Step by step
The question
Question: find the correlation coefficient r for (1, 10), (2, 8), (3, 9), (4, 5), (5, 3).
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 10 minus 7 is 3.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−2 times 3 is −6.
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.
3 squared is 9.
The squares show how spread out x and y are.
We add the last three columns: −17, 10 and 34.
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: −17 on top.
10 times 34 under the root.
Careful!
Do not forget the square root.
Without it, r comes out far too small.
4 Work it out
10 times 34 is 340.
With a calculator, the square root of 340 is about 18.439.
So r is about −17/18.439.
Divided: about −0.922, to 3 decimal places.
5 What r tells us
On a scale from negative 1 to 1.
r is here, at −0.922.
Negative: as x goes up, y goes down.
Close to negative 1: strong.
A strong negative correlation.
6 Check
Let's check.
The products add up to −17, so r must be negative.
It is.
And r is between negative 1 and 1.
So it makes sense!
7 Pick the choice
Our answer is r ≈ −0.922.
That is choice A.
8 Why not the others
B is 0.922.
Wrong: y goes down as x goes up: r is negative.
C is −1.7.
Wrong: that is the slope: r is between −1 and 1.
D is 0.85.
Wrong: that is r².
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.