In many undergraduate experiments, you will be attempting to measure a value to be compared with the literature.
In a standard experiment where all data points are subject to the same random error, this is done using the mean and standard error.
In Physics, it is often important to repeat measurements multiple times and to take an average of the result. The most important method for calculating an average is by calculating the mean of the data, as you'll have come across in school.
For a set of N measurements of the value x, the arithmetic mean of x is defined as
The mean is defined by the sum of the points divided by the number of points
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.
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 in the mean 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
Equation for the standard error
where sigma is the standard deviation, given by
Equation for standard deviation
The correct use of significant figures is important in showing the precision of your data.
Results should be quoted to the number of significant figures so that the last digit is the same order of magnitude as the uncertainty. For example, a reading of 0.245 V with an uncertainty of ±0.01 V should be quoted as 0.25±0.01 V.
The rules for significant figures are as follows:
Excel has built-in functions called AVERAGE and STDEV which make this process significantly simpler