Lower Quartile and Upper Quartile

The quartiles split the set of data at three points: the lower quartile (Q1), the median (Q2) and the upper quartile (Q3).

These quartiles together with the minimum and maximum values split the data set into 4 equal groups:

1. from the minimum data point to the lower quartile (Q1)
2. from Q1 up to the median (Q2)
3. from the median (Q2) to the upper quartile (Q3)
4. from Q3 to the maximum data point.

The lower and upper quartiles are the first and third quarters of the data points.

In effect, they are the middle values of the first half and second half of the data.

Lower Quartile (1st Quartile) also called Q1

You can find the lower quartile (1st quartile) by putting the set of data in order, then finding the median value.

Once you have found the median value, you need to find the middle value between the minimum and the median value.

About ¼ (25%) of the data values are lower than the 1st quartile and ¾ (75%) of the data values are higher than the 1st quartile.

Upper Quartile (3rd Quartile) also called Q3

You can find the upper quartile (3rd quartile) by putting the set of data in order, then finding the median value.

Once you have found the median value, you need to find the middle value between the median value and the maximum.

About ¾ (75%) of the data values are lower than the 3rd quartile and ¼ (25%) of the data values are higher than the 3rd quartile.

Note that we have excluded the median value when finding the lower and upper quartiles.

There are two methods for finding the lower and upper quartile, one which excludes the median (which we are using) and one which includes the median.

There is a quick formula to help you find the correct terms in the set for the lower and upper quartiles.

• Count the number of data points in the set - call this number N.
• Next, arrange all the data points in order from smallest to largest.
• The lower quartile (Q1) is one-quarter of the way along the number of data points + 1.
• \[ Q1 = {1 \over 4} (N + 1) th \; term \]
• The median (Q2) is halfway (or two-quarters) along the number of data points + 1.
• \[ Q2 = {2 \over 4 } (N + 1) th \; term\]
• The upper quartile (Q3) is three-quarters of the way along the number of data points + 1.
• \[ Q3 = {3 \over 4} (N + 1) th \; term\]

The median is the midpoint (or middle value) of a set of numbers.

It is found by ordering the set of numbers and then finding the middle value in the set.

The median can also be thought of as the 2nd quartile or Q2.

About ½ (50%) of the data values are lower than the median and ½ (50%) of the data values are higher than the median.

The interquartile range is the difference between the Upper Quartile (Q3) and Lower Quartile (Q1).

So Interquartile range = Q3 - Q1.

In the example above, the interquartile range is 58 - 31 = 27.

Example 1) Find the median and interquartile range of these temperatures taken across the UK in March:

Step 1) Put the data in order.

Step 2) Identify the median (Q2). There are 11 data points, so the median is the 6th point which is -1°C.

Step 3) There are now 5 data points on each side of the median point.

• The lower quartile will be the 3rd point which is -2°C.
• The upper quartile will be the 9th point which is 3°C.

Step 4) To find the interquartile range, we subtract the lower quartile from the upper quartile.

• Interquartile range = 3°C - (-2°C) = 5°C.

Example 2) Find the median and interquartile range of these History test scores:

Step 1) Put the data in order.

Step 2) Identify the median (Q2). There are 13 data points, so the median is the 7th point which is 30 marks.

Step 3) There are now 6 data points on each side of the median point.

• The lower quartile will be halfway between the 3rd and 4th point = (24 + 26) ÷ 2 = 25.
• The upper quartile will be halfway between the 9th and 10th point = (32 + 35) ÷ 2 = 33 ½.

Step 4) To find the interquartile range, we subtract the lower quartile from the upper quartile.

• Interquartile range = 33 ½ - 25 = 8 ½

Example 3) Sally does a traffic survey in the street outside her house at different times of the day. Find the lower quartile and upper quartile of her survey, and also the interquartile range.

Step 1) Put the data in order.

Step 2) Identify the median (Q2). There are 10 data points, so the median is halfway between the 5th and 6th point which is (42 + 47) ÷ 2 = 44 ½.

Step 3) There are now 5 data points on each side of the median point.

• The lower quartile will be the 3rd point which is 35 cars.
• The upper quartile will be the 8th point which is 62 cars.

Step 4) To find the interquartile range, we subtract the lower quartile from the upper quartile.

• Interquartile range = 62 - 35 = 27 cars

We have created 2 worksheets to help you practice this skill.

The first worksheet is easier and includes sets of data with up to 9 terms.

The second worksheet is harder and has data with up to 13 terms.

Find links to our Median worksheets below.

Using this webapge will help you to:

• find the median of a set of data;
• find the median of both odd and even numbers of data points;
• show you worked examples of how to find the median.

The sheets in this section will help you to find the mean of a range of numbers, including negative numbers and decimals.

There are a range of sheets involving finding the mean, and also finding a missing data point when the mean is given.

The sheets in this section will help you to find the mode and range of a set of numbers, including negative numbers and decimals.

There are easier sheets involving fewer data points, and harder ones with more data points.

The sheets in this section will help you to find the mean, median, mode and range of a set of numbers, including negative numbers and decimals.

There are easier sheets involving fewer data points, and harder ones with more data points.

We also have a selection of box plot worksheets.

The worksheets will help you to practice creating and interpreting box plots.

The link below will open the quiz in a dedicated quiz page or you can preview a copy of the quiz underneath.

The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

If you have any questions or need any information about our site, please get in touch with us using the 'Contact Us' tab at the top and bottom of every page.