In the second year of a computer
game tournament, there were 42 participants and the number of Bonus Bugs each one
won in level one was recorded. The data is shown in the graph
below where each bug represents one participant. Find the median number of Bonus
Bugs won and the lower and upper quartiles, Q one and Q three.
There is also a second part to this
question that we will look at later. We can see from the graph that
there was one student who achieved 13 Bonus Bugs. There was also one student who
achieved 15 Bonus Bugs. Two students achieved 19 Bonus
Bugs, and two students achieved 20. The maximum number of Bonus Bugs
achieved by any student was 38.
In order to calculate the median
and quartiles, we could write all of these numbers out in ascending order. 13, 15, 19, 19, 20, 20, and so
on. This would be very
time-consuming. So a quicker method is to work out
which position the median and quartiles would be in. The median position can be
calculated using the formula 𝑛 plus one divided by two. 𝑛 is the number of data values, in
this case 42. Substituting this into the formula
gives us an answer of 21.5. This means that the median position
is between the 21st and 22nd number.
By calculating the running total or
cumulative frequency, we can see that the 19th, 20th, 21st, and 22nd number are all
26. This means that the median number
of bugs is 26. We can calculate the Q one or lower
quartile position using a similar method. This time, we divide 𝑛 plus one by
four, giving us an answer of 10.75. As this is past halfway between 10
and 11, we round up to the 11th number. The 11th and 12th numbers are equal
to 23. Therefore, Q one equals 23.
To calculate the Q three or upper
quartile position, we multiply the lower quartile position by three. This gives us 32.25. As this is less than halfway
between 32 and 33, we round down. We’re looking for the 32nd
number. This is equal to 29.
The second part of the question
wants us to calculate what score the top 25 percent of participants achieved. The quartiles split our data into
quarters or 25 percent. This means that 25 percent of the
scores will go from the upper quartile to the maximum. A score of 29 or more would put a
student in the top 25 percent.