The scatter graph shows the weights of cars and the average miles per gallon they achieve. Part a) What type of correlation is there between weight and average miles per gallon? Circle your answer: is it strong negative, weak negative, strong positive or weak positive?
When two variables have a correlation, that means there’s a relationship between them. And there are three types of correlation we look for. Two variables are said to be positively correlated if as one increases so does the other. They have a negative correlation if as one increases the other decreases. And we say that there’s no correlation if there’s no real pattern between the points. We can see by looking at the pattern of the point on this graph that as the weight of the car increases the miles per gallon decrease.
This means there is a negative correlation. And if the points have a strong relationship or a strong correlation, they’ll lie close to the line of best fit. Let’s draw this in and see what it looks like. Remember the line of best fit is a straight line that passes as centrally as possible through the coordinates. It should follow the same slope as the points, and it will look a little bit like this. Notice how the coordinates here are very close to the line. So this means that they have a strong negative correlation.
Part b) By drawing a line of best fit, estimate the average miles per gallon of a car that weighs 2.5 tonnes.
We’ve already included our line of best fit. In fact, we should try to draw one whenever we have a scatter graph question as we can’t interpret or read information from the graph without one. Before we can read off the graph, let’s check the scale. On the vertical scale, we can see that five small squares represents five miles per gallon. This means that one small square must represent one mile per gallon. The horizontal axis isn’t so straightforward. Four small squares here represent one tonne.
Since we’re trying to find the average miles per gallon of a car that weighs 2.5 tonnes, we need to halve these to get the value for 0.5 tonnes. That gives us that two small squares is equal to 0.5 tonnes. So let’s find that on our horizontal axis. 2.5 is two small squares right of the number two. It’s actually halfway between two and three as we’d expect. To estimate we draw a vertical line until we hit our line of best fit. We then draw a horizontal line until we hit the 𝑦-axis. Here that tells us that the average miles per gallon of a car that weighs 2.5 tonnes is 25 miles per gallon.
Remember the line of best fit is drawn by eye, so there will be a little bit of leeway on this. In the mark scheme, the line of best fit is ever so slightly different and that gives us a value of 25.5 miles per gallon.
Part c) Thomas says I can use the line of best fit to find out the weight of a car which will do zero miles to the gallon. Comment on his statement.
Well there’s actually several things wrong with his statement. Firstly, when reading information from the line of best fit, we are only ever getting an estimate. Secondly, it doesn’t really make a lot of sense for a car to do zero miles to the gallon. That would mean for every gallon you fill the car with, it would travel zero miles which just doesn’t make sense.
What’s most important though is considering the data that we’ve been given. We have information about cars that weigh between one and five tonnes. We can estimate based on values within this dataset. That’s called interpolation. When we extend a pattern, which we’d need to do here, and then try to make estimations, that’s called extrapolation. It’s not accurate to go outside of our dataset though. So an estimate for the weight of a car which would do zero miles to the gallon would likely be inaccurate.