Original subtitles

Instructor: All right, excellent.

We've covered all univariate measures.

Now it is time to see measures that are used when we work

with more than one variable.

In the next two lessons, we'll explore measures

that can help us explore the relationship between variables.

Our focus will be on covariance

and the linear correlation coefficient.

Let's zoom out a bit and think

of an example that is very easy to understand

and will help us grasp the nature

of the relationship between two variables a bit better.

Think about real estate, which is one

of the main factors that determine house prices?

Their size, right?

Typically, larger houses are more expensive,

as people like having extra space.

The table that you can see here

shows us data about several houses.

On the left side, we can see the size of each house,

and on the right we have the price

at which it's been listed in a local newspaper.

We can present these data points in a scatter plot.

The x-axis will show a house's size,

and the Y-axis will provide information about its price.

We can certainly notice a pattern.

There is a clear relationship between these variables.

We say that the two variables are correlated,

and the main statistic to measure this correlation

is called covariance.

Unlike variance, covariance may be positive,

equal to zero, or negative.

To understand the concept better,

I would like to show you the formulas that allow us to

calculate the covariance between two variables.

It is formulas with an S, because once again

there is a sample and a population formula.

Here they are.

Since this is obviously sample data

we should use the sample covariance formula.

Let's apply it in practice

for the example that we saw earlier.

X will be house size and Y stands for house price.

First, we need to calculate

the mean size and the mean price.

I will also compute the sample standard deviations

in case we need them later on.

Okay, done.

Now let's calculate the nominator

of the covariance function.

Starting with the first house

I'll multiply the difference between its size

and the average house size,

by the difference between the price

of the same house and the average house price.

Once we're ready, we have to perform this calculation

for all houses that we have in the table,

and then sum the numbers we've obtained.

See?

Great.

Our sample size is five.

Now we have to divide the sum above

by the sample size minus one.

The result is the covariance.

It gives us a sense of the direction

in which the two variables are moving.

If they go in the same direction,

the covariance will have a positive sign.

While if they move in opposite directions,

the covariance will have a negative sign.

Finally, if their movements are independent,

the covariance between the house size

and its price will be equal to zero.

There is just one tiny problem with covariance though,

it could be a number like five or 50,

but it can also be something like 0.0023456,

or even over 30 million as in our example.

Values of a completely different scale.

How could one interpret such numbers?

Proceed to the next lecture to find out how

the correlation coefficient can help us with this issue.

Thanks for watching.