The readings from an experiment are often used to test a relationship between two quantities, typically whether two quantities are proportional or inversely proportional.
You should know that if two quantities y and x are directly proportional:
the formula that relates them is $y = kx$, where k is a constant if a graph is plotted of y against x then the graph is a straight line through the origin and the gradient is the value of k.
If the two quantities are inversely proportional then $y = \frac{k}{x}$ and a graph of y against gives a straight line through the origin.
These statements can be used as a basis for a test. If a graph of y against x is a straight line through the origin, then y and x are directly proportional. If you know the values of y and x for two points, you can then calculate two values of k with the formula $y = \frac{k}{x}$ and see whether these two values of k are actually the same. But what if the points are not exactly on a straight line or the two values of k are not exactly the same – is the relationship actually false or is it just that errors caused large uncertainties in the readings?
Later in this chapter, we will look at how to combine the uncertainties in the values for y and x to find an uncertainty for k. However, you can use a simple check to see whether the difference in the two values of k may be due to the uncertainties in the readings. For example, if you found that the two values of k differ by $5\% $ but the uncertainties in the readings of y and x are $5\% $, then you cannot say that the relationship is proved false. Indeed, you are able to say that the readings are consistent with the relationship.
You should first write down a criterion for checking whether the values of k are the same. This criterion is just a simple rule you can invent for yourself and use to compare the two values of k with the uncertainties in the readings. If the criterion is obeyed you can then write down that the readings are consistent with the relationship.

Criterion 1

A simple approach is to assume that the percentage uncertainty in the value of k is about equal to the percentage uncertainty in either x or y; choose the larger percentage uncertainty of x or y.
You first look at the percentage uncertainty in both x and y and decide which is bigger. Let us assume that the larger percentage uncertainty is in x. Your stated criterion is then that ‘if the difference in the percentage uncertainty in the two values of k is less than the percentage uncertainty in x, then the readings are consistent with the relationship’.
If the percentage difference in k values is less than the percentage uncertainty in x (or y), the readings are consistent with the relationship.

Criterion 2

Another criterion is to state that the k values should be the same within $10\% $ or $20\% $, depending on the experiment and the uncertainty that you think sensible. It is helpful if the figure of $10\% $ or $20\% $ is related to some uncertainty in the actual experiment.
Whatever criterion you use, it should be stated clearly and a clear conclusion given. The procedure to check whether two values of k are reasonably constant is as follows:
- Calculate two values of the constant k. The number of significant figures chosen when writing down these values should be equal to the least number of significant figures in the data used. If you are asked to justify the number of significant figures you give for your value of k, state the number of significant figures that x and y were measured to and that you will choose the smallest. Do not quote your values of k to one significant figure to make them look equal when x and y were measured to two significant figures.
- Calculate the percentage difference in the two calculated values of k. It is worthwhile using one more significant figure in each actual value of k than is completely justified in this calculation.
Compare the percentage difference in the two values of k with your clearly stated criterion. You could compare your percentage difference in k values with the larger of the percentage differences in x and y.