When should I use median instead of mean?

Use the median when your data is skewed or contains significant outliers: income, house prices, wealth, survival times, waiting times, and any data where extreme values are possible. The median is unaffected by extreme values. If Bill Gates joins a group of 10 people, the mean wealth changes dramatically; the median barely moves.

What does bimodal distribution mean?

A bimodal distribution has two peaks (two modes). It often indicates two distinct subgroups within your data. For example, test scores might be bimodal if the class contains two groups with very different preparation levels. A bimodal distribution can make the mean misleading — it might fall between the two peaks where few actual data points exist.

How is the median calculated for an even number of values?

Average the two middle values. Sort the data, then take the mean of the n/2 and (n/2 + 1) values. For data: 3, 7, 9, 12: median = (7 + 9) ÷ 2 = 8. For data: 5, 10, 15, 20, 25, 30: median = (15 + 20) ÷ 2 = 17.5.

What is the relationship between mean, median, and mode in a normal distribution?

In a perfectly normal (symmetric, bell-shaped) distribution, mean = median = mode — they all coincide exactly at the centre. This is why the mean is ideal for normally distributed data like heights, blood pressure, and exam scores across large populations. When the distribution is skewed, these three measures diverge in a predictable pattern.

Statistics Basics: Mean, Median, Mode Calculator Guide

Mean, median, and mode are the three measures of central tendency in statistics — each describes the "centre" of a dataset in a different way. Choosing the wrong measure can lead to genuinely misleading conclusions, which is why understanding when to use each is a critical statistical skill. This guide explains the calculation for each measure, when it's appropriate, and how to avoid being misled by poorly chosen averages.

The Mean (Arithmetic Average)

Mean = Sum of all values ÷ Number of values

x̄ = Σxᵢ ÷ n

Calculation Example

Dataset: 12, 15, 18, 22, 25, 28
Sum = 120 | n = 6
Mean = 120 ÷ 6 = 20

When to Use the Mean

The mean is the most mathematically useful average and is appropriate when:

The data is symmetrically distributed (roughly normal/bell-shaped)
There are no extreme outliers
The data is continuous (heights, temperatures, exam scores)
You need to use the value in further calculations (e.g. standard deviation)

When NOT to Use the Mean

The mean is severely distorted by outliers. Classic example: average income. In the UK, a small number of very high earners pull the mean average salary well above the level most workers experience. When Bill Gates walks into a pub, the average wealth of everyone in the room suddenly becomes millions — but nobody is actually richer.

Dataset with outlier: 18, 19, 20, 21, 22, 23, 250
Mean = 373 ÷ 7 = 53.3 — far above 6 of the 7 values.

The Median

The median is the middle value when data is sorted in order. Half the values lie above it, half below.

For odd n: Median = value at position (n+1)/2

For even n: Median = mean of values at positions n/2 and (n/2)+1

Calculation Examples

Odd dataset: 3, 7, 9, 12, 15 → Median = 9 (middle value, position 3)

Even dataset: 3, 7, 9, 12, 15, 18 → Median = (9 + 12) ÷ 2 = 10.5

With outlier: 18, 19, 20, 21, 22, 23, 250 → Median = 21 — unaffected by the outlier of 250.

When to Use the Median

Income and wealth distributions (skewed by high earners)
Property prices (skewed by luxury properties)
Survival times and waiting times (often skewed)
Any data with significant outliers
Ordinal data (e.g. ranked satisfaction scores 1–5)

The median UK salary (around £34,000 in 2024/25) is more representative of what a typical worker earns than the mean (around £37,000), which is pulled up by high earners.

The Mode

The mode is simply the most frequently occurring value in a dataset. A dataset can have:

One mode (unimodal)
Two modes (bimodal)
Multiple modes (multimodal)
No mode (all values appear equally frequently)

Example: 3, 5, 7, 5, 9, 5, 11, 3 → Mode = 5 (appears 3 times)

Bimodal: 2, 4, 4, 5, 7, 7, 9 → Modes = 4 and 7

When to Use the Mode

Categorical data (most common colour, most common job title)
Shoe or clothing sizes (retail stocking decisions)
Survey responses (most common rating given)
Any situation where you want the most common value rather than the average

Comparing the Three Measures

For a symmetric distribution (e.g. heights of adults), mean = median = mode. They all give the same answer.

For a right-skewed distribution (e.g. income):

Mode: lowest (most common, typical value)
Median: middle
Mean: highest (pulled up by outliers)

This is why politicians arguing about "average" wages should always be asked: which average? Mean or median?

Beyond Central Tendency: Range and Standard Deviation

The three averages tell you where the centre is — but not how spread out the data is. A class where everyone scored 70% on an exam is very different from a class where scores range from 10% to 100% with a mean of 70%.

Range: Maximum − Minimum. Quick but sensitive to outliers.
Interquartile range (IQR): Q3 − Q1 (the middle 50% of data). More robust than range.
Standard deviation (σ): The average distance from the mean. The most complete measure of spread.

Worked Example: House Price Data

Seven house sales in a street: £180k, £195k, £205k, £210k, £215k, £225k, £890k

Mean: (180+195+205+210+215+225+890) ÷ 7 = 2,120 ÷ 7 = £302,857
Median: 4th value (sorted) = £210,000
Mode: None (all unique)

The mean of £303k makes this street appear far more expensive than it is for most properties, due to one outlier. The median of £210k is far more representative.

Summary

Mean = sum ÷ count — best for symmetric data without outliers. Median = middle value — best for skewed data and outlier-resistant summary. Mode = most frequent value — best for categorical data. Always question which "average" is being quoted in news stories and reports — the choice between them can dramatically change the story.