Step by step
The question
Question: find the sample standard deviation of 15, 9, 12, 18 and 6.
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 60.
60 divided by 5 is 12.
2 Distances from the mean
A table: how far each number is from the mean.
15 minus 12 is 3.
15 is above the mean.
9 minus 12 is −3.
Below the mean.
And the same for the rest.
Look: the distances add up to 0.
The ones below the mean cancel the ones above.
3 Square them
A new column: each distance squared.
3 squared is 9.
−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 9 plus 0 plus 36 plus 36 is 90.
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.
There are 5 numbers, so we divide by 5 minus 1, which is 4.
90 divided by 4 is 22.5.
Careful!
For a sample we divide by 4, not by 5.
6 Square root
The variance is in squared units.
The square root brings it back.
The square root of 22.5 is 4.7434 and more digits.
The books round to one more decimal place than the data.
So the SD is about 4.7.
So the numbers are usually about 4.7 away from the mean, 12.
7 Check
Let's check: 4.7 squared is 22.09.
That is about the variance, 22.5.
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.