# Standard Error

## What is the Standard Error?

The standard error is a measure of the random error in a set of data. Repeat measurements in an experiment will be distributed over a range of possible data, scattered about the mean. So how can this be calculated? Firstly we have to calculate the standard deviation of the data.

Fig 1: A qualitative representation of the standard error. The mean could lie anywhere in the red region of the curve. As we take more data measurements (shown by the histogram) the uncertainty on the mean reduces. The standard error quanifies this uncertainty.

## Standard Deviation

For a set of N measurements of the value x, the standard deviation is defined as

This is effectively the root mean squared of the average of
the deviations of all the different measurements. It gives a quantified
measure of the *spread* of the data. How can we use this to
calculate the error related to a mean?

## Standard Error

If we were to take the error of the mean to be the standard deviation, it would be rather pessimistic! More importantly, if we were to repeat the measurement more times, there would be little change to the standard deviation. However, we would expect the random error to reduce significantly. So how do we take this into account?

For a set of N data points, the random error can be estimated using the standard error approach, defined by

## Using Excel

Similarly to calculating the mean, it would be impractical to calculate the standard deviation and standard error by hand. Thankfully in Excel this can solved in a few simple commands.

Consider again the data from the mean webpage, but this time calculate the standard deviation and standard error. The data, with the mean, can be found in this spreadsheet.

This time the important function that needs to be used is the "STDEV()" function, which will calculate the standard deviation of a set of data. This is shown in figure 2.

Fig 2: How to calculate the standard deviation and standard error of a set of data.

Try this yourself! The solution is on the second sheet of the example spreadsheet.

The mean and the standard error together give us the best value for the current in this system. This would be quoted as

(1.05 ± 0.03) A.

## Significant Figures of the Standard Error

You will normally only need to **quote the standard error to one significant figure**. The exceptions to this are

when the first significant figure of the error is a 1 consider quoting to two significant figures, and

when you have around 10,000 or more data points consider quoting to two significant figures.

The reason for the first exception is that, for example, rounding 0.14 to 0.1 represents a change in the error of almost 30%! The reason for the second exception is that the error in the error (errors have errors too!) does not fall to a few percent until we have around 10,000 data points, and only then can we justify that second signifcant figure (remember that the second figure in a number represents less than 10% of that number). See section 2.7.1 of Hughes and Hase for more detail.

For more information on significant figures, see this page.

## Other Resources

- A more detailed analysis into calculating the mean and standard deviation
- Calculating Mean and Standard Deviation in Excel
- A similar applet
- Applets illustrating mean and standard deviation

Figure 1 was modified from Measurements and their Uncertainties, Hughes and Hase..