Step by step
The question
Question: find the correlation coefficient r for (1, 2), (2, 3), (3, 5), (4, 4), (5, 6).
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: 20 divided by 5 is 4.
2 Make the table
A table with the points.
How far each point is from the means.
1 minus 3 is −2, and 2 minus 4 is −2.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−2 times −2 is 4.
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.
Both are −2: squared, 4.
The squares show how spread out x and y are.
We add the last three columns: 9, 10 and 10.
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: 9 on top.
10 times 10 under the root.
Careful!
Do not forget the square root.
Without it, r comes out far too small.
4 Work it out
10 times 10 is 100.
The square root of 100 is 10.
So r is 9/10.
9 divided by 10 is 0.9.
5 What r tells us
On a scale from negative 1 to 1.
r is here, at 0.9.
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 9, 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 = 0.9.
That is choice D.
8 Why not the others
A is 1.
Wrong: the points are not on one line.
B is 0.81.
Wrong: that is r².
C is −0.9.
Wrong: y goes up with x: r is positive.
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.