# Median - GMAT Math Study Guide

## Definitions

• Median - The number that is located in the middle of a set of numbers when that set is ordered sequentially from the smallest to the largest.
For example, the median of the set 2, 4, 10 is 4 and the median of the set -200, 2, 4, 100, 150 is also 4.

## Calculating the Median

1. Arrange the numbers in order from least to greatest.
2. With an odd number of numbers, the median is the middle number.
3. With an even number of numbers, the median is the average of the two middle numbers.

### Common Trap: Not Arranging From Least to Greatest

Order numbers from smallest to largest before finding the median.

Perhaps the easiest mistake one can make when calculating the median of a list of numbers is forgetting to order the numbers from smallest to largest. When this mistake is made, the value computed as the median is not correct. Consider the following example:

4, 1, 5, 3, 7, 6, 10

Without ordering the numbers, the median is 3
However, the numbers must first be ordered:
1, 3, 4, 5, 6, 7, 10

The correct median is: 5

### Basic Examples

There are two types of basic median problems: those with an even number of items and those with an odd number of items. As noted previously, when calculating the median of a list with an even number of items, take the average of the two numbers located in the middle (after ordering sequentially from smallest to largest). With an odd number of items in the list, the median is simply the middle number in the list.

4, 2, 6, 5
Order sequentially:
2, 4, 5, 6
With an even number of items in the list (i.e., with 4 numbers), the median is the average of the two middle numbers:4.5

Another Example:

What is the median of the set that consists of the numbers 5, 2, 8, 9, 2?
Order sequentially:
2, 2, 5, 8, 9
Median = 5

### More Complex Examples

If x < 5, what is the median of the set consisting of the following numbers:
6, 5, 6, 7, 9, 10, 2, x

Order the items sequentially:
2, x, 5, 6, 6, 7, 9, 10
You do not know whether x > 2 or x < 2. However, it does not matter.
With an even number of items in the list, the median is the average of the two middle numbers: 6

Another Example:

If w < y, x < z, and y < x, what is the median of the set y, x, z, x, w?
Order the items sequentially:
w, y, x, x, z
With an odd number of items, the median is the middle number: x