How to Find the Median
The median is the middle value of a data set when the values are arranged in order. It's one of the three most common measures of center — alongside the mean and mode — and it has a useful property that the mean doesn't: it ignores extreme values.
What the Median Tells You
The median splits a data set exactly in half. Half the values fall at or below it, and half fall at or above it. That's why it's often described as the "typical" value — it represents the middle of the distribution, not the mathematical average.
This distinction matters when data has outliers. Consider household incomes in a neighborhood where most families earn around $60,000, but one household earns $4,000,000. The mean income gets pulled sharply upward by that one value — it might come out to $250,000, a number that doesn't reflect the experience of anyone actually living there. The median, though, would stay right around $60,000.
This is exactly why income and housing statistics are almost always reported as medians rather than means.
How to Find the Median
Step 1: Sort the values from smallest to largest.
Step 2: If there's an odd number of values, the median is the middle one. If there's an even number, the median is the average of the two middle values.
Odd number of values
Find the median of: 14, 7, 3, 21, 9
Sort: 3, 7, 9, 14, 21
Five values, so the middle is the 3rd one. Median = 9.
Even number of values
Find the median of: 8, 3, 15, 6, 11, 2
Sort: 2, 3, 6, 8, 11, 15
Six values, so the median is the average of the 3rd and 4th:
$$\text{Median} = \frac{6 + 8}{2} = 7$$
Median = 7 — even though 7 doesn't actually appear in the data set.
Finding the middle position
With a large data set it helps to use a formula to locate the middle. For \(n\) values sorted in order:
- If \(n\) is odd, the median is at position \(\dfrac{n+1}{2}\)
- If \(n\) is even, average the values at positions \(\dfrac{n}{2}\) and \(\dfrac{n}{2} + 1\)
For example, with 19 values: position \(\frac{19+1}{2} = 10\) — the 10th value is the median. With 20 values: average positions 10 and 11.
Median vs. Mean
Neither measure is universally better — they answer different questions.
The mean accounts for every value in the data set equally. It's the right choice when the data is roughly symmetric and there are no extreme outliers (test scores, production measurements, etc.).
The median is better when the data is skewed or contains outliers. Real estate prices, salaries, and response times are commonly reported as medians for this reason.
When the mean and median are close together, the data is probably fairly symmetric. When they're far apart, that's a sign the distribution is skewed or contains outliers worth investigating.
Practice Problems
1. Find the median of: 5, 12, 3, 8, 7, 19, 1 Show answerSorted: 1, 3, 5, 7, 8, 12, 19. Seven values, middle is the 4th. Median = 7.
2. Find the median of: 44, 17, 62, 8, 31, 55 Show answerSorted: 8, 17, 31, 44, 55, 62. Six values, average the 3rd and 4th: \(\frac{31+44}{2} = 37.5\). Median = 37.5.
3. A class of 9 students scored: 72, 85, 91, 68, 74, 88, 95, 71, 80. Find the median score. Show answerSorted: 68, 71, 72, 74, 80, 85, 88, 91, 95. Nine values, middle is the 5th. Median = 80.
4. Which measure — mean or median — better represents the typical salary in a company where 10 employees earn $45,000–$55,000 and the CEO earns $2,000,000? Show answerThe median. The CEO's salary is a strong outlier that would pull the mean far above what most employees earn. The median would fall in the $45,000–$55,000 range, which is more representative.
5. The median of five consecutive even integers is 14. What are the five integers? Show answerFive consecutive even integers centered on 14: 10, 12, 14, 16, 18.