>> This video is going to be
about the correlation coefficient,
the small r. Now let's do an example
of correlation coefficient.
An agricultural research organization tested a particular
chemical fertilizer to try to find out whether an increase
in the amount of fertilizer used would lead
to a corresponding increase in the food supply.
Let's look at the data; fertilizer in pounds and bushels
of beans, and that's the data.
Now we need to think about who's going to be x and who's going
to be y. In this case the fertilizer is going to be x
because that's independent on how many beans will be produced.
So this will be x and this will be y. Y is called the dependent
variable because it depends on the fertilizer.
Okay let's continue.
And what we want to do is we want to write this instead
of horizontal we want to write it vertically,
and now that we know which one's x and which one's y we're going
to write the data vertically.
Now for the correlation coefficient we're going
to need more variables; we're going to need x times y,
we're going to need x square, and also y square.
We're going to n-- we're going to need those as well.
So let's write down the data here,
put in the vertical setup here.
Alright, now let's multiply x times y. So 2 times 4 is 8,
1 times 3 is 3, 3 times 4 is 12, 2 times 3 is 6, 4 times 6 is 24,
and 5 times 5 is 25, and 3 times 5 is 15.
So we filled that column up.
Let's also now do x squared.
Make sure you get the x variable.
So 2 square is 4, 1 square is 1, 3 square is 9, 2 square is 4,
4 square is 16, 5 square is 25, and 3 square is 9.
Alright now let's do the y square column.
Don't forget we need to utilize the y column, so let's do that.
So 4 square is 16, 3 square is 9, 4 square is 16,
3 square is 9, 6 square is 36, 5 square is 25,
and 5 square again is 25.
Great, so let's complete the table by writing
down some vertical lines here so I can see the columns better
and let's find the sums; we'll add column xy, xy, x square
and y square so we can find the sums.
So the sum of x, we add that column, gives you 20.
The sum of y, we call--
we do that, we add it all up, gives you 30.
The sum of xy, and when we sum that we get 93.
The sum of x square we sum that, we get 68.
And the sum of y square, we add that up, we get 136.
Okay now let's write down the formula for r,
the correlation coefficient.
r is n times the sum of x times y; take away the sum
of x times the sum of y.
All that we're going to put-- we're going to divide that
and the denominator will be n times the sum of x square,
take away the sum of x, and when you have the sum
of x then square it all this is going to be one operation,
we're going to put brackets on it, and then another bracket;
n times the sum of y square--
y squared, and I'm going to subtract that.
I'm going to take the sum of y and when we find the sum
of y we're going to square it, and we're going
to close the bracket,
and we want to take the square root of it.
Now here is an important point to--
to realize is that a couple of things.
First of all, the best way to do this problem is to do the math.
So best way to do, you know, after you substitute,
multiply these two out, multiply these two out,
and here's a subtraction, right, write it out.
Another thing you'll notice is the--
that there's a bracket here and then there's a bracket there.
So it's best after you place the numbers in here
when you substitute, you know, multiply, square,
and then subtract, very important to do the stepwise.
Also don't forget to take the square root in the end
that sometimes we forget.
Now let's substitute the values.
n is seven, that's how many data points we have.
The sum of xy, where's that at?
It's right here, it's 93, right.
And then we're going to have a subtraction,
the sum of x times the sum of y, and those you can find,
right here's the sum of x, here's the sum of y,
making sure also that we don't forget the subtraction sign,
right, and we keep the parenthesis, right;
very important so we can remember
that we need to multiply.
Alright we're going to divide that,
we're going to keep substituting,
and the denominator we have n, right we have the bracket n,
which is seven, the sum of x square is 68, take away the sum
of x. We look in the column, which is 20,
and then we square that, and then we close the bracket;
very important to keep everything intact.
Then the new bracket and a seven,
the sum of y square is 136, and subtract that
and we find the sum of y and then we square it
and we close the bracket; don't forget the square root.
Now the next step is going to be to start multiplying
and simplifying this a little bit.
So 7 times 93 is going to give you 651;
don't forget the subtraction.
20 times 30 is going to be 600,
divided by 7 times 68 will give you 476.
Subtract 20 square is going to be 400,
and then make sure you put the brackets.
Then the next bracket, 7 times 136 will give you 9--
952, and subtract that from 30 square which is 900.
Close the bracket; don't forget the square root.
So the next step will be 651 take away 600; that's going
to give you 51, divided by 476 take away 400; that's going
to give you 76; don't forget the bracket.
952 take away 900; that's going to give you 52.
Close the bracket; don't' forget the square root.
Alright let's rewrite; 51 divided by 76 times 52 is going
to give you 3952; don't forget the square root.
Then again 51 divided by the square root of 300-- 3952.
That's going to be 62.8649.
And then we do the division and give you 8.11, which is a very--
is a very strong correlation,
a strong relationship between x and y.