What type of linear correlation might exist between the mass loaded onto a spring and its extension.
Let’s begin by reminding ourselves in general about the different types of linear correlation that exist. Firstly, we have positive linear correlation. If the two variables whose correlation we’re assessing are 𝑥 and 𝑦, then a positive linear correlation means that, in general, as 𝑥 increases, 𝑦 also increases.
If we were to consider a scatter plot of 𝑥 against 𝑦, then if the two variables are positively linearly correlated, then, in general, the points will lie relatively close to a straight line with positive gradient, that is, sloping upwards from left to right. The strength of the positive linear correlation will be determined by how closely the points lie to the straight line. And we can also use words like moderate, strong, or weak to describe the degree of positive correlation.
The second type of linear correlation that exists is negative correlation. This time, in general, as one variable increases, the other decreases. And if we consider a scatter plot of the two variables, we’ll see that the points lie relatively close to a straight line with a negative slope. That’s a line sloping downwards from left to right. Again, we can describe the degree, or strength, of correlation depending on how closely the points lie to the straight line.
Finally, although it’s not really a type of correlation, we could say that there is no linear correlation between the two variables if they don’t look like they can be closely modelled by a straight line. It may be the case that the pattern can be modelled by another type of graph, for example, a quadratic curve. But for now, we’ll just say that if the points can’t be modelled by a straight line, then there’s no linear correlation. It means there’s no tendency for one to increase as the other decreases, or vice versa.
So, now, let’s consider the specific example in this question. Which is the type of linear correlation that might exist between the mass loaded onto a spring and the amount that the spring extends? Well, let’s think about this from a practical point of view. We would expect that as we load a greater mass onto our spring, then its extension would increase because there is a greater force pulling down on the spring causing it to extend.
So, we would expect that as the mass loaded onto the spring increases, its extension will also increase. And if we look back at our definitions of positive, negative, and no correlation, we see that this is consistent with what happens when there is positive correlation between the two variables.
This may only be true up to a certain point. Presumably, we’ll reach a point where the spring is at its maximum extension. And loading further mass onto it won’t cause any further increase. In fact, it may even cause the spring to snap. At least initially though, we can say that we expect there to be a positive linear correlation between the mass loaded onto a spring and its extension.