0:21

He worked for technical consulting company in Florida,

Â and one of their clients was a company that he didn't name so

Â I called it Florida Elevated Roadway, FERY.

Â This company, FERY, makes or builds pillars and

Â elevated roadways in Florida.

Â Why, do they have to do this?

Â Florida has a lot of swamp, so you can't put a street on the ground,

Â in many areas, the way we do it here in Switzerland.

Â Instead, you have to put a solid foundation into the swamp,

Â put up a pillar and then put the road on top.

Â 1:02

The construction of these pillars of course,

Â is key to make this safe, a safe road.

Â So after pillar is constructed, it is measured and tested in various ways.

Â One, particular parameter let's call it just x here, is the following.

Â What you're worried about is that this pillar may sink into the swamp.

Â So what companies then do, they put a lot of pressure on the foundation

Â of the pillar, push it into the ground, and then take the pressure away and

Â see whether the pillar has actually sunk, or whether it's stood up to the pressure.

Â Here is a related picture I got from a student in Switzerland.

Â You see, they put a lot of pressure on this pillar here in the middle.

Â And this is the key test, a key stress test for the pillar.

Â Of course it's important for me that the pillar can hold up the street.

Â Because I don't want to drive on the street and

Â then suddenly there's no pillar and bam, and then I become alligator food.

Â I don't want that to happen!

Â So this is a key test.

Â Now, these pillars are allowed to sink a little bit, but not too much.

Â Now lets abstract from the unit's here, our variable x,

Â that describes this particular physical feature has an average value of 0.8.

Â But this doesn't always read just exactly the average, it varies.

Â Think about this, these are pillars put into swampy ground,

Â perhaps in 100 or 110 degree Fahrenheit weather.

Â There're mosquitoes,

Â there's lots of uncertainty in the construction of these pillars.

Â As a result, the physical goal of a 0.8 isn't always reached, there's variation.

Â Here, the student told me, let's assume the variation is about 0.1.

Â So in our language, the standard deviation is 0.1.

Â Now, the pillar is classified as a failure if the extra value,

Â that x takes on, is one or larger.

Â Because then the pillar is not considered safe.

Â 3:19

So now the question is, how likely is it that there's a failure in the pillar?

Â That's important, maybe they make lawsuits,

Â I may have extra construction costs, all kinds of trouble may result.

Â Now, the student said, that when they calculate such probabilities, they use

Â a normal distribution because they've had good experience with that distribution.

Â So, what is the probability that a pillar will fail under these assumptions?

Â 3:50

We can use the NORM.DIST function, you find it in the same menu in Excel,

Â where we found the BINOMDIST function.

Â So here we have NORM.DIST, the critical cut-off value is number 1.

Â Our mean is 0.8 the standard deviation, 0.1, and again the word TRUE.

Â And we see if you type that into Excel, you get the answer 0.97725.

Â But remember NORM.DIST is a cumulative distribution,

Â it gives us by definition, the area to the left of the cut off where you want.

Â But we care about the area to the right, numbers above 1, because that's a failure.

Â So we use once again the complement rule,

Â 1 minus the probability we just calculated.

Â And the final answer, the probability of a pillar failure, is about 2.3%.

Â That's of course rather large if you build hundreds and hundreds of pillars.

Â So you actually want to improve the process relative to

Â the numbers I gave you.

Â 5:40

The difference between x and the mean divided by the standard deviation is

Â always the same number in this case 2.

Â And that's a key reason why they all get the same probability.

Â And so this is a key takeaway.

Â The actual number, x, mu, and sigma, do not matter.

Â 6:03

The key figure that matters is this ratio, x- mu, divided by sigma.

Â That's really what determines the probability.

Â By the way, there was nothing special about that ratio being 2 here,

Â the ratio is -1.

Â Again, all five probabilities are the same.

Â 6:24

And this is actually now a key property that all normal distributions have and

Â it leads us to the so called standardized normal distribution.

Â We can actually transform any normal distribution with any mean mu and

Â any standard deviation sigma into the so-called standard normal distribution,

Â which has a mean 0, and the standard deviation 1.

Â Any one of you who has heard the concept of a z score, or a z value,

Â or zed value, has worked with standard normally distributed random variables.

Â The nice thing is, there's a special function for this in Excel,

Â which is NORM.S.DIST, S for standard normal.

Â Here now, I have to warn you.

Â In the very latest version of Excel that I'm using here,

Â this function has two arguments, the z and the true.

Â If you have an older version of Excel, 2011 or earlier,

Â there is no true it's just z.

Â But otherwise, it's all the same.

Â And so here now instead of using the NORM.DIST of the original values,

Â you could also first calculate Z and

Â then use the NORM.S.DIST, you will get the same answer.

Â No big deal, here's a nice illustration.

Â 7:53

You see on the right is a standard normal distribution.

Â On the left is your original distribution.

Â And you see on the original distribution with a mean of 0.8 and

Â a standard deviation of 0.1, the probability to the left of

Â x=1 is the same as in the standard normal to the left of z=2.

Â And I can use the standard normal distribution for any value of z.

Â Again, nothing special about z=2.

Â Here are the numbers for z equals negative 1.

Â 8:30

To sum up, we saw a first cool application of a normal distribution from engineering.

Â Then we learned about the standard normal distribution, and

Â the function, NORM.S.DIST, in Excel,

Â that allows us to calculate the standard normal probabilities.

Â