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Okay. We are in the middle of about half hour,

45 minute of Statistics. Actually, it's more, because we'll keep

going back to it, after we introduce the concepts.

But I'm also assuming, by the way, if you have never done Statistics before, you

will, would have before watching the next step, made yourself very comfortably at

calculating means and standard deviations which includes calculating variants.

Before each two relationships, because the reason I'm saying that is because I've

given you a note on the side which goes through numbers.

I shouldn't spend time in the video going through numbers doing mean and variance.

I believe that is taking up more valuable time, doing and talking about things that

are, I think, more important. Means and variances are more difficult.

Having said that, please do examples and the easier number crunching in your

assessment. Okay.

Now, I think I tried to convince you a little bit about why relationships matter.

If you are not fully convinced, believe me after you are done about portfolio theory.

In fact, by the end of next week, you will see that that's the only thing that

matters. Not just in life, relationships not just

matter in life. They matter in Finance.

No wonder I love it. Okay, so let's start with relationships.

There are two things you listen, there are multiple ways you listen, but two

fundamental things about relationships is this, how do you measure them, right?

So, suppose I have y over here and I have x over here.

As soon as I have relationships, I have two, right?

At least two. So, I'm going to call this y and you'll

see in a second why I'm calling this y and x.

Because, in Finance, in Statistics there is a fundamental structure and symbols

which are very similarly used in almost every textbook.

And y and x is very common. So, let's make y now.

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What is bushels of corn? The amount, it's a, it's a measurement

unit of corn. And let's call this x inches of rainfall.

Again, I'm doing an example which you can relate to whether really it's important to

Finance or not, right? Because that's the beauty of Statistics.

I sometimes feel like Statistics is may go slightly ahead of Finance in the love

hierarchy of things. I can't believe I said it.

But anyways, so, that's the way, okay. So now, why are you interested in this

relationship? You are interested in this relationship

simply to figure out whether rainfall has impact on the production of corn, one, and

what is the nature of the impact, okay? So, the reason why we are interested in

the relationships in a portfolio in the Finance context will become more apparent

once we have started doing the financial application.

First thing you do is calculate the average value, y bar.

Second thing you do here is calculate the average value, x bar.

Another way of saying the x bar and y bar are the values you expect to happen.

So, on average, how many bushels of corn are you expected to produce?

And on average, how many inches of rainfall are you expected to get?

And we can throw in numbers there if you want, you know, 120, 60, whatever.

I'm, I'm, I'm sure I'm just 60 inches of rainfall, maybe a lot, a little, I don't

have a clue, right? You're not interested in the averages.

Stand alone you are, but now what you are asking is the following question.

How are these two related? So, the very simple way of thinking about

is rain falling. Ask yourself, when rainfall is below the

average? And remember, here are the probabilities.

When rainfall is below the average, what the heck is corn doing?

Let's assume that for this particular data point, corn, it happens to be above its

average, right? For this data point, how would you measure

that deviation, this deviation? It will be y, sorry, x.

Xi - x bar is the amount, measurement of this.

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Then, you multiply that by yi - y bar. In this case, what is xi - x bar?

It happens to be positive, and but accidentally or maybe somehow, the yi

minus y bar is negative. But when you do most of these, what will

you tend to see? That this tends to be positive on this

side, right? So, you, what will you do?

You'll then sum all these up and you'll get what is called covariance.

And what it's measuring is, multiplied by the probabilities, of course.

What it's measuring? What is the tendency of corn to be when

rainfall is off its normal behavior? Corn, off it's normal behavior, too.

So, I took a perverse example here, when rainfall was positive, corn became little

negative. That could happen, right?

If there's excess rainfall. However, on average, what do you expect

this to be? This number to be greater than zero.

Why? Because hopefully, rainfall helps the

production of corn. Maybe if I took rice here and I am just

out of my league, I am talking about agriculture, maybe the relationship is

more strong, positive, right, because it needs more water.

But you understand whatever I measure, I measured.

When I deviate from the average in this, how does this deviate exactly for that

situation? So, point by point, point by point, that's

where the probabilities are about. This is the fundamental measure of

relationships, okay? And its called covariance.

How do you co-vary with something, right? So, let's ask, lets assume, every time I

turn, Come on, on life and I have a smile on my face, you also automatically have

smile on your face. So, you tend to smile.

We have a positive covariance going on. On the other hand, if you dislike my guts

for some reason and every time I smile, you kind of frown.

Then maybe we don't have something positive going on.

So, what I am saying is you can use this same phenomena and the normal is no smile

at all, right? So, covariance captures that relationship.

But there's a tragedy with covariance, and that is, two things are wrong, wrong with

it, and I'll just emphasize that in a second.

Let me just start off with a [unknown]. So, covariance has one issue.

One issue is, magnitude is not communicated.

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Let me take a corn example. So, it's called sigma y, x.

Covariance of y with x is this. Summation Pi (yi - y bar) (xi - x bar),

right? It's very important to take deviations

from normal behavior. That's why I said, when I smile, it's,

what is my normal behavior? No smile, right?

Okay that's, that's [unknown]. And the probabilities are in there, right?

Look, what will, what will happen to this? Right now, I had this in bushels, if I

believe, yep. And I had this rainfall in inches.

Could I change this number, and suppose this number was 55, whatever, can I make

this number larger Just, without changing anything dramatically fundamental?

Answer is yes. Start measuring your rainfall instead of

inches, in millimeters. What will happen?

This number will become big. Because there are a lot more millimeters

in an inch, right? So, it does, it doesn't have magnitude.

It doesn't tell me anything. The only thing that's good is, it's

telling sign is okay. But magnitude is not reflective of

strength or weakness. The second tragedy with it is this, it is

unit-dependent. So, what are the units of covariance?

The units of the covariance are the units of both the things being measured,

alright? [laugh] In, when we do return analysis,

both the returns are measured in percentage.

So, it's not a big deal, but I wanted that's why to show you, why Statistics is

so awesome and why you have to deal with things which are more difficult than in

finance? So, bushel inches, what it is suppose to

mean? I mean, it shouldn't be.

An ideal measure of relationship shouldn't have units.

So, how can I compare bushel inches with, say, the productivity of people and

whether they have had a school education or not.

That units are totally different, correlation, it, the measurement, the

relationship should be comparable across different phenomena.

So, here's the tragedy of covariance. Though it's trying to measure the right

thing, its magnitude doesn't mean much and it's unit-dependent.

So, what did we do? What does the, the statisticians do?

A very little about this to talk about that's important to us, but this one is

extremely important. So, we took covariance of y and x, which

units was what? Bushels, four inches.

And magnitude was what? Didn't affect much, but sign was okay.

In other words, the, that it's positive or negative relationship is being reflected

because you're taking deviations from the mean and then multiplying.

So, what did we do? We came up with a measure called

correlation. And what is correlation?

Correlation simply takes sigma yx. Remember, what are the units?

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And a second thing happens, which is little bit, I am not going to go into a

correlation which is written like this, yx is between -one and +one.

So, you see the awesomeness of correlation.

It's taken covariance, retained its value of positive negative sign being important,

got rid of its issues of what, the units by dividing by the standard deviations of

the two, and creating a number that can be compared across different phenomena, okay?

So, let's take a break again and I'll come back with one last statistical and an

extremely important statistical concept called regression.

Which is needed in Finance and almost in any other discipline.

Again, a way of capturing relationships, but an important one for us.

See you.