The number of Bonus Bugs won by
each of 15 students in the first level of a computer game tournament was
recorded. The results are in the table
below. Find the median, Q two, and the
lower and upper quartiles, Q one and Q three, for the number of Bonus Bugs won. If the organizers of the tournament
decide that the top 25 percent of students can compete in level two, above what
number of Bonus Bugs must a student win to go on to the next level?
In order to calculate the median
and quartiles of any data set, we firstly need to sort the data into ascending
order. The lowest number of Bonus Bugs
that a student won was 14. The next lowest was 15. The completed list in ascending
order is as shown. Once our data is in order, we can
calculate the median by crossing off one number from either end until we reach the
middle. We would cross off 14 and 35. We would then cross off 15 and 32
and repeat this process until we arrived at the middle. If there were two middle numbers,
we would find the midpoint of these two.
When dealing with a large data set,
there is a quicker way of finding the median position. We do this using the formula 𝑛
plus one divided by two, where 𝑛 is the number of data values. In this question, there are 15 data
values. We add one to 15 and then divide by
two. This is equal to eight. Therefore, the median will be the
eighth number in our list. This is equal to 22. So the median number of Bonus Bugs
We can work out the lower quartile
and upper quartile positions in a similar way. The lower quartile or Q one
position is calculated by dividing 𝑛 plus one by four. 15 plus one is equal to 16, and
dividing this by four gives us four. We can therefore say that the
fourth number in our list, in this case 17, is the lower quartile.
An alternative way to find the
lower quartile would be to find the center of the bottom half of our list. There are seven values below the
median, and the middle one of these is 17, the fourth value. To find the upper quartile or Q
three position, we multiply 𝑛 plus one by three-quarters or multiply 𝑛 plus one by
three and then divide by four. This is equal to 12. Notice that this is three times the
Q one position. The 12th number in our list is 29,
so this is the upper quartile.
As the upper quartile is the center
of the top half of data values, we could once again have found this by finding the
middle of the seven values above the median. The median number of Bonus Bugs is
22, the lower quartile is 17, and the upper quartile is 29.
We will now clear some space to
work out the second part of the question.
The second part of the question was
interested in the top 25 percent of students. We recall that one of the reasons
for calculating the quartiles is to split our data into quarters. One-quarter is the same as 25
percent. This means that the top 25 percent
of students will lie between Q three and the maximum inclusive. As Q three or the upper quartile
was equal to 29, any student will be in the top 25 percent if they achieve 29 Bonus
Bugs or more.