When you're given a set of numbers, what kind of metrics or measurements can you use to learn more about the data set? One simple yet important idea is breaking the set into **quartiles** or roughly breaking it into fourths and examining what the breakdown tells us about the numbers in the set.

The **first quartile**, often written **q1**, is the median of the lower half of the set (the numbers must be listed in increasing order). About 25 percent of the numbers will be smaller than the first quartile while about 75 percent will be larger.

#### TL;DR (Too Long; Didn't Read)

The **first quartile** is the median of the lower half of the set when the numbers are listed in increasing order.

## How to Find the First Quartile

To find the first quartile, first put the numbers in the set in order.

Say you're given a set of numbers: {1, 2, 15, 8, 5, 9, 12, 42, 25, 16, 20, 23, 32, 28, 36}.

Rewrite the numbers in increasing order, like this: {1, 2, 5, 8, 9, 12, 15, 16, 20, 23, 25, 28, 32, 36, 42}.

Next, find the **median**. The median is the middle number in the set when the numbers are listed in order. We have 15 numbers in our set, so the middle number is going to be in the 8th spot: There will be 7 numbers on either side of it.

The median for our set is 16. Sixteen is the "half-way" mark. Any number smaller than 16 is in the "lower half" of the set, and all the numbers bigger than 16 are in the "upper half" of the set.

Now that we've split our set in half, let's look at the lower half. We have 1, 2, 5, 8, 9, 12, and 15 in the lower half of our set. The **first quartile** is going to be the median of these numbers. In this case, the median is 8, since it's the middle number with three numbers on either side of it. So our q1 is 8.

Keep in mind that if we had an even number of numbers, there wouldn't be an obvious "middle," or median. In that case, we would take the middle two numbers and find the average of them (add them together and divide by two).

To find the third quartile, we'll do the same thing to the upper half of the set. The **third quartile**, often written **q3**, is the median of the upper half of the set.

The upper half of our set is all the numbers after 16, so: {20, 23, 25, 28, 32, 26, 42}.

The median of these is 28, so 28 is called the third quartile, or q3. It's approximately the 75 percent mark in the set: It's larger than about 75 percent of the numbers in the set but smaller than the final 25 percent.

## Quartile Calculator

This website has a useful quartile calculator. If you enter the numbers in your set, it will tell you the first quartile, median and third quartile.

## Interquartile Range

The **interquartile range** is the difference between the first quartile and the third quartile; that is, q3 - q1.

In our example set, the interquartile range is 28 - 16, which equals 12.

The interquartile range is useful for finding out the "spread" of most of numbers in the set. Are the middle ones mostly clustered together, or is everything very spread out? The interquartile range allows us to look at what most of the numbers in the set are doing, without getting skewed by outliers at the far end of the set. In that sense, it can be more useful than the **range**, which is the highest number minus the lowest number.

## Box and Whiskers

On a box and whiskers plot, the box starts at q1 and ends at q3. The "whiskers" go from either side of the box all the way to the highest and lowest numbers. But our first quartile and the interquartile range are the stars of the show.

References

About the Author

Elise Hansen is a journalist and writer with a special interest in math and science.