as a probability, but as a
defect for a million parts. So we have to multiply this, with a
million and we see that the number here is roughly 0.002.
In other words, you're gonna have two defects for billion dot billion parts.
So a quality target is typically expressed in defect probabilities of parts per
million. We see in this table, that, that can be
matched to a capability score. This allows me to ask myself, for a given
specification limit, what is the amount of variability in the process that I can
tolerate, before violating my quality goal?
Let me go back to the example of the M and M's.
So we said we had the USL, minus an LSL. Divide it by six sigma and that had to be,
if I'm aiming for six sigma operation, that would have to equate to two.
Now, in our example, the difference here between the USL and the LSL was four.
So that gives me an equation that I can solve.
Four divided by six sigma is equal. To two.
And so, in other words, sigma is equal to one-third.
