Step by step
The question
Question: find the correlation coefficient r for (0, 7), (1, 6), (2, 6), (3, 3), (4, 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: 10 divided by 5 is 2.
The mean of the y numbers: 25 divided by 5 is 5.
2 Make the table
A table with the points.
How far each point is from the means.
0 minus 2 is −2, and 7 minus 5 is 2.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−2 times 2 is −4.
And each distance squared.
−2 squared is 4.
2 squared is 4.
We add the last three columns: −11, 10 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: −11 on top.
10 times 14 under the root.
Careful!
Do not forget the square root.
Without it, r comes out far too small.
4 Work it out
10 times 14 is 140.
With a calculator, the square root of 140 is about 11.832.
So r is about −11/11.832.
Divided: about −0.93, to 3 decimal places, like the books.
5 What r tells us
On a scale from negative 1 to 1.
r is here, at −0.93.
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 −11, 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.