Step by step
The question
Question: find the correlation coefficient r for (1, 9), (3, 6), (5, 4), (7, 1).
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: 16 divided by 4 is 4.
The mean of the y numbers: 20 divided by 4 is 5.
2 Make the table
A table with the points.
How far each point is from the means.
1 minus 4 is −3, and 9 minus 5 is 4.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−3 times 4 is −12.
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.
−3 squared is 9.
4 squared is 16.
The squares show how spread out x and y are.
We add the last three columns: −26, 20 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: −26 on top.
20 times 34 under the root.
Careful!
Do not forget the square root.
Without it, r comes out far too small.
4 Work it out
20 times 34 is 680.
With a calculator, the square root of 680 is about 26.077.
So r is about −26/26.077.
Divided: about −0.997, to 3 decimal places.
5 What r tells us
On a scale from negative 1 to 1.
r is here, at −0.997.
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 −26, 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 ≈ −0.997.
That is choice B.
8 Why not the others
A is 0.997.
Wrong: y goes down as x goes up: r is negative.
C is −1.3.
Wrong: that is near the slope: r is between −1 and 1.
D is −0.5.
Wrong: the points are almost on a line.
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.