Step by step
The question
Question: find the sample standard deviation of 6, 7, 8, 5, 9, 7, 6 and 8.
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 56.
56 divided by 8 is 7.
2 Distances from the mean
A table: how far each number is from the mean.
6 minus 7 is −1.
Negative: 6 is below the mean.
7 minus 7 is 0.
8 minus 7 is 1.
Above the mean.
Look: the distances add up to 0.
The ones below the mean cancel the ones above.
So adding the distances tells us nothing: it is always 0.
We need every distance to count as positive.
So we square each distance.
A square is never negative.
3 Square them
A new column: each distance squared.
−1 squared is −1 times −1.
Negative times negative is positive: 1.
And the rest.
4 Add the squares
We add the squares: 1 plus 0 plus 1 plus 4 plus 4 plus 0 plus 1 plus 1 is 12.
5 Divide
These numbers are a sample: only part of the whole group.
A sample is usually a bit closer to its own mean than the whole group is.
So we divide by one less than how many: n minus 1, where n is how many numbers.
That makes the answer a little bigger, to make up for it.
There are 8 numbers, so we divide by 8 minus 1, which is 7.
12 divided by 7 is 1.7143 and more digits.
Careful!
For a sample we divide by 7, not by 8.
6 Square root
We squared the distances, so the variance is in squared units.
The square root brings it back to the units of the numbers.
The square root of 1.7143… is 1.3093 and more digits.
We round to one more decimal place than the data, so the answer does not look more exact than the numbers.
So the SD is about 1.3.
So the numbers are usually about 1.3 away from the mean, 7.
7 Check
Let's check: 1.3 squared is 1.69.
That is about the variance, 1.71.
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.