Median (Statistics) — Definition, Formula & Examples

The median is the middle value when you arrange a set of numbers from least to greatest. If there are two middle values (an even number of data points), the median is the average of those two values.

For an ordered data set of $n$ observations, the median is the value at position $\frac{n+1}{2}$ when $n$ is odd. When $n$ is even, the median is the arithmetic mean of the values at positions $\frac{n}{2}$ and $\frac{n}{2}+1$ .

Key Formula

\text{Median position} = \frac{n + 1}{2}

Where:

$n$ = Total number of data values in the set

How It Works

First, sort all the values from smallest to largest. Then count how many values you have. If the count is odd, the median is the single middle number. If the count is even, find the two middle numbers and average them. Unlike the mean, the median is resistant to outliers — one extremely large or small value barely shifts it.

Worked Example

Problem: Find the median of this data set: 12, 7, 3, 15, 9, 21, 5.

Order the data: Arrange the values from least to greatest.

3, \; 5, \; 7, \; 9, \; 12, \; 15, \; 21

Find the middle position: There are 7 values, so the median is at position (7 + 1) ÷ 2 = 4.

\frac{7+1}{2} = 4

Identify the median: The 4th value in the ordered list is 9.

\text{Median} = 9

Answer: The median is 9.

Why It Matters

When reporting typical home prices or household incomes, analysts use the median because a few extremely high values would inflate the mean. The median gives a more honest picture of what is typical in skewed distributions. You will rely on it heavily in statistics courses and in any field that works with real-world data.

Common Mistakes

Mistake: Finding the middle value without sorting the data first.

Correction: Always arrange the numbers in order from least to greatest before locating the middle. Picking the middle value from an unsorted list gives the wrong answer.