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The Standard Deviation

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The standard deviation is a measure of how far each piece of data is, on average, from the mean. Consider the following data:

x = 3,6,2,7,4,6,7

The mean of these numbers is

We can therefore write out a list of x-x

x-x = -2,1,-3,2,-1,1,2

The mean of these numbers is

The numbers cancel each other out. You should expect this, as there ought to be equal numbers above and below the mean.

We need to look at how far a piece of data is from the mean, regardless of sign. There are two possibilities. We can either take the modulus of the difference, or we can square the difference, so removing any minus signs.

The first option gives you a quantity called the deviation. This process is, however, difficult to put into a simple formula.

The second option leads to values called the variance and standard deviation and gives a very neat formula.

Variance

The variance is defined as the average square deviation from the mean value.

By expanding the bracket and simplifying, a simple formula for the variance can be found.

Expand the brackets, to obtain 3 different sums.

Since x is the same for all data, we can just remove this from the middle bracket. The right hand sum is simply counting the squared mean, n times.

In the middle term, the sum divided by n just gives the mean.

Tidying up gives the final expression

Therefore the variance is given by the formula:

Let's use this formula on the data at the top of the page

x = 3,6,2,7,4,6,7

When calculating standard deviations, you should always tabulate the results

x	x²
3	9
6	36
2	4
7	49
4	16
6	36
7	49

199

Therefore the variance is

Substitute in the sums from above.

write over a common denominator.

The Standard Deviation

The standard deviation is defined as the square root of the variance, and is denoted by

So for the above example

When you are asked for a measure of dispersion, it is the standard deviation that you should quote, not the variance. If you think about it, the square root undoes the previous squaring, giving an indication of the average deviation from the mean.

The next few pages will look at more examples of calculating the standard deviation, in particular from grouped data. The standard deviation is very important.

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