Step by step
The question
Question: find the correlation coefficient r for (5, 60), (10, 70), (15, 65), (20, 85), (25, 95).
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: 75 divided by 5 is 15.
The mean of the y numbers: 375 divided by 5 is 75.
2 Make the table
A table with the points.
How far each point is from the means.
5 minus 15 is −10, and 60 minus 75 is −15.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−10 times −15 is 150.
And each distance squared.
−10 squared is 100.
−15 squared is 225.
We add the last three columns: 425, 250 and 850.
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: 425 on top.
250 times 850 under the root.
Careful!
Do not forget the square root.
Without it, r comes out far too small.
4 Work it out
250 times 850 is 212500.
With a calculator, the square root of 212500 is about 460.977.
So r is about 425/460.977.
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.
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 425, 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.