Step by step
The question
Question: find the correlation coefficient r for (1, 3), (3, 5), (5, 4), (7, 8).
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: 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 3 minus 5 is −2.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−3 times −2 is 6.
And each distance squared.
−3 squared is 9.
−2 squared is 4.
We add the last three columns: 14, 20 and 14.
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: 14 on top.
20 times 14 under the root.
Careful!
Do not forget the square root.
Without it, r comes out far too small.
4 Work it out
20 times 14 is 280.
With a calculator, the square root of 280 is about 16.733.
So r is about 14/16.733.
Divided: about 0.837, to 3 decimal places, like the books.
5 What r tells us
On a scale from negative 1 to 1.
r is here, at 0.837.
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 14, so r must be positive.
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.