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68-95-99.7 Rule

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Hi. This tutorial covers the 68-95-99.7 rule. So let's start with the definition of this rule. Sometimes this 68-95-99.7 rule is also called the empirical rule.

And what it states is that for every normal distribution-- so this works for any normal distribution-- 68% of the data values are within one standard deviation. 95% of the data values are within two standard deviations of the mean. And 99.7% of data are within three standard deviations. So, again, 68 within one standard deviation-- that can be above or below the mean-- 95 within two standard deviations, 99.7 within three standard deviations.

So let's take a look at what this distribution is going to look like. So let's start by drawing it nice and normal here. Down the center here, this is going to be your mean. And let's mark off a couple standard deviations on both sides. So that's mu plus 1 sigma. This is mu plus 2 sigma, and out here-- extend my picture a little bit-- mu plus 3 sigma.

We'll do the same thing on the other side. So now this is mu minus 1 sigma, mu minus 2 sigma, and mu minus 3 sigma. So that gives us a pretty good picture of where all of those important values are going to lie.

Now, if we continue to mark these off-- I'm going to extend my lines here-- within one standard deviation, we're going to have 68% of our data. Within two standard deviations-- I'm going to mark those lines here-- we're going to have 95% of our data. And if we go all the way out to three standard deviations, that's almost all of the data-- 99.7%. OK. So that marks it out like that.

Now, we can also figure out what value-- or what percent kind of lies within each of these little subdivisions here. So we know that a normal distribution is symmetric. So the 68% in the middle here, within one standard deviation, can easily be broken down into 34% and 34%.

Now, to figure out what goes in these regions, these two regions on their own, they're also symmetric. So these values are going to be both the same. And to determine what percent will be in each, what I can do is I can take the 95% within two standard deviations and subtract what's within one standard deviation. So what this tells me is that 27% of the data lies outside of the one standard deviation boundaries but still inside the two. And then if I want that split up into two regions, that's going to give me 13.5% in each of these two regions.

And then I can do the same thing for that region and that region. So we'll take now 99.7 minus 95, divide that by 2. And we end up with 2.35%. So that goes here. And 2.35% here.

And then finally, this region over here, that's going to be outside of three standard deviations. So really it's going to be 100% minus 99.7%. So if I do that, that's going to give me 0.3%, so less than 1%. Divide that by 2 and that's going to give me 0.15%.

So that's what's going to go over in this region. 0.15% is three standard deviations above the mean or bigger. And then same thing over here except for now we're going to go smaller. All right. So that's kind of a good picture of where all of that data will lie on a normal distribution.

So now let's take a look at another example here. So now we're looking at the distribution of the duration of human pregnancies was known to be approximately normal with a mean of 270 and a standard deviation of 15. And these are both in days.

So let's draw the distribution. We're going to center this at 270. One standard deviation above the mean is going to be at 285, 300. And then this is going to be at three standard deviations out at 315. This would be at 255, this 240, and 225.

So then, again, I'm going to have my same values in each of these. I'll just mark the first couple. So this is 34%, 34%, 13.5%, 13.5%, 2.35%, and 2.35%.

Now based on this picture, let's answer some questions involving percentages. We'll come back to this picture in a minute. So what percent of the pregnancies last between 240 and 300 days?

So we'll go back to our picture. We want 240 and 300-- so here to here. So we want all of these values.

So really, if we just add those up-- well, we could add them up. But we also know that that's going to be two standard deviations below. That's two standard deviations above. So by our rule, we know that this has to be 95%.

Now what percent of pregnancies last between 255 and 270? Again, going back to the picture, between 255 and 270, well, that's pretty easy. That's that 34%. That's the half of the 68. So that's 34%.

What percent of pregnancies last more than 280 days? OK. So let's check that out. Now, we want more than-- or excuse me-- more than 285 days. So we want really this region, all of this.

So one way to do it is I could add that number, that number, and then whatever is above. Or I know that below 270 is 50, so above 270 is also 50. But I don't want this whole 50%. I want to not include the 34%. So if I take the 50% above the mean minus 34%, that's going to leave me simply with 16%.

And finally, if we go to what percent of pregnancies last less than 300 days? So again we go to our picture. 300 days-- that's here. We want to know what percent is less than 300.

So less than 300-- I could add all of these values up. Or I know that above 300 is 2 and 1/2%-- so the 2.35 plus the 0.15%. The other way I know that is that I know 95% are within these two boundaries. So that means that 2 and 1/2% has to be here. 2 and 1/2% is here.

So I know that this is going to be 2 and 1/2% are above 300. So that means 97 and 1/2% are less than 300. So my answer here is 97.5%.

OK. So that is a way of answering some common questions that might come up given a normal distribution with a certain mean and standard deviation. So the big things to get out of this tutorial is, again, what that 68-95-99.7 rule means and then how to manipulate those percentages to answer questions similar to this.

So that has been your tutorial on the 68-95-99.7 rule. Thanks for watching.