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
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
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.