Step by step
The question
Question: find the population standard deviation of 4, 8, 6, 5 and 7.
The standard deviation, SD, tells 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.
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.
Careful!
Adding the distances always gives 0.
So we square them first.
Squares are never negative.
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.
5 Divide
These numbers are the whole group, a population.
So we divide by how many there are.
10 divided by 5 is 2.
This is the variance.
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.
The books round to one more decimal place than the data.
So the SD is about 1.4.
So the numbers are usually about 1.4 away 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!
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.