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In this class, we start to
look at the definition

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of Six Sigma and what a Six Sigma process is.

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To start off with though, we need to
understand the process capability ratio, PCR.

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The process capability ratio can be calculated
for any final quality attribute of your product.

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It can and should be used also for intermediate
products, products that go from one stage

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to another stage within your process.

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Capability ratios do not have to be
calculated only on those products that end

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up leaving your facility
and go to your customers.

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To define the capability ratio, we
have to know the upper specification

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and the lower specification
limits for that attribute.

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Let's say you're measuring
the viscosity of a liquid.

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There's typically an upper and a lower
specification limit that you've reported

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to your customer, or if we are looking at
some step in the middle of your flow sheet,

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these would specification limits
that you have set internally.

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Specification limits, remember
from the earlier class,

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will typically be outside the
upper and lower control limits.

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The upper and the lower specification
limits should not be confused with those.

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The simple definition for process capability is

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to take the upper specification limit
minus the lower specification limit

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and divide by six sigma.

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Let's also be specific about
how sigma is estimated.

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Sigma is estimated by taking data from your
process when you know that it is stable.

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You can prove that by using
a Shewhart monitoring chart

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and only using data from in-control operation.

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In other words, there should be no special
causes happening, that you are aware of.

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To interpret the process capability ratio,
we have to make the additional assumption

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that the process is exactly centered between
the upper and lower specification limit

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and secondly, that the attribute you're
measuring has a normal distribution.

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That's easy to check.

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We've used the q-q plot for that in the past.

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That's all there is to it.

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You sub in the upper and lower
specification limits and divide

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by six times the estimated standard deviation.

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You get a single process capability
ratio number for that attribute.

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Let's try this out.

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Assume the mean of our process is 80.

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The lower specification limit is 65.

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The upper specification limit is 95.

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Our estimate of 'sigma' is 10.

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Calculate the process capability
ratio for that system.

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You should have gotten a value of 0.5.

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Let's interpret that.

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It is very helpful to draw a
diagram to illustrate the system.

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A system where the mean is 80 and
has standard deviation of 10 units,

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assuming it's normally distributed,
would have this shape.

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Let's now superimpose those lower and
upper specification limits on there.

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I've emphasized the region below the lower spec
limit and the region above the upper spec limit.

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That's a lot of area that's shaded, indicating
a lot of the product we are producing

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on our process is outside specification.

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This is actually a really bad process.

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A PCR of 0.5 is very undesirable.

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   In fact, under the assumption of normal
distributions, we can calculate the z-value

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for the lower spec limit
and the upper spec limit.

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Can you do that before I show you the answer?

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You should have found values
of plus and minus 1.5.

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Using the pnorm(...)

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function or looking this up on
tables, we will find that 13.4 percent

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of our process production is outside the limits.

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That's 13.4 percent of our product that we
are not able to pass on to our customers

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because they don't meet specification.

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Let's try it with a different system.

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This time, the mean is still 80.

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The lower spec limit is still 65.

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The upper spec limit is 95.

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Let's see what happens if we're able to reduce
the variability in our process four times.

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That's a sigma, this time, of two and a half.

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Go ahead and calculate the PCR for that case.

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You should have gotten a value of two.

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Let's illustrate that again.

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This time, notice how narrow
that distribution is

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and how well it lies within
the specification limits.

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The z-value for the lower and the upper
spec limits are plus and minus six units.

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The shaded area is almost unobservable.

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What we've learned from these
two case studies is

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that a higher process capability
ratio is more desirable.

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Perhaps you can now even see why it is
called the "process capability ratio."

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It is a measure of how capable our process is,

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how capable we are of staying
within those limits.

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Remember, the process has a mean
of 80, but it's never constant.

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That 80 will drift to the left and the right.

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Sigma occasionally will get a little
bit wider and sometimes narrower.

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No process is ever stable, where
those numbers stay exactly the same.

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A high process capability ratio means that
we've got the room to move within our lower

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and upper specification limits
and still not produce bad product.

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I would like you to also notice one other
feature about the process capability ratio.

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In this case, our standard deviation is 2.5.

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The distance from the upper spec limit
to the lower spec limit is 30 units.

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This implies that the width of our
specification is 12 standard deviations.

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We can fit 12 standard deviations, side by
side, from the lower specification limit

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up to the upper specification limit.

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A process capability ratio has two times
six sigma, in other words, 12 sigma widths.

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A process capability of one implies that
the width of the process is six sigma,

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three standard deviations from the
mean to the lower specification limit

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and three standard deviations from the
mean to the upper specification limit,

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giving you a total width of six sigma.

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Let's illustrate that.

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Here I've drawn a process
which has PCR equal to one.

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The lower and the upper specification limits are
the same as in the prior example, but this time,

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sigma is equal to five standard deviations.

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The width of the process is six sigma.

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What is the area of probability
under that distribution

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that lies outside the specification limits?

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You should have found the z values for the lower

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and upper specification limits
to be plus and minus three.

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Using the pnorm(...)

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function in R or from tables you
should have found that 99.73 percent

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of regular operation lies within
the limits and 0.27 percent

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of your operation lies outside the limits.

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That means if you've produced
1,000 products on your process,

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only three of those 1,000
products lie outside the limits.

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That's still a high number.

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PCR equal to one unit isn't a desirable process.

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Even though it sounds quite high that
six standard deviations fit in the gap

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from the lower spec limit to
the upper spec limit, actually,

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what we term a "Six Sigma process" is a
process where there are 12 standard deviations,

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six sigmas to the left and six sigmas
to the right, giving a total of 12.

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You can see then that a Six Sigma process
is one where the capability ratio is 2.0.

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Such a process produces almost
no off-specification product.

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In fact, a minimum requirement on a process
is where the capability ratio is 1.3.

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Processes where the attribute being measured
has a critical application or is there

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for safety purposes, we typically
look for values of 1.7.

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A capability of two units is
the standard that we aim for,

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as far as we possibly can,
within reasonable costs.

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There is one shortcoming of the
definition that we've looked at.

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That is, of course, the assumption that
we are operating right in the middle

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of the lower and upper specification limits.

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Almost no processes have this exact midpoint.

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Mostly, we are operating closer to
one of the specification limits.

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We redefine the process capability ratio.

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We call it "Cpk," as follows.

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We choose the worst option.

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Are we closer to the upper specification limit?

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Then we use this first equation.

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If we're closer to the lower specification
limit, we use the second equation.

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Or we can combine it, together, in one
equation and use the minimum function.

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In other words, we choose to report
the capability that is the lowest.

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Let's emphasize, once more, two of the
important functions in the Cpk value.

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The attribute being measured has to be
assumed normally distributed in order

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to interpret Cpk in the way shown so far.

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Secondly, the assumption is
that the process is stable,

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that no special causes are
operating on the process.

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If there are special causes, we need
to wait till the process is stable,

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then collect the data and
calculate sigma from those data.

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Sigma calculated when the process is not
stable is likely to be a much higher value.

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You don't want to be using that sort of
data anyway to calculate your capability.

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I don't want you to go away and think
that Cpk is a fairly artificial number.

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This number is widely requested by customers.

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A customer wanting to purchase a
significant quantity of product produced

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on your process will often come in before they
sign the contract and inspect your process.

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They will want you to prove that you have
the capability of running your process

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in a reliable way that produces quality
products within the specification limits.

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This is a number you're going
to see regularly in your career.

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Even in cases where you are not
requested to provide this number,

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it is worth calculating it anyway.

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It's very quick.

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It's easy.

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It's a great way to show your boss that you've
made a significant improvement in your process.

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You just need to show that you've gone
from a baseline to a higher value and that

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that higher value has improved the
quality and stability of your process.

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If you can get to a Cpk of 2.0, you've
done a great job of improving the stability

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and the quality produced on that process.

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I've even seen cases in companies where
bonuses and rewards are given to employees

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who can achieve Cpk targets for the
processes that they are responsible for.

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That's the key that you need
to understand about Six Sigma,

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is that single definition, the Cpk number.

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Six sigma is more than just that number, Cpk.

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It also encapsulates the philosophy of
producing good quality product on your process.

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How do we measure that quality?

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What is the error in our measurements?

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How reproducible are these measurements?

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Can we use linear prediction models to
troubleshoot and find problems in our process?

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Can we use design of experiments to identify
bottlenecks and problems in our systems

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and move our systems to find
a better operating point?

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If you ever go into the area of study
of Six Sigma, you will see all the tools

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that we've learned in this course
used in that Six Sigma course.

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They're often wrapped up under
the global umbrella of Six Sigma,

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but what Six Sigma does effectively
is bring these tools together

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and make them operate coherently to achieve
high-quality production on your process.

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It is time for some examples.

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We'll look at those in the next video.