Worked examples
The question
Question: find the population standard deviation of 4, 8, 6, 5 and 7.
Population SD of 4, 8, 6, 5, 7
The standard deviation, SD, tells how far the numbers usually are from the mean.
SD = how far the numbers usually are from the mean
1 Find the mean
First the mean: add them all, then divide by how many.
The total is 30.
30 divided by 5 is 6.
mean = 4 + 8 + 6 + 5 + 75 = 305 = 6
2 Distances from the mean
A table: how far each number is from the mean.
4 minus 6 is −2.
Negative: 4 is below the mean.
8 minus 6 is 2.
Above the mean.
And the same for the rest.
Look: the distances add up to 0.
The ones below the mean cancel the ones above.
−2 + 2 + 0 − 1 + 1 = 0
Careful!
Adding the distances always gives 0.
So we square them first.
Squares are never negative.
Careful: the distances always add to 0: square them first
3 Square them
A new column: each distance squared.
−2 squared is −2 times −2.
Negative times negative is positive: 4.
And the rest.
4 Add the squares
We add the squares: 4 plus 4 plus 0 plus 1 plus 1 is 10.
4 + 4 + 0 + 1 + 1 = 10
5 Divide
These numbers are the whole group, a population.
So we divide by how many there are.
population: ÷ n
10 divided by 5 is 2.
This is the variance.
variance = 105 = 2
6 Square root
The variance is in squared units.
The square root brings it back.
The square root of 2 is 1.4142 and more digits.
SD = √2 = 1.4142…
The books round to one more decimal place than the data.
So the SD is about 1.4.
SD ≈ 1.4
1 decimal place: 1 more than the data
So the numbers are usually about 1.4 away from the mean, 6.
the numbers are about 1.4 from the mean 6
7 Check
Let's check: 1.4 squared is 1.96.
That is about the variance, 2.
So it is right!
1.42 = 1.96 ≈ 2
Let's look back.
First: find the mean.
Then: distances from the mean.
Then: square them.
Then: add the squares.
Then: divide.
Then: square root.
Last: check.
Answer 1 decimal place: 1 more than the data
▶ Watch it solved
The question
Question: find the sample standard deviation of 3, 5, 7 and 9.
Sample SD of 3, 5, 7, 9
The standard deviation, SD, tells how far the numbers usually are from the mean.
SD = how far the numbers usually are from the mean
1 Find the mean
First the mean: add them all, then divide by how many.
The total is 24.
24 divided by 4 is 6.
mean = 3 + 5 + 7 + 94 = 244 = 6
2 Distances from the mean
A table: how far each number is from the mean.
3 minus 6 is −3.
Negative: 3 is below the mean.
5 minus 6 is −1.
7 minus 6 is 1.
Above the mean.
Look: the distances add up to 0.
The ones below the mean cancel the ones above.
−3 − 1 + 1 + 3 = 0
3 Square them
A new column: each distance squared.
−3 squared is −3 times −3.
Negative times negative is positive: 9.
And the rest.
4 Add the squares
We add the squares: 9 plus 1 plus 1 plus 9 is 20.
9 + 1 + 1 + 9 = 20
5 Divide
These numbers are a sample: only part of the whole group.
For a sample we divide by one less than how many, n minus 1.
sample: ÷ (n − 1)
There are 4 numbers, so we divide by 4 minus 1, which is 3.
20 divided by 3 is 6.6667 and more digits.
variance = 204 − 1 = 203 = 6.6667…
Careful!
For a sample we divide by 3, not by 4.
Careful: not ÷ 4 ✗
6 Square root
The variance is in squared units.
The square root brings it back.
The square root of 6.6667… is 2.5820 and more digits.
SD = √6.6667… = 2.5820…
The books round to one more decimal place than the data.
So the SD is about 2.6.
SD ≈ 2.6
1 decimal place: 1 more than the data
So the numbers are usually about 2.6 away from the mean, 6.
the numbers are about 2.6 from the mean 6
7 Check
Let's check: 2.6 squared is 6.76.
That is about the variance, 6.67.
So it is right!
2.62 = 6.76 ≈ 6.67
Let's look back.
First: find the mean.
Then: distances from the mean.
Then: square them.
Then: add the squares.
Then: divide.
Then: square root.
Last: check.
Answer 1 decimal place: 1 more than the data
▶ Watch it solved
The question
Question: find the population standard deviation of 70, 85, 90, 75 and 80.
Population SD of 70, 85, 90, 75, 80
The standard deviation, SD, tells how far the numbers usually are from the mean.
SD = how far the numbers usually are from the mean
1 Find the mean
First the mean: add them all, then divide by how many.
The total is 400.
400 divided by 5 is 80.
mean = 70 + 85 + 90 + 75 + 805 = 4005 = 80
2 Distances from the mean
A table: how far each number is from the mean.
70 minus 80 is −10.
Negative: 70 is below the mean.
85 minus 80 is 5.
Above the mean.
And the same for the rest.
Look: the distances add up to 0.
The ones below the mean cancel the ones above.
−10 + 5 + 10 − 5 + 0 = 0
Careful!
Adding the distances always gives 0.
So we square them first.
Squares are never negative.
Careful: the distances always add to 0: square them first
3 Square them
A new column: each distance squared.
−10 squared is −10 times −10.
Negative times negative is positive: 100.
And the rest.
4 Add the squares
We add the squares: 100 plus 25 plus 100 plus 25 plus 0 is 250.
100 + 25 + 100 + 25 + 0 = 250
5 Divide
These numbers are the whole group, a population.
So we divide by how many there are.
population: ÷ n
250 divided by 5 is 50.
This is the variance.
variance = 2505 = 50
6 Square root
The variance is in squared units.
The square root brings it back.
The square root of 50 is 7.0711 and more digits.
SD = √50 = 7.0711…
The books round to one more decimal place than the data.
So the SD is about 7.1.
SD ≈ 7.1
1 decimal place: 1 more than the data
So the numbers are usually about 7.1 away from the mean, 80.
the numbers are about 7.1 from the mean 80
7 Check
Let's check: 7.1 squared is 50.41.
That is about the variance, 50.
So it is right!
7.12 = 50.41 ≈ 50
Let's look back.
First: find the mean.
Then: distances from the mean.
Then: square them.
Then: add the squares.
Then: divide.
Then: square root.
Last: check.
Answer 1 decimal place: 1 more than the data
▶ Watch it solved
