Descriptive statistics

From: Jon Walsh
Date: 15 Feb 1999

Subject: Which standard deviation formula?

I am confused about all the different formulae for standard deviation. Which one should I use?

Maths Help suggests:

Yes, it can be confusing, especially as different textbooks sometimes use different notation.

Basically there are two different standard deviations:
one has DIVISOR n, the other has DIVISOR n-1
NB we have omitted subscripts for clarity

The "divisor n" formula is used if the data set is known to be the whole population, i.e. the complete set of data you are interested in.
(This formula is used at GCSE and in introductory courses.)
The "divisor n-1" formula is used when you are working with a sample data set, and wish to estimate the standard deviation of the parent population.
(This is a more sophisticated concept, met when studying statistical inference.)

All the other formulae which you might see are modifications of the above two, for example to make the calculations easier to perform by hand, or to use if the data is given in the form of a frequency table.

Since there are too many modifications to list here, we strongly suggest you ask your tutor which ones to use, and ignore any others you might come across. For example, if you are working towards an exam, you might be given a Formula Sheet to refer to, so just practise using the particular formulae included there.

Nowadays, most pocket calculators include statistical functions built in. Such calculators have two buttons for standard deviations, one for the "divisor n" form and one for the "divisor n-1" form. Make sure you know which is which!

HINT
If in doubt, note that the standard deviation when working with a sample (divisor n-1) will be slightly greater that the standard deviation when working with a population (divisor n) because we are only dividing by n-1 rather than n.

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