Video: AQA GCSE Mathematics Higher Tier Pack 2 • Paper 2 • Question 18

Data was collected on the total monthly rainfall in the UK (in millimetres) in the year 2000. 50% of the months had between 73 mm and 147 mm of rain. The lower quartile is 73 mm. 50% of the months had more than 120 mm of rain. The month with the most rain had 194 mm of rain. The month with the least rain had 57 mm of rain. (a) Use this information to draw a box plot. (b) A prediction about the total monthly rainfall in the UK for the year 2100 is given in the table. Make two comments about the predicted changes to the total monthly rainfall in the year 2100 compared with the year 2000.

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Video Transcript

Data was collected on the total monthly rainfall in the UK in millimetres in the year 2000. 50 percent of the months had between 73 millimetres and 147 millimetres of rain. The lower quartile is 73 millimetres. 50 percent of the months had more than 120 millimetres of rain. The month with the most rain had 194 millimetres of rain. The month with the least rain had 57 millimetres of rain. Part (a) is to use this information to draw a box plot.

So what is a box plot? Well, it’s something that looks a bit like this. Notice that it has five vertical lines, which line up with a number line below the box plot.

The vertical lines at either end of the box plot corresponds to the minimum and maximum values of the data. And the middle vertical line corresponds to the median of the data. That’s a value that exactly half of the data lies below and half of the data lies above.

The remaining two vertical lines correspond to the lower and upper quartiles. Just as the median lies in the middle of the data separating it into two halves, the quartiles separate the data further into quarters, hence their name.

So 25 percent of the data lies between the minimum and the lower quartile. 25 percent of the data lies between the lower quartile and the median. 25 percent of the data lies between the median and the upper quartile. And 25 percent of the data lies between the upper quartile and the maximum value of the data.

Okay, so our task is to draw a box plot for the data summarised in the question. We, therefore, need to find the minimum lower quartile, median, upper quartile, and maximum of the data. So let’s go through the question sentence by sentence.

The first sentence just tells us what the data is about. The second sentence tells us that 50 percent of the months had between 73 millimetres and 147 millimetres of rain. On its own, this statement doesn’t help us much because we don’t know whether this 50 percent is the most rainy or least rainy 50 percent of the data or somewhere in between.

But looking at the next sentence, we can see that the lower quartile is 73 millimetres. The lower quartile is one of the values that we need for our box plot. So let’s to draw a vertical line on our diagram at 73 millimetres.

We first have to identify 73 millimetres on our number line. Obviously, it is going to be between 70 and 80 millimetres. But we have to be slightly careful here. There are five squares between 70 and 80 millimetres. And so, each square represents two millimetres.

If we counted three squares along from 70, we won’t be at 73, we would be at 76. 73 actually lies one and a half squares along from 70. And so, we should draw our vertical line to represent the lower quartile here.

So we’ve represented the lower quartile of 73 millimetres on the diagram. And we can look at the previous sentence again. 50 percent of the months had between 73 millimetres and 147 millimetres of rain.

Now, from the next sentence, we know that 73 millimetres is the lower quartile of the data. Looking at our sketch of the box plot, we can see that 50 percent of the data lies between the lower quartile and the upper quartile.

And so, as 73 millimetres is our lower quartile, 147 millimetres must be our upper quartile. We, therefore, need another vertical line at 147 millimetres on our diagram. And counting the squares carefully, we put it here three and half squares along from 140.

And now, we’ve successfully used the information in this sentence. 50 percent of the months had more than 120 millimetres of rain. This is half of the months. Half of them had more than 120 millimetres of rain.

And so, presumably, half of them will have less than or perhaps equal to 120 millimetres of rain. This is our median value. So we draw another line at 120 millimetres to represent our median. So that sentence is done.

And we move on to the last two sentences, which tell us that the month with the most rain had 194 millimetres of rain. So that’s the maximum value in our dataset. And the month with the least rain had 57 millimetres of rain. That’s the minimum value in our dataset.

So we draw our maximum line at 194 millimetres, which is two squares along from 190. And we draw our minimum line at 57 millimetres, which is three and a half squares along from 50.

Now, we’ve got all five vertical lines that we need. Now, we just need to draw the horizontal lines as well. And there, we have it. We can use this box plot to help us answer part b of the question. But first we need to clear some room.

Part (b) says, “A prediction about the total monthly rainfall in the UK for the year 2100 is given in the table.”

So the table tells us that the lower quartile is predicted to be 84 millimetres. The median is predicted to be 130 millimetres. And the upper quartile is predicted to be 171 millimetres. We’re asked to make two comments about the predicted changes to the total monthly rainfall in the year 2100 compared with the year 2000.

Now, any two sensible comments comparing the datasets would be acceptable. But generally, when comparing two datasets, we like to focus on two things: the averages of the datasets and their spreads.

So does this new dataset have a larger or smaller average? So in our context, we want to know whether the average rainfall in 2100 is predicted to be greater than or less than it was in 2000. And is this new data more spread out or less spread out?

So is it predicted to be more varied rainfall, lots of rain in some months and much less in other months? Or is rainfall predicted to be much more regular in the future with very similar amounts of rainfall in each month? So how can we compare these two datasets to tell whether the average rainfall goes up or down?

Well, we use their medians. We were told in the question and we drew on our box plot that the median rainfall in the year 2000 was 120 millimetres. And we’re told that the median rainfall in the year 2100 is predicted to be 130 millimetres.

Our first comment is that median monthly rainfall is predicted to be 10 millimetres greater in the year 2100 than it was in the year 2000. So average rainfall is predicted to go up. So that was our comment about the averages of the datasets.

Now, how about their spreads? Now, one measure of spread is the interquartile range or IQR. This is the difference between the upper quartile and the lower quartile.

So in the year 2100, the interquartile range is predicted to be 171 minus 84 millimetres. And performing this subtraction, we get an interquartile range of 87 millimetres in 2100, whereas in the year 2000, our interquartile range was 147, that’s the upper quartile, minus 73, that’s lower quartile in millimetres.

And performing this subtraction, we get an interquartile range of 74 millimetres. The interquartile range in the year 2100 is predicted to be greater than it was in the year 2000.

And as interquartile range is a measure of spread of data, a greater interquartile range means that the rainfall is predicted to be more varied.

So these two comments comparing the averages and spread of the datasets would do for part b. So together with the boxplot we drew in part a, they form the answer to our question.

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