Median Notes

Published on: September 29, 2026

This Median Calculator finds the middle of a list of numbers and shows the working line by line. The values are sorted first, because the middle only means something once they are in order. An odd count leaves one value in the middle; an even count leaves two, which are added and halved. Unlike the mean, a single extreme value barely moves it. Following the same pattern every time is what makes the median easy to check and repeat.

Step-by-step method

  1. Set up the problem.
  2. Sort the values from smallest to largest.
  3. Find the middle position or positions.
  4. Add the values at those positions.
  5. Divide by how many middle values there were.
See Example 1 Hide Example 1

Problem

\(5,\; 8,\; 12,\; 15\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Sort the values from smallest to largest.
  3. Find the middle position or positions.
  4. Add the values at those positions.
  5. Divide by how many middle values there were.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the middle of the values \(5,\; 8,\; 12,\; 15\).

\(5,\; 8,\; 12,\; 15\)

Step 2

Step 2 - Sort the values from smallest to largest.

In this problem: Putting the values in order from smallest to largest gives \(5,\; 8,\; 12,\; 15\).

\(5,\; 8,\; 12,\; 15\)

Step 3

Step 3 - Find the middle position or positions.

In this problem: There are \(4\) values, which is an even count, so no single value sits in the middle. The two closest to it are at positions \(2\) and \(3\).

\(\text{middle positions} = 2,\; 3\)

Step 4

Step 4 - Add the values at those positions.

In this problem: Adding the two middle values gives \(8 + 12 = 20\).

\(\text{Median} = \frac{8 + 12}{2}\)

Step 5

Step 5 - Divide by how many middle values there were.

In this problem: There were two middle values, so the total is halved: \(20 \div 2 = 10\).

\(\text{Median} = \frac{20}{2} = 10\)

Final Answer

\(\text{Median} = 10\)
See Example 2 Hide Example 2

Problem

\(6,\; 6,\; 7,\; 10\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Sort the values from smallest to largest.
  3. Find the middle position or positions.
  4. Add the values at those positions.
  5. Divide by how many middle values there were.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the middle of the values \(6,\; 6,\; 7,\; 10\).

\(6,\; 6,\; 7,\; 10\)

Step 2

Step 2 - Sort the values from smallest to largest.

In this problem: Putting the values in order from smallest to largest gives \(6,\; 6,\; 7,\; 10\).

\(6,\; 6,\; 7,\; 10\)

Step 3

Step 3 - Find the middle position or positions.

In this problem: There are \(4\) values, which is an even count, so no single value sits in the middle. The two closest to it are at positions \(2\) and \(3\).

\(\text{middle positions} = 2,\; 3\)

Step 4

Step 4 - Add the values at those positions.

In this problem: Adding the two middle values gives \(6 + 7 = 13\).

\(\text{Median} = \frac{6 + 7}{2}\)

Step 5

Step 5 - Divide by how many middle values there were.

In this problem: There were two middle values, so the total is halved: \(13 \div 2 = 6.5\).

\(\text{Median} = \frac{13}{2} = 6.5\)

Final Answer

\(\text{Median} = 6.5\)
See Example 3 Hide Example 3

Problem

\(4,\; 9,\; 11,\; 16\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Sort the values from smallest to largest.
  3. Find the middle position or positions.
  4. Add the values at those positions.
  5. Divide by how many middle values there were.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the middle of the values \(4,\; 9,\; 11,\; 16\).

\(4,\; 9,\; 11,\; 16\)

Step 2

Step 2 - Sort the values from smallest to largest.

In this problem: Putting the values in order from smallest to largest gives \(4,\; 9,\; 11,\; 16\).

\(4,\; 9,\; 11,\; 16\)

Step 3

Step 3 - Find the middle position or positions.

In this problem: There are \(4\) values, which is an even count, so no single value sits in the middle. The two closest to it are at positions \(2\) and \(3\).

\(\text{middle positions} = 2,\; 3\)

Step 4

Step 4 - Add the values at those positions.

In this problem: Adding the two middle values gives \(9 + 11 = 20\).

\(\text{Median} = \frac{9 + 11}{2}\)

Step 5

Step 5 - Divide by how many middle values there were.

In this problem: There were two middle values, so the total is halved: \(20 \div 2 = 10\).

\(\text{Median} = \frac{20}{2} = 10\)

Final Answer

\(\text{Median} = 10\)
See Example 4 Hide Example 4

Problem

\(13,\; 2,\; 21,\; 8,\; 17\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Sort the values from smallest to largest.
  3. Find the middle position or positions.
  4. Add the values at those positions.
  5. Divide by how many middle values there were.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the middle of the values \(13,\; 2,\; 21,\; 8,\; 17\).

\(13,\; 2,\; 21,\; 8,\; 17\)

Step 2

Step 2 - Sort the values from smallest to largest.

In this problem: Putting the values in order from smallest to largest gives \(2,\; 8,\; 13,\; 17,\; 21\).

\(2,\; 8,\; 13,\; 17,\; 21\)

Step 3

Step 3 - Find the middle position or positions.

In this problem: There are \(5\) values, which is an odd count, so one value sits in the middle at position \(\frac{5 + 1}{2} = 3\).

\(\text{middle position} = \frac{5 + 1}{2} = 3\)

Step 4

Step 4 - Add the values at those positions.

In this problem: Only \(13\) is in the middle, so there is nothing to add to it and the total stays \(13\).

\(\text{Median} = \frac{13}{1}\)

Step 5

Step 5 - Divide by how many middle values there were.

In this problem: There was one middle value, so the total is divided by \(1\), and \(13 \div 1 = 13\) leaves it unchanged.

\(\text{Median} = \frac{13}{1} = 13\)

Final Answer

\(\text{Median} = 13\)