Step by step
The question
Question: find the correlation coefficient r for (1, 4), (3, 5), (5, 9), (7, 8), (9, 14).
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: 25 divided by 5 is 5.
The mean of the y numbers: 40 divided by 5 is 8.
2 Make the table
A table with the points.
How far each point is from the means.
1 minus 5 is −4, and 4 minus 8 is −4.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−4 times −4 is 16.
And each distance squared.
−4 squared is 16.
−4 squared is 16.
We add the last three columns: 46, 40 and 62.
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: 46 on top.
40 times 62 under the root.
Careful!
Do not forget the square root.
Without it, r comes out far too small.
4 Work it out
40 times 62 is 2480.
With a calculator, the square root of 2480 is about 49.800.
So r is about 46/49.800.
Divided: about 0.924, to 3 decimal places, like the books.
5 What r tells us
On a scale from negative 1 to 1.
r is here, at 0.924.
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 46, 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.