Worked examples
The question
Question: find the correlation coefficient r for (1, 2), (2, 4), (3, 6), (4, 8), (5, 10).
(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.
r: from −1 to 1, how close to a line
1 Find the means
The mean of the x numbers: 15 divided by 5 is 3.
mean x = 1 + 2 + 3 + 4 + 55 = 3
The mean of the y numbers: 30 divided by 5 is 6.
mean y = 2 + 4 + 6 + 8 + 105 = 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.
r = products√x squares × y squares
each word = the total of its column
We put the totals in: 20 on top.
10 times 40 under the root.
r = 20√10 × 40
Careful!
Do not forget the square root.
Without it, r comes out far too small.
Careful: not 2010 × 40 ✗
4 Work it out
10 times 40 is 400.
r = 20√400
The square root of 400 is 20.
10 × 40 = 400
√400 = 20
So r is 20/20.
r = 2020
20 divided by 20 is 1.
r = 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.
= perfect positive correlation
near ±1: strong, near 0: none
6 Check
Let's check.
The products add up to 20, so r must be positive.
It is.
total of products 20 > 0 → r > 0
And r is between negative 1 and 1.
So it makes sense!
−1 ≤ 1 ≤ 1
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.
▶ Watch it solved
The question
Question: find the correlation coefficient r for (1, 5), (2, 4), (3, 6), (4, 3), (5, 7).
(1, 5) (2, 4) (3, 6) (4, 3) (5, 7)
r is a number from negative 1 to 1.
It tells how close the points are to a straight line, going up or down.
r: from −1 to 1, how close to a line
1 Find the means
The mean of the x numbers: 15 divided by 5 is 3.
mean x = 1 + 2 + 3 + 4 + 55 = 3
The mean of the y numbers: 25 divided by 5 is 5.
mean y = 5 + 4 + 6 + 3 + 75 = 5
2 Make the table
A table with the points.
How far each point is from the means.
1 minus 3 is −2, and 5 minus 5 is 0.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−2 times 0 is 0.
And each distance squared.
−2 squared is 4.
0 squared is 0.
We add the last three columns: 3, 10 and 10.
3 Put them in the rule
The rule: the total of the products, over the square root of the two totals of squares multiplied.
r = products√x squares × y squares
each word = the total of its column
We put the totals in: 3 on top.
10 times 10 under the root.
r = 3√10 × 10
Careful!
Do not forget the square root.
Without it, r comes out far too small.
Careful: not 310 × 10 ✗
4 Work it out
10 times 10 is 100.
r = 3√100
The square root of 100 is 10.
10 × 10 = 100
√100 = 10
So r is 3/10.
r = 310
3 divided by 10 is 0.3.
r = 0.3
5 What r tells us
On a scale from negative 1 to 1.
r is here, at 0.3.
r is close to 0: no correlation.
The points do not follow a line.
= no correlation
near ±1: strong, near 0: none
6 Check
Let's check.
The products add up to 3, so r must be positive.
It is.
total of products 3 > 0 → r > 0
And r is between negative 1 and 1.
So it makes sense!
−1 ≤ 0.3 ≤ 1
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.
▶ Watch it solved
The question
Question: find the correlation coefficient r for (1, 3), (2, 5), (3, 4), (5, 6), (6, 9), (7, 9).
(1, 3) (2, 5) (3, 4) (5, 6) (6, 9) (7, 9)
r is a number from negative 1 to 1.
It tells how close the points are to a straight line, going up or down.
r: from −1 to 1, how close to a line
1 Find the means
The mean of the x numbers: 24 divided by 6 is 4.
mean x = 1 + 2 + 3 + 5 + 6 + 76 = 4
The mean of the y numbers: 36 divided by 6 is 6.
mean y = 3 + 5 + 4 + 6 + 9 + 96 = 6
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 6 is −3.
These distances always add up to 0.
A good check.
The product: the two distances multiplied.
−3 times −3 is 9.
And each distance squared.
−3 squared is 9.
−3 squared is 9.
We add the last three columns: 28, 28 and 32.
3 Put them in the rule
The rule: the total of the products, over the square root of the two totals of squares multiplied.
r = products√x squares × y squares
each word = the total of its column
We put the totals in: 28 on top.
28 times 32 under the root.
r = 28√28 × 32
Careful!
Do not forget the square root.
Without it, r comes out far too small.
Careful: not 2828 × 32 ✗
4 Work it out
28 times 32 is 896.
r = 28√896
With a calculator, the square root of 896 is about 29.933.
28 × 32 = 896
√896 ≈ 29.933
So r is about 28/29.933.
r ≈ 2829.933
Divided: about 0.935, to 3 decimal places, like the books.
r ≈ 0.935
5 What r tells us
On a scale from negative 1 to 1.
r is here, at 0.935.
Positive: as x goes up, y goes up.
Close to 1: strong.
A strong positive correlation.
= strong positive correlation
near ±1: strong, near 0: none
6 Check
Let's check.
The products add up to 28, so r must be positive.
It is.
total of products 28 > 0 → r > 0
And r is between negative 1 and 1.
So it makes sense!
−1 ≤ 0.935 ≤ 1
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.
▶ Watch it solved
