National Physical Laboratory

Statistical calculations

The two most important calculations are to find the average or arithmetic mean, and the spread or standard deviation.

Arithmetic mean (average)

To get the best estimate of the ‘true value’ of a measurement you need to take the average of a number of readings.

Just because repeated measurements give you different answers, it may not mean that you are doing anything wrong. It may be due to natural variations in what is going on or it may be because your measuring instrument does not behave in a completely stable way - a tape measure may stretch and give different results.

An average or arithmetic mean is usually shown by a symbol with a bar above it, e.g.  x̄ is the mean value of x.

Blob Plot

This is a ‘blob plot’ showing an example set of values and the mean.

Example of how to calculate arithmetic mean

Calculating the average or arithmetic mean of a number of values

Suppose you have a set of 10 readings. To find the average, add them together and divide by the number of values (10 in this case).

The readings are: 16, 19, 18, 16, 17, 19, 20, 15, 17 and 13

The sum of these is: 170

The average of the 10 readings is: 170/10 = 17

Standard Deviation (spread)

Calculate standard deviation to show the spread of repeated readings. This tells you about the uncertainty of the results.

Roughly two thirds of all readings will fall plus and minus one standard deviation of the average. Roughly 95% of all readings will fall within two standard deviations.

The symbol s is used for the estimated standard deviation.

It is standard practice to use a calculator to work this out but in order to understand the process involved look at an example of how to calculate standard deviation. 

Calculating the estimated standard deviation of a set of values

It is rarely convenient to calculate standard deviations by hand, with pen andpaper alone. But it can be done as follows:

Suppose you have a set of n readings

Start by finding the average:

For the set of readings we used before, 16, 19, 18, 16, 17, 19, 20, 15, 17 and 13.

The average is 17.

Next, find the difference between each reading and the average

i.e. -1, +2, +1, -1, 0, +2, +3, -2, 0, –4

and square each of these

i.e. 1, 4, 1, 1, 0, 4, 9, 4, 0, 16

Next, find the total and divide by n-1 (in this case n is 10, so n-1 is 9)

i.e. (1+4+1+1+0+4+9+4+0+16)/9 = 4.44

The estimated standard deviation, s, is found by taking the square root of the total

i.e. s=4.44 square root = 2.1 (correct to one decimal place).

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