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So here, what you see is the standard normal distribution,

and the standard normal distribution has a mean of zero and a standard deviation of one.

So, whatever other distribution that you have and if you

have the population mean of that distribution and if it's a normal distribution,

or if it can be assumed to be normal,

you can put the mean at where the z of zero is.

So you're sort of taking that distribution and

superimposing it on the standard normal distribution.

You're saying the population mean represents a z value of zero.

What does that help us do?

It helps us use the properties of the standard normal distribution. How do we use these?

Well, if you have a sample mean and you have the standard deviation of this distribution,

what you can do is you can compute

the z value and you have the formula right on top of the slide,

which says, take the sample mean,

subtract from that the population mean,

divide that by the standard error,

standard deviation divided by square of n,

and that gives you a z value.

And here's an illustration of how you would use the z value.

So if you do get a z value of one,

what it is telling you essentially is that your sample mean is

one standard deviation away from the population mean.

So, a z value of one is indicating that it's

one standard deviation from the population mean.

Z is simply the number of standard deviations that it is away from that mean.

So, if it's a positive one,

it's on the right side of the population mean.

And then if we plug these values into the norms dist function of Excel,

and the new Excel has it with a period after norm and after s,

so norms dist function of one being

the z value and the second one in

that function is indicating that it's a cumulative frequency.

So it's taking it all the way from the left tail

to a positive value of one and it's giving us

the area under the curve and which indicates the probabilities of these values.

So what does that indicate to us?

That with a norms dist one,

one turning out to be 0.84,

it's telling us that to the left of the z value of one lies 84% of the area,

therefore 84% of the observations are

toward the left of the z value or of the sample mean,

and to the right of the z value lie the rest,

which is 16%, because the whole thing is going to be 100%.

So to the left, 84%; to the right,

which is the pink area in this particular normal distribution,

that is going to be 16%.

So if you think about calculating defects and things like that,

then you can start thinking about what is

the probability of getting a defect if it's going to be in the tail,

and that's going to be 16% in this case.

So this is how we start applying the norms dist.

Now just for completion purposes,

how would you Inverse this?

How would you say, well,

if I know that I want to find a point at

which 84% of the values are going to be to it's left,

I can use the norms inverse function.

So the norms inverse function has probabilities in it,

and you're simply saying 84% of the area from the left,

please give me the z value,

and norms inverse is giving me a z value that's 0.9945.

Practically speaking, that's a z value of one.

So you've seen here that we're getting the opposite of both of

those things when we use norms dist and norms inverse.

What does all this mean when we're thinking about Six Sigma,

the initiative that we're talking about here and the

metric that we've talked about quite a bit earlier?

So how this translates into Six Sigma is the fact that we're getting a z value of

4.5 when we put 3.4 defects per million opportunities in that little tail over there.

So we're saying that the tail is going to be really

small and it's going to be 3.4 defects per million opportunities,

and that would give us a sigma value,

a z value of 4.5.

And if you remember,

we add 1.5 to this to make it look like a sigma value of 6 when it's actually 4.5.

So from the statistics point of view,

it's still a 4.5 sigma value. All right.

So, one more use of this whole idea off

the standard normal distribution and the idea of

z values is how we use it for conducting hypothesis tests.

And whether we know it or not,

and sometimes we blindly use statistical tests like analysis of variance and regression

and we use the p-value and the alpha value

and we don't know where exactly they're coming from.

So just to talk about the origins a little bit of

what we are talking about there in terms of a p-value.

So whenever you do a statistical test,

you say that I get a certain significance value.

What does that significance value mean?

We also call it the p-value.

So the p-value is the minimum error rate at which the hypothesis will be rejected.

It's basically you're finding

the z score of a particular value that you're using in order to conduct the test,

and you're saying from that z score,

can I get the area under the curve to the right of it,

to the error side of it,

which is typically the right side of it as being the error side.

So you get that as the p-value.

Now what is the alpha value?

The alpha value, although we're talking about this after the p-value,

the alpha value is something that you're supposed to actually put

in stone or decide before you conduct the hypothesis test.

And I say supposed to because sometimes we tend to say,

well, at this alpha value,

it will be rejected but at the other one, it won't be rejected.

Strictly speaking, or if we were to do this absolutely correctly,

we should be stating the alpha value.

And what is the alpha value?

There you're stating on that standard normal curve

what is the level of error that you're willing to tolerate.

So you are stating that level of error in terms of the alpha value,

and you state that upfront even before you set up the hypothesis test.

When you do the hypothesis test,

you do some statistical analysis,

you get a p-value from that statistical analysis,

and then you compare the p-value with the alpha value.

And then the simple rule that we use in order to reject our null hypothesis is we say,

if p is less than the alpha,

the decision is going to be reject the null hypothesis.

If p-value is less than the alpha value,

we will reject the null hypothesis,

and we also call that as being a significant result.

So we got a significant result because the level at which we

found our test to be working at is lower than the level that we had set,

which is the alpha value.

So that's another use of the normal distribution.

May not mean much at this point if you have not seen analysis of variance,

or regression, or t-test,

or things like that.

So we'll take a look at this when we come back

to analysis of variance and regression in a different session.

Okay, so in closing,

how do we use the normal distribution when we're talking about Six Sigma projects?

So a number of uses,

we use it to compute sigma levels from defects per million opportunities.

Once we have the defects per million opportunities,

we compute the z value,

and that gives as a sigma level for a process.

We can use this in the measure phase.

We can use the standard normal distribution and the normal distribution in

the measure phase in order to partition the variation that we see in

our data and how much of it is coming from measurement and how much of it is

actually from the material itself or the process itself.

So we can parse out that variation based on some tests that we can do,

and these are related to analysis of variance that we'll be looking at later.

We also use this for statistical process control,

and we also use it for process capability analysis.

So what is statistical process control?

It's the idea that every process is assumed to vary under normal circumstances,

under usual circumstances within plus or minus three standard deviations.

So we use that property to indicate what is the inherent potential of a process.

That's what we can use statistical process control for,

is establishing what is the waste of the process.

How is a process performing under usual ordinary circumstances?

And then we can take that that notion of this is the inherent capability of the process,

can we compare it to what the customer is expecting?

So, waste of the process compared with waste of the customer gives us process capability.

We can see how the process is doing in comparison

to what the customer of the process is expecting from this process,

and for that we do something called

the process capability analysis and we can use that even to come up with sigma values.

We can indirectly use the process capability analysis and use that ratio to

come up with sigma values and we'll talk about that in a later session.

We can also use the normal distribution or we do also use the normal distribution,

we use the z distribution and the t distribution,

the other student distribution for looking at analysis of variance and regression.

For regression, we use both the normal and the t distribution.

So, we're going to use some properties of

the normal distribution for both of those kinds of analysis,

which we'll see in terms of using them for root cause analysis in subsequent sessions.