Question Video: The Median and Quartiles of a Data Set Mathematics

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 (Q2) and the lower and upper quartiles (Q1 and Q3) for the number of Bonus Bugs won. If the organizers of the tournament decide that the top 25% of students can compete in level 2, above what number of Bonus Bugs must a student win to go on to the next level?

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Video Transcript

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 is 22.

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.

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