Step by step
The question
Question: find the correlation coefficient r for (2, 10), (4, 8), (6, 9), (8, 5), (10, 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.
1 Find the means
The mean of the x numbers: 30 divided by 5 is 6.
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.
2 minus 6 is −4, and 10 minus 7 is 3.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−4 times 3 is −12.
And each distance squared.
−4 squared is 16.
3 squared is 9.
We add the last three columns: −34, 40 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.
We put the totals in: −34 on top.
40 times 34 under the root.
Careful!
Do not forget the square root.
Without it, r comes out far too small.
4 Work it out
40 times 34 is 1360.
With a calculator, the square root of 1360 is about 36.878.
So r is about −34/36.878.
Divided: about −0.922, to 3 decimal places, like the books.
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 −34, so r must be negative.
It is.
And r is between negative 1 and 1.
So it makes sense!
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.
Last: check.