Explainer: Quartiles of a Data Set

In this explainer, we will learn how to find the median and upper and lower quartiles of a data set.

Our goal in organizing and analyzing a data set is to gain information from it, and there are a number of “summary statistics” that can help us with this. For example, the mean, median, and mode of a data set give us an indication of where the center of the data or the most frequently occurring values lie. And the range, interquartile range, variance, and standard deviation are all measures of how the data spread out from, or cluster around, the central values.

We can also split the data into quarters, so that 25% of the data values lie within a particular quarter.

The picture below represents a data set with 16 data values.

Q1 The first (or lower) quartile (Q1) marks the center of the lowest half of a data set. So, 25% of the data sit below Q1 and 75% of the data sit above Q1.

Q2 The second quartile (Q2), which is the median, marks the middle of a data set. So, 50% of the data set is below the median and 50% is above the median.

Q3 The third (or upper) quartile (Q3) marks the center of the top half of a data set. So, 75% of the data set is below Q3 and 25% is above it.

Let us see how this all works with an example.

Example 1: The Median and Quartiles of a Data Set

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.

  1. Find the median (Q2) and the lower and upper quartiles (Q1 and Q3) for the number of Bonus Bugs won.
  2. 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 to the next level?

Answer

Part 1

To find the median and the quartiles, Q1 and Q3, we first need to order the data from smallest to largest. The least number of bugs won was 14, so this is our first item in an ordered list. The second lowest number of Bonus Bugs won was 15, and the next highest was 16, and so on, until the highest number of Bonus Bugs won was 35. Our ordered list then looks like this: 141516171820212223252729313235.

We can see how the number of Bonus Bugs won is spread out on the number line by putting the ordered data into a line plot. (Each bug in the plot below represents one student.)