Calculating the Standard Deviation - Formula 1

Well, there it is! The first formula for calculating the standard deviation. Firstly, I had better explain those curious heiroglyphs, and then we will go through how the formula is turned into a set of instructions:
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Explanation of the Terms

x. Normally in mathematics, letters such as "x" stand for single numbers. In this case, the letter actually stands for an entire list of numbers - every single number in the list whose standard deviation we are trying to find. We use each of these numbers in turn when we calculate the standard deviation, although we just write them as one letter in the formula. In some text books you may see this written as xi, or something similar, to show that it stands for a list of numbers: x1, x2, x3 etc.

. The little line on top of the x is called a "bar", so we pronounce this symbol as "x bar". It simply means the arithmetic mean of all the numbers in the list - i.e. add them together and divide by N, where N is how many numbers there are.

N. Didn't I just tell you this one? N stands for how many numbers there are in the list, i.e. if we are trying to find the standard deviation of 1420 numbers, then N = 1420.

S. This is the other version of the Greek "sigma" - the upper case letter. It has a standard meaning in mathematics, namely "add up a list of numbers". (They use the Greek letter S because it represents Sum, i.e. add together).

What the formula means

This is where we interpret the formula step by step. There are six basic steps. As I list each one, I shall demonstrate it with the following list of numbers:

4, 1, 0, 2, 1, 5, 2, 3, 4, 3

The first thing we must do is to calculate the arithmetic mean of the list of numbers.

=
4 + 1 + 0 + 2 + 1 + 5 + 2 + 3 + 4 + 3
10
 		=
25
10
 = 2.5

Now we subtract the mean value from all the numbers in the list. This gives us the following list of numbers:

1.5, -1.5, -2.5, -0.5, -1.5, 2.5, -0.5, 0.5, 1.5, 0.5

You will notice that some of the numbers are negative, and some positive. This is to be expected, as the mean value is a value which is roughly half way through the list of numbers, so when you subtract it from every number in the list, you would expect it to produce a negative answer roughly half the time.

In algebraic terms, we write the following:

x -
To counteract all the effects of those negative signs, we square the numbers in that list that we just obtained. Squaring a negative number has the effect of removing the minus sign, but we square all the numbers, not just the negative ones:

2.25, 2.25, 6.25, 0.25, 2.25, 6.25, 0.25, 0.25, 2.25, 0.25

You will notice that squaring the first two numbers of the list (1.5 and -1.5) gives exactly the same thing. Again, we can add this squaring to the formula that we are gradually building up:

(x - )2
This is where we come to the "Sigma" part. We now have to add all those squared differences together to get a single number:

2.25 + 2.25 + 6.25 + 0.25 + 2.25 + 6.25 + 0.25 + 0.25 + 2.25 + 0.25 = 22.5

We symbolise adding the numbers together by putting the "Sigma" sign in front of the list of numbers in the algebraic expression:

S(x - )2
The penultimate step is to divide the sum that we got in the previous step by N (or N - 1 if you are using the unbiased version), which you may recall, is the number of items in the list. This gives a number which we call the Variance, and which we represent by the symbol s2. In this case, there are 10 items in the list, so we divide 22.5 by 10:

Variance, s2 =
22.5
10
 = 2.25

Again, we can write all this algebraically:

Variance, s2 =
S(x - )2
N
You will probably have noticed that the symbol for the variance is the same sigma that we used to represent the standard deviation, except that it is squared. That is perfectly true - the variance is the standard deviation squared, and all we need to do to the variance is square root it:

Standard deviation, s = ÖVariance = Ö2.25 = 1.5

Applying the square root to the formula for the variance gives us the complete formula for standard deviation which you saw at the top of this page:

In this case, it gives a fairly simple number - the standard deviation happens to be 1.5 - but in most cases, it will give a nasty never-ending square root.

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