Step by step
The question
Question: find the sample standard deviation of 5, 7, 8 and 12.
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 32.
32 divided by 4 is 8.
2 Distances from the mean
A table: how far each number is from the mean.
5 minus 8 is −3.
Negative: 5 is below the mean.
7 minus 8 is −1.
And the same for the rest.
12 minus 8 is 4.
Above the mean.
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 −3 times −3.
Negative times negative is positive: 9.
And the rest.
4 Add the squares
We add the squares: 9 plus 1 plus 0 plus 16 is 26.
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 4 numbers, so we divide by 4 minus 1, which is 3.
26 divided by 3 is 8.6667 and more digits.
Careful!
For a sample we divide by 3, not by 4.
6 Square root
The variance is in squared units.
The square root brings it back.
The square root of 8.6667… is 2.9439 and more digits.
The books round to one more decimal place than the data.
So the SD is about 2.9.
So the numbers are usually about 2.9 away from the mean, 8.
7 Check
Let's check: 2.9 squared is 8.41.
That is about the variance, 8.67.
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.