# Video: Pack 2 • Paper 1 • Question 12

Pack 2 • Paper 1 • Question 12

07:10

### Video Transcript

Roger, who is nine years old, wrote down the ages of his 11 cousins. They were four, six, eight, nine, 13, 17, 17, 21, 22, 22, and 23. Part a) Draw a box plot for this information.

Roger’s friend Brendan, who is the same age, also has 11 cousins. He worked out the following information based on the ages of his cousins. The median age is 13. The interquartile range of ages is five. The range of ages is 18. Part b) Is there more variability in the ages of Roger’s or Brendan’s cousins?

In order to draw a box plot, we need to work out five bits of information: the lowest value, the lower quartile, the median, the upper quartile, and the highest value. As the ages are already in order, the lowest value is the first value and the highest value is the last value. In this case, the lowest age of one of Roger’s cousins is four and the highest age is 23.

The median is the middle value. The position of the median can be calculated using the formula 𝑛 plus one divided by two. As Roger had 11 cousins, the value for 𝑛 in this example is 11. 11 plus one is equal to 12 and 12 divided by two is equal to six. This means that the median will be the sixth number in our list. In this question, the median age of Roger’s cousins is 17.

We can work out the position of the lower quartile in the list by adding one to 𝑛 and dividing by four. This is because the lower quartile is the 25th percentile. 11 plus one is equal to 12 and 12 divided by four is equal to three. This means that the lower quartile is the third number in our list. As the third number is equal to eight, the lower quartile of ages equals eight.

Finally, we can find the position of the upper quartile by multiplying the position of the lower quartile by three. This is because the upper quartile is the 75th percentile and 75 percent is equal to three-quarters. Three multiplied by 12 over four or three multiplied by three is equal to nine. This means that the upper quartile is the ninth age in our list from left to right. The upper quartile of the ages of Roger’s cousins is 22.

The lowest age of Roger’s cousins was four and the highest age was 23. Therefore, when we draw a box plot, we need to ensure that our scale goes from below four to above 23. In this case, we’ve drawn it from zero to 24. It is also important that the rest of the numbers are to scale. Usually in the exam, you’ll be given graph paper to draw your box plot. In this case, we have used equal spacing, going up in fours: zero, four, eight, 12, 16, 20, and 24.

Our next step is to draw two vertical lines at the lowest value, in this case four, and the highest value, 23. Next, we can draw another two vertical lines at the lower quartile, in this case eight, and the upper quartile, which is equal to 22. Our fifth vertical line is that the median, which in this case was 17. We then need to draw two horizontal lines from the lower quartile to the upper quartile to create the box. This box shows the interquartile range from the upper quartile to the lower quartile.

Finally, we complete the box plot with two horizontal lines: one from the lower quartile to the lowest value and the other from the upper quartile to the highest value. This is now our completed box plot.

Before we move on to part b, we need to calculate the range and the interquartile range for Roger’s cousins. The range is calculated by subtracting the lowest value from the highest value. It is the distance from one end of the box plot to the other. In this case, we need to subtract four from 23. 23 minus four is equal to 19. Therefore, the range of ages of Roger’s cousins is equal to 19.

The interquartile range is calculated by subtracting the lower quartile from the upper quartile, as shown on the box plot. The lower quartile was eight and the upper quartile was 22. Therefore, we need to subtract eight from 22. This gives us an interquartile range of 14.

Part b of the question asked us to compare the variability in the ages of Roger and Brendan’s cousins, given that Brendan’s cousins had a median age of 13. They had an interquartile range of five and they had a range of 18. The variability means the spread of the ages. This means that in this question we can ignore the median age as this has no impact on the spread.

We need to compare the range and interquartile range of Roger and Brendan. The range in ages was pretty similar. Roger’s cousins had a range of 19 years and Brendan’s cousins had a range of 18 years. Whilst the range of Roger’s cousins is greater, this is not enough to make a conclusion. We also need to look at the interquartile range, where in Roger’s case, it was equal to 14 and in Brendan’s case, it was equal to five.

This means that the middle 50 percent from the lower quartile to the upper quartile will more spread out for Roger than they were for Brendan. We can, therefore, conclude that there is a greater variability in the ages of Roger’s cousins than Brendan’s as the ages of Roger’s cousins have a greater range and interquartile range.