Step by step
The question
Question: find the correlation coefficient r for (1, 3), (2, 5), (3, 6), (4, 9), (5, 12).
r is a number from negative 1 to 1.
It tells how close the points are to a straight line, going up or down.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find the means
The mean of the x numbers: 15 divided by 5 is 3.
The mean of the y numbers: 35 divided by 5 is 7.
2 Make the table
A table with the points.
How far each point is from the means.
1 minus 3 is −2, and 3 minus 7 is −4.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−2 times −4 is 8.
Why multiply?
Up and right of the means, both distances are positive: the product is positive.
Down and left, both are negative: positive again.
Those points make an uphill line.
The other points give negative products: downhill.
And each distance squared.
−2 squared is 4.
−4 squared is 16.
The squares show how spread out x and y are.
We add the last three columns: 22, 10 and 50.
3 Put them in the rule
The rule: the total of the products, over the square root of the two totals of squares multiplied.
Dividing by the spread keeps r between negative 1 and 1, whatever the numbers.
We put the totals in: 22 on top.
10 times 50 under the root.
Careful!
Do not forget the square root.
Without it, r comes out far too small.
4 Work it out
10 times 50 is 500.
With a calculator, the square root of 500 is about 22.361.
So r is about 22/22.361.
Divided: about 0.984, to 3 decimal places.
5 What r tells us
On a scale from negative 1 to 1.
r is here, at 0.984.
Positive: as x goes up, y goes up.
Close to 1: strong.
A strong positive correlation.
6 Check
Let's check.
The products add up to 22, so r must be positive.
It is.
And r is between negative 1 and 1.
So it makes sense!
7 Pick the choice
Our answer is r ≈ 0.984.
That is choice D.
8 Why not the others
A is 0.968.
Wrong: that is r².
B is −0.984.
Wrong: y goes up with x: r is positive.
C is 2.2.
Wrong: that is the slope: r is between −1 and 1.
Let's look back.
First: find the means.
Then: make the table.
Then: put them in the rule.
Then: work it out.
Then: what r tells us.
Then: check.
Then: pick the choice.
Last: why not the others.