Basic Data Analysis

Introduction

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.

The Mean

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

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.

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 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

Quoting your result

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:

1. All nonzero digits are significant: 1.234 g has 4 significant figures. 1.2 g has 2 significant figures.
2. Zeroes between nonzero digits are significant: 1002 kg has 4 significant figures, 3.07 mL has 3 significant figures.
3. Zeroes to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point: 0.001 ^oC has only 1 significant figure, 0.012 g has 2 significant figures.
4. Zeroes which follow a number after the decimal point are significant: 0.023 mL has 2 significant figures, 0.200 g has 3 significant figures.
5. When a number ends in zeroes that are not to the right of a decimal point, the zeroes may not be significant: 190 miles may be 2 or 3 significant figures, 50,600 calories may be 3, 4, or 5 significant figures.

Using Excel

Excel has built-in functions called AVERAGE and STDEV which make this process significantly simpler