"Average" is one of the most used and most misused words in mathematics. In everyday language it usually means one specific thing — add up the numbers and divide. But in statistics there are three distinct types of average, each appropriate for different situations. Choosing the wrong one leads to misleading conclusions.
The Three Types of Average
1. Mean (Arithmetic Average)
The mean is what most people mean by "average." Add all values and divide by how many there are.
Mean = Sum of all values / Number of values
Example: Test scores: 72, 85, 91, 68, 77, 95, 82
Sum = 72 + 85 + 91 + 68 + 77 + 95 + 82 = 570
Count = 7
Mean = 570 / 7 = 81.4
When to use it: When data is roughly symmetrical with no extreme outliers. Works well for heights, test scores, temperatures.
When NOT to use it: When outliers exist. One billionaire in a room of average earners makes the mean income extremely misleading.
2. Median (Middle Value)
The median is the middle value when data is sorted in order. Half the values are above it, half below.
For an odd number of values: sort and take the middle one.
For an even number: sort and take the mean of the two middle values.
Example (odd): 72, 68, 85, 91, 77, 95, 82
Sort: 68, 72, 77, 82, 85, 91, 95
Median = 82
Example (even): 68, 72, 77, 82, 85, 91
Middle two: 77 and 82
Median = (77 + 82) / 2 = 79.5
When to use it: When data has outliers or is skewed. House prices, salaries, and income distributions always use median because a handful of extreme values would distort the mean.
3. Mode (Most Frequent Value)
The mode is the value that appears most often. A dataset can have one mode (unimodal), two (bimodal), or more (multimodal). If no value repeats, there is no mode.
Example: Shoe sizes sold in a week: 6, 7, 7, 8, 8, 8, 9, 9, 10
Mode = 8 (appears 3 times)
When to use it: Categorical data, survey responses, or when you need the most common value rather than a mathematical centre. A shoe manufacturer cares about the mode, not the mean shoe size.
Weighted Mean
When some values count more than others, use the weighted mean:
Weighted mean = Σ(value × weight) / Σ(weights)
Example: University module grades with different credit weightings:
| Module | Grade | Credits | |--------|-------|---------| | Maths | 72 | 30 | | English | 85 | 15 | | History | 68 | 15 | | Science | 91 | 40 |
Weighted mean = (72×30 + 85×15 + 68×15 + 91×40) / (30+15+15+40)
= (2,160 + 1,275 + 1,020 + 3,640) / 100
= 8,095 / 100
= 80.95
This is different from the simple mean of 79.0 — the Science module's higher credit weighting pulls the average up.
GPA calculations, investment portfolio returns, and exam marking all use weighted means.
Geometric Mean
For quantities that compound or multiply (growth rates, investment returns), use the geometric mean:
Geometric mean = (x₁ × x₂ × ... × xₙ)^(1/n)
Example: Annual investment returns of +50%, −30%, +20%
Simple mean = (+50 − 30 + 20) / 3 = +13.3% — misleadingly optimistic
Geometric mean = (1.50 × 0.70 × 1.20)^(1/3) − 1
= (1.26)^(1/3) − 1
= 1.0797 − 1
= +7.97% per year
This reflects the actual compounding: £1,000 → £1,500 → £1,050 → £1,260, giving 7.97% annualised growth — not 13.3%.
Which Average Should You Use?
| Situation | Best average | |-----------|-------------| | Symmetric data, no outliers | Mean | | Skewed data or outliers present | Median | | Most common value needed | Mode | | Values have different importance | Weighted mean | | Rates, ratios, or compounding | Geometric mean | | Salary / income comparisons | Median | | House price statistics | Median | | Sports batting averages | Mean (or specific formula) | | Investment return over years | Geometric mean |
Common Mistakes
Assuming "average" always means mean. When you see "average salary" in news reports, ask whether it's mean or median. The mean is typically 20–30% higher than the median due to high earners skewing the data.
Averaging percentages without weighting. If your portfolio has £1,000 in Fund A (+10%) and £9,000 in Fund B (+2%), the average return is NOT 6%. It's (£100 + £180) / £10,000 = 2.8%.
Ignoring the distribution. The mean can be the same for very different datasets. A class where everyone scores 70% and a class where half score 40% and half score 100% have the same mean — but very different learning outcomes.
Use our Mean, Median, Mode Calculator and Weighted Average Calculator to calculate any type of average from your own data.