In these lessons, we will learn how to find the median, quartiles and percentiles of ungrouped data (discrete data).

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Frequency Table

Cumulative Frequency Graph

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We have learned that the median is the middle value when a set of data is arranged in order of increasing magnitude. We will now consider lower quartiles and upper quartiles.

The **median** divides the data into a lower half and an upper half.

The **lower quartile** is the middle value of the lower half.

The **upper quartile **is the middle value of the upper half.

The following figure shows the median, quartiles and interquartile range. Scroll down the page for examples and solutions.

**Example:**

Find the median, lower quartile and upper quartile of the following numbers.

12, 5, 22, 30, 7, 36, 14, 42, 15, 53, 25

**Solution:**

First, arrange the data in ascending order:

Median (middle value) = 22

Lower quartile (middle value of the lower half) = 12

Upper quartile (middle value of the upper half) = 36

If there is an even number of data items, then we need to get the average of the middle numbers.

**Example:**

Find the median, lower quartile, upper quartile, interquartile range and range of the following numbers.

12, 5, 22, 30, 7, 36, 14, 42, 15, 53, 25, 65

**Solution:**

First, arrange the data in ascending order:

Lower quartile or first quartile =

Median or second quartile =

Upper quartile or third quartile =

Interquartile range = Upper quartile – lower quartile

= 39 – 13 = 26

Range = largest value – smallest value

= 65 – 5 = 60

When evaluating the quartiles, always remember to first arrange the data in increasing order.

**How to compute the interquartile range for a set of data?**

Remember to reorganize the data so that you can find the median values easier.

**Median, Quartiles and Interquartile Range**

How to calculate median, quartiles and interquartile range for discrete data.

**How to summarize data using quartiles and interquartile range?**

**How to Estimate Mean, Median and Quartiles from Continuous Grouped Data?**

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