Median Calculator

What is the median?

The median is the middle value in a sorted list of numbers. It divides a data set into two equal halves, with 50% of values above and 50% below.

Unlike the mean (average), the median is resistant to outliers, making it a better measure of central tendency for skewed distributions.

How to find the median

For an odd number of values

1. Sort the numbers from smallest to largest
2. The median is the middle number

Example: For 3, 7, 9, 12, 15

The middle value is 9.

For an even number of values

1. Sort the numbers from smallest to largest
2. Find the two middle numbers
3. Calculate their average

Example: For 2, 5, 8, 11

The two middle values are 5 and 8.

Median = (5 + 8) ÷ 2 = 6.5

The formula

For a sorted list of n values:

Odd n:

$$\text{Median} = x_{\frac{n+1}{2}}$$

Even n:

$$\text{Median} = \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2}$$

Median vs. mean

Measure	Median	Mean
Calculation	Middle value	Sum ÷ count
Affected by outliers	No	Yes
Best for	Skewed data	Symmetric data
Example use	Income data	Test scores

When to use median

Use the median when:

• Your data has significant outliers
• The distribution is skewed (not symmetric)
• You want a typical value that isn't pulled by extremes

Example of outlier effect

Consider salaries: $40k, $45k, $50k, $55k, $500k

• Mean: $138,000 (pulled up by the outlier)
• Median: $50,000 (represents the typical salary better)

Related statistics

Mode

The most frequently occurring value. A dataset can have:

• No mode (all values appear once)
• One mode (unimodal)
• Multiple modes (bimodal, multimodal)

Range

The difference between the maximum and minimum values:

$$\text{Range} = \text{Max} - \text{Min}$$

Quartiles

Quartiles divide data into four equal parts:

• Q1 (25th percentile): Median of lower half
• Q2 (50th percentile): The median
• Q3 (75th percentile): Median of upper half

Practical applications

Real estate

Home prices are often reported as medians because a few luxury homes would dramatically skew the mean upward.

Income statistics

Median household income is preferred over mean income because wealth concentration affects the average.

Performance metrics

Website load times often use median values because occasional timeouts would distort averages.

Scientific research

Medians are used when data doesn't follow a normal distribution or when outliers are expected.

Properties of the median

1. Minimizes absolute deviations: The sum of |x - median| is minimized
2. Positional measure: Depends only on the position, not the exact values
3. Unique for continuous data: Always produces a single value
4. Scale invariant: Multiplying all values by a constant multiplies the median by that constant

Limitations

1. Doesn't use all data points (ignores exact values)
2. Less efficient statistically than mean for normal distributions
3. Can be affected by how ties are handled in discrete data
4. Doesn't have nice algebraic properties (median of sums ≠ sum of medians)