Step by step
The question
Question: find the correlation coefficient r for (1, 2), (2, 4), (3, 6), (4, 8), (5, 10).
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: 15 divided by 5 is 3.
The mean of the y numbers: 30 divided by 5 is 6.
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 6 is −4.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−2 times −4 is 8.
And each distance squared.
−2 squared is 4.
−4 squared is 16.
We add the last three columns: 20, 10 and 40.
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: 20 on top.
10 times 40 under the root.
Careful!
Do not forget the square root.
Without it, r comes out far too small.
4 Work it out
10 times 40 is 400.
The square root of 400 is 20.
So r is 20/20.
20 divided by 20 is 1.
5 What r tells us
On a scale from negative 1 to 1.
r is here, at 1.
Positive: as x goes up, y goes up.
Exactly 1: all the points are on one straight line.
A perfect positive correlation.
6 Check
Let's check.
The products add up to 20, 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.