Mean, median, mode and the full descriptive set from any list of numbers. Also flags outliers, and tells you which average your data actually needs — because for rates of return the arithmetic mean is simply the wrong answer, and for speeds it is wrong in a different way.
Average Calculator
LiveThree different averages, and when each is correct
"Average" usually means the arithmetic mean, and for most data it is right. For two specific kinds of data it produces an answer that is not merely imprecise but wrong — and the error is not small.
| Mean | Use for | Why the arithmetic mean fails |
|---|---|---|
| Arithmetic sum ÷ n | Most data — heights, scores, temperatures, prices | It doesn't. This is the default for a reason |
| Geometric nth root of the product | Growth rates, investment returns, anything that compounds | Returns multiply rather than add. Averaging them arithmetically overstates the result |
| Harmonic n ÷ sum of reciprocals | Rates over a fixed quantity — speeds over the same distance, price-to-earnings ratios | The quantity being averaged sits in a denominator, so the reciprocals are what add |
You drive somewhere at 60 km/h and return along the same road at 40 km/h. What was your average speed?
The arithmetic mean says 50. The correct answer is 48. You spent more time at the slower speed, so it weighs more heavily — and the harmonic mean captures that automatically. Over a 120 km leg each way: 2 hours out, 3 hours back, 240 km in 5 hours = 48 km/h.
An investment returns +10%, +20% and +5% over three years. The arithmetic mean says 11.67% a year.
The geometric mean is 11.49% — and it is the one that reproduces the actual final balance. Compounding 11.67% for three years overshoots what you really have. The gap widens with volatility: a year of +50% followed by −50% averages to 0% arithmetically, while you have actually lost 25%.
Mean or median?
The single most consequential choice in describing data, and the answer turns on one question: is the distribution skewed, or are there outliers?
| Situation | Use | Reason |
|---|---|---|
| Roughly symmetric data | Mean | Uses every value, and is more efficient statistically |
| Skewed data — income, house prices, wait times | Median | A few extreme values drag the mean away from where most of the data sits |
| Outliers present | Median | The median barely moves; the mean can shift enormously |
| Categorical data | Mode | The only average that works when values are labels rather than quantities |
The demonstration is stark. For the set 1, 2, 3, 4, 100 the mean is 22 and the median is 3. Four of the five values sit below 4 — the mean describes none of them. This is exactly why income statistics are reported as medians: a handful of very high earners make the mean unrepresentative of almost everybody.
A practical heuristic: if the mean and median differ by more than about 10% of the range, your data is skewed and the median is probably the honest figure to report.
How outliers are identified
The calculator flags outliers using the interquartile range method, which is the standard approach and does not assume the data is normally distributed:
Sample or population?
Standard deviation and variance come in two forms, and the choice is not cosmetic:
| Version | Divides by | Use when |
|---|---|---|
| Sample (s) | n − 1 | Your data is a subset used to infer something about a larger group. This is almost always the case |
| Population (σ) | n | Your data includes every member of the group being described — every employee, every item in a batch |
The n−1 correction exists because a sample's spread systematically underestimates the population's. Dividing by the smaller number nudges the estimate upward to compensate — this is Bessel's correction. The difference matters most with small samples: with n = 5 it changes the result by around 12%; with n = 500 it is negligible.
When the arithmetic mean is simply the wrong tool
Averaging rates, ratios and growth is where the arithmetic mean stops being conservative and starts being wrong:
| What you are averaging | Correct mean | Why |
|---|---|---|
| Investment returns | Geometric | +50% then −50% averages to zero arithmetically and to −13.4% geometrically. The balance really is down a quarter |
| Speeds over equal distances | Harmonic | 60 km/h out and 20 back is 30 km/h overall, not 40 — you spend three times as long on the slow leg |
| Ratios and multiples | Geometric | Averaging P/E ratios or index numbers arithmetically overweights the large values |
| Inflation over years | Geometric | Prices compound, so the average must too |
| Totals and budgets | Arithmetic | When the sum is what matters, the arithmetic mean is exactly right |
Averages of averages
Averaging a set of averages gives the right answer only when every group is the same size, and that is rarely true:
Class A: 30 students, average 60
Class B: 10 students, average 80
Average of the averages: (60 + 80) ÷ 2 = 70 — wrong
Weighted by size: (30×60 + 10×80) ÷ 40 = 65 — correct
The unweighted figure is five marks too high, because it treats ten students as equal in weight to thirty. This is the same error behind Simpson’s paradox, where a treatment can look better in every subgroup and worse overall — and it appears wherever percentages from different-sized populations are averaged together.
Common mistakes
Frequently asked questions
Should I use the mean or the median?
Mean for roughly symmetric data with no extreme values; median when the data is skewed or contains outliers. The quick test is to compare them — if they differ substantially, the data is skewed and the median is the more honest summary. This is why income, house prices and waiting times are almost always reported as medians: a few very large values pull the mean away from where most observations actually sit.
What is the geometric mean for?
Anything that compounds — investment returns, growth rates, population change. Because these multiply rather than add, the arithmetic mean overstates them. Returns of +10%, +20% and +5% give an arithmetic mean of 11.67% but a geometric mean of 11.49%, and only the second reproduces the actual final balance. The gap grows with volatility.
When would I need the harmonic mean?
When averaging rates over a fixed quantity. Travelling out at 60 km/h and back at 40 km/h over the same road gives an average speed of 48 km/h, not 50 — you spend more time at the slower speed, so it weighs more. The same logic applies to averaging price-to-earnings ratios across a portfolio, and to combining precision and recall into an F1 score.
Can I average two averages?
Only if the groups behind them are the same size. A class of 10 averaging 90% and a class of 100 averaging 60% combine to 62.7%, not 75% — the larger class dominates. To do it correctly, weight each average by its group size, or work from the raw totals. The weights field above handles this.
What counts as an outlier?
The standard rule flags anything more than 1.5 interquartile ranges beyond the first or third quartile. It is a convention rather than a law — Tukey chose 1.5 because it captures about 99.3% of normally distributed data. Being flagged means "look at this", not "delete this". An outlier can be a typo or it can be the single most informative point in the set.
Sample or population standard deviation?
Sample, almost always. Population is correct only when your data includes every member of the group you are describing. Sample divides by n−1 rather than n, correcting for the fact that a subset systematically understates the true spread. With small datasets the difference is substantial — around 12% at n = 5.
What if my data has no mode, or several?
Both are normal. If every value appears once there is no mode, which is common with continuous measurements. If several values tie for the highest frequency the data is multimodal, and the calculator lists all of them. Multimodality is worth noticing — it often means two different populations have been mixed into one dataset.
Is my data stored?
No. Everything runs in your browser with no server request, and works offline once the page has loaded. Nothing you paste is transmitted or saved.
Related calculators
See the full list of Math calculators, or try:
- Standard Deviation Calculator — spread in detail
- Percentile Calculator — position within a distribution
- Z-Score Calculator — how far a value sits from the mean
- Weighted Grade Calculator — averages with category weights
- CAGR Calculator — the geometric mean applied to growth
- Confidence Interval Calculator — uncertainty around a mean