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

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

Â So, whatever other distribution that you have and if you

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

Â or if it can be assumed to be normal,

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

Â So you're sort of taking that distribution and

Â superimposing it on the standard normal distribution.

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

Â What does that help us do?

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

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

Â what you can do is you can compute

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

Â which says, take the sample mean,

Â subtract from that the population mean,

Â divide that by the standard error,

Â standard deviation divided by square of n,

Â and that gives you a z value.

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

Â So if you do get a z value of one,

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

Â one standard deviation away from the population mean.

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

Â one standard deviation from the population mean.

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

Â So, if it's a positive one,

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

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

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

Â so norms dist function of one being

Â the z value and the second one in

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

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

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

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

Â So what does that indicate to us?

Â That with a norms dist one,

Â one turning out to be 0.84,

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

Â therefore 84% of the observations are

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

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

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

Â So to the left, 84%; to the right,

Â which is the pink area in this particular normal distribution,

Â that is going to be 16%.

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

Â then you can start thinking about what is

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

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

Â So this is how we start applying the norms dist.

Â Now just for completion purposes,

Â how would you Inverse this?

Â How would you say, well,

Â if I know that I want to find a point at

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

Â I can use the norms inverse function.

Â So the norms inverse function has probabilities in it,

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

Â please give me the z value,

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

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

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

Â those things when we use norms dist and norms inverse.

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

Â the initiative that we're talking about here and the

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

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

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

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

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

Â and that would give us a sigma value,

Â a z value of 4.5.

Â And if you remember,

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

Â So from the statistics point of view,

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

Â So, one more use of this whole idea off

Â the standard normal distribution and the idea of

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

Â And whether we know it or not,

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

Â and we use the p-value and the alpha value

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

Â So just to talk about the origins a little bit of

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

Â So whenever you do a statistical test,

Â you say that I get a certain significance value.

Â What does that significance value mean?

Â We also call it the p-value.

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

Â It's basically you're finding

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

Â and you're saying from that z score,

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

Â to the error side of it,

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

Â So you get that as the p-value.

Â Now what is the alpha value?

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

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

Â in stone or decide before you conduct the hypothesis test.

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

Â well, at this alpha value,

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

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

Â we should be stating the alpha value.

Â And what is the alpha value?

Â There you're stating on that standard normal curve

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

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

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

Â When you do the hypothesis test,

Â you do some statistical analysis,

Â you get a p-value from that statistical analysis,

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

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

Â if p is less than the alpha,

Â the decision is going to be reject the null hypothesis.

Â If p-value is less than the alpha value,

Â we will reject the null hypothesis,

Â and we also call that as being a significant result.

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

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

Â which is the alpha value.

Â So that's another use of the normal distribution.

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

Â or regression, or t-test,

Â or things like that.

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

Â to analysis of variance and regression in a different session.

Â Okay, so in closing,

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

Â So a number of uses,

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

Â Once we have the defects per million opportunities,

Â we compute the z value,

Â and that gives as a sigma level for a process.

Â We can use this in the measure phase.

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

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

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

Â actually from the material itself or the process itself.

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

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

Â We also use this for statistical process control,

Â and we also use it for process capability analysis.

Â So what is statistical process control?

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

Â under usual circumstances within plus or minus three standard deviations.

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

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

Â is establishing what is the waste of the process.

Â How is a process performing under usual ordinary circumstances?

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

Â can we compare it to what the customer is expecting?

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

Â We can see how the process is doing in comparison

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

Â and for that we do something called

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

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

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

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

Â we use the z distribution and the t distribution,

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

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

Â So, we're going to use some properties of

Â the normal distribution for both of those kinds of analysis,

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

Â