0:21

He worked for technical consulting company in Florida,

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

I called it Florida Elevated Roadway, FERY.

This company, FERY, makes or builds pillars and

elevated roadways in Florida.

Why, do they have to do this?

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

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

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

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

1:02

The construction of these pillars of course,

is key to make this safe, a safe road.

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

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

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

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

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

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

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

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

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

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

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

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

I don't want that to happen!

So this is a key test.

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

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

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

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

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

perhaps in 100 or 110 degree Fahrenheit weather.

There're mosquitoes,

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

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

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

So in our language, the standard deviation is 0.1.

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

that x takes on, is one or larger.

Because then the pillar is not considered safe.

3:19

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

That's important, maybe they make lawsuits,

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

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

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

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

3:50

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

where we found the BINOMDIST function.

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

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

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

But remember NORM.DIST is a cumulative distribution,

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

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

So we use once again the complement rule,

1 minus the probability we just calculated.

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

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

So you actually want to improve the process relative to

the numbers I gave you.

5:40

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

always the same number in this case 2.

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

And so this is a key takeaway.

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

6:03

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

That's really what determines the probability.

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

the ratio is -1.

Again, all five probabilities are the same.

6:24

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

it leads us to the so called standardized normal distribution.

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

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

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

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

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

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

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

Here now, I have to warn you.

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

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

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

there is no true it's just z.

But otherwise, it's all the same.

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

you could also first calculate Z and

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

No big deal, here's a nice illustration.

7:53

You see on the right is a standard normal distribution.

On the left is your original distribution.

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

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

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

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

Again, nothing special about z=2.

Here are the numbers for z equals negative 1.

8:30

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

Then we learned about the standard normal distribution, and

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

that allows us to calculate the standard normal probabilities.