# Mean vs Median vs Mode: Nine Salaries Where the Average Beats Eight of Them

One data point moves the mean by $26,292 and the median by $1,000. Here is the same dataset through all three averages, the four situations where each is the right one, and the two averages nobody teaches that you probably need.

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- **Canonical URL:** https://dothecalculation.com/blog/math/mean-vs-median-vs-mode
- **Category:** Math
- **Author:** Do The Calculation Team
- **Published:** 2026-08-03
- **Reading time:** 11 min read
- **Publisher:** Do The Calculation (https://dothecalculation.com)
- **Methodology:** https://dothecalculation.com/methodology

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## Mean vs Median vs Mode

All three are called the average and they answer different questions. The mean is the balance point of the data. The median is the middle value. The mode is the most common one. On symmetric data they nearly coincide; on real data they often do not.

Choosing wrongly is not a technicality. Report a mean salary on a payroll with one founder in it and you produce a number that describes nobody in the building.

Tool: [Try the average calculator](https://dothecalculation.com/calculators/average-calculator) — Paste a dataset to get the mean, median, mode, and range together rather than one at a time.

## Nine salaries

A small firm pays, in ascending order: $38,000, $41,000, $45,000, $45,000, $47,000, $52,000, $58,000, $61,000, and $285,000 to the founder.

**Three averages of the same nine numbers**
| Measure | Value | How it was found |
| --- | --- | --- |
| Mean | $74,667 | $672,000 total ÷ 9 |
| Median | $47,000 | The fifth value, four above and four below |
| Mode | $45,000 | The only value that appears twice |

> **What the mean is claiming** — The mean salary is $74,667, and eight of the nine people earn less than that. Only the founder earns more. As a description of what someone at this firm is paid, the mean is not merely imprecise; it is wrong about every single employee.

Now remove the founder and recompute. This is the clearest demonstration of what each measure is sensitive to.

**Effect of removing one value**
| Measure | With founder | Without | Change |
| --- | --- | --- | --- |
| Mean | $74,667 | $48,375 | $26,292 |
| Median | $47,000 | $46,000 | $1,000 |
| Mode | $45,000 | $45,000 | None |

The mean uses every value, so every value can move it, and an extreme one moves it a long way. The median only cares about position, so a single extreme value shifts it by one place in the ordering. This is what statisticians mean by calling the median robust.

## The gap between mean and median is the skew

Comparing the two tells you the shape of the data without plotting anything.

**Reading the relationship**
| Relationship | Shape | Typical examples |
| --- | --- | --- |
| Mean well above median | Right skewed, a long high tail | Income, house prices, sales per customer |
| Mean roughly equals median | Roughly symmetric | Heights, measurement error, test scores |
| Mean below median | Left skewed, a long low tail | Exam scores with a ceiling, age at retirement |

On the salary data the mean is 59% above the median, which is severe right skew. That single comparison is enough to know the mean should not be the headline figure, before you look at a single individual value.

## Which one to use

**Matching the measure to the question**
| Situation | Use | Why |
| --- | --- | --- |
| Typical income, house price, or wait time | Median | These distributions are always right skewed |
| Anything you need to multiply back up to a total | Mean | Total equals mean times count; the median has no such property |
| Symmetric data with no extremes | Mean | Uses all the information and is more efficient |
| Categorical data: colour, size, model | Mode | There is nothing to add or order |
| Deciding how many of each shirt size to stock | Mode | You want the most common, not a mathematical centre |
| Small dataset with a known error in it | Median | One bad value does not contaminate the answer |
| Growth rates or returns over time | Geometric mean | Rates compound rather than add |
| Rates from groups of different sizes | Weighted mean | A simple average of percentages is not a percentage |

> **The one thing only the mean can do** — Total = mean × count. If you know the mean salary is $74,667 across nine people, payroll is $672,000. The median tells you nothing about the total, which is why finance reports means and journalism reports medians. Both are behaving correctly.

## The average of averages is not an average

This is the most common and most expensive mistake in the whole subject. Two stores run a promotion. Store A converts 10% of 100 visitors. Store B converts 4% of 900 visitors.

**Simple mean against weighted mean**

```
Simple: (10% + 4%) ÷ 2 = 7%    ·    Weighted: (10 + 36) ÷ 1,000 = 4.6%
```
- Store A produced 10 conversions, Store B produced 36.
- 46 conversions from 1,000 visits is 4.6%.
- The simple average overstates the real rate by 52%.

The rule is that you can only average rates directly when the denominators are equal. Otherwise, add the numerators, add the denominators, and divide. Anything else gives a number that does not correspond to any actual outcome.

## And for anything that compounds, the mean is wrong

An investment gains 50% one year and loses 50% the next. The arithmetic mean return is 0%. The actual outcome is a 25% loss: $100 becomes $150, then $75.

**Geometric mean of growth rates**

```
Geometric mean = ( ∏ (1 + rᵢ) )^(1/n) − 1
```
- √(1.50 × 0.50) − 1 = √0.75 − 1 = −13.4% per year
- Check: 0.866 × 0.866 = 0.75, which is the actual two-year outcome.
- The geometric mean is never above the arithmetic mean, and the gap widens with volatility.

Anything that multiplies rather than adds needs the geometric mean: investment returns, population growth, compound inflation, year-over-year revenue. Reporting an arithmetic average annual return systematically overstates what an investor actually experienced.

Tool: [Try the compound interest calculator](https://dothecalculation.com/calculators/compound-interest-calculator) — See how a sequence of returns compounds, which is the arithmetic the geometric mean summarises.

## A middle option: the trimmed mean

If the mean is too fragile and the median throws away too much information, drop a fixed percentage from each end and take the mean of the rest. Removing the highest and lowest salary from the nine leaves seven values averaging $49,857.

That figure uses more of the data than the median while resisting the founder's salary entirely. Trimmed means are standard in sports judging, in some inflation measures, and anywhere a small number of extreme values is expected and not interesting.

## Edge cases worth knowing

- A dataset can have no mode, if every value is unique, or several, if two or more tie. The mode is the only one of the three that is not always a single number.
- With an even count, the median is the mean of the two middle values, which can produce a median that is not in the dataset at all.
- The mode is the only measure that works on categorical data, and the only one that is always an actual observed value.
- On a perfectly symmetric single-peaked distribution, all three coincide exactly. That is the case textbooks draw and real data rarely provides.
- The mean minimises the sum of squared distances from the data; the median minimises the sum of absolute distances. That is the formal reason squared-error methods produce means and absolute-error methods produce medians.

## What no average tells you

- Spread. Two datasets can share all three averages and look completely different. Always report a measure of dispersion alongside: standard deviation, interquartile range, or simply the minimum and maximum.
- Shape. A single peak, two peaks, or a uniform spread can all produce the same mean. A bimodal dataset is precisely the case where every average is misleading, because the centre is where nothing is.
- Sample size. A mean of 47 from three observations and from three thousand are very different claims, and the number alone does not distinguish them.
- Whether the data is any good. An average of a biased sample is a precise summary of a biased sample.
- Change over time. A stable average can hide two groups moving in opposite directions at the same rate.
- Which individual cases matter. In cost, risk, and capacity planning, the tail is often the whole question and the average is a distraction from it.

> **A defensible default** — Report the median with the interquartile range for skewed data, the mean with the standard deviation for symmetric data, and both when you are unsure. If the mean and median differ by more than about 10%, say so, because your reader will assume they do not.

**Why is median income always lower than mean income?**

Because income is bounded below at zero and unbounded above. A small number of very high earners pull the mean up while the median, being a position, barely moves. Every income distribution in every country has this shape.

**Which average should I report by default?**

The median for anything skewed, which includes most money, time, and count data. The mean when the data is symmetric or when you need the total. Reporting both is the most honest option and costs one extra column.

**Can a dataset have more than one mode?**

Yes. If two values tie for most frequent, the data is bimodal, and that is usually a signal that you have two populations mixed together. Splitting them and describing each separately is almost always more useful than any single average.

**How do I average percentages correctly?**

Do not average the percentages. Add the numerators, add the denominators, and divide. Ten conversions from 100 visits and 36 from 900 is 46 from 1,000, which is 4.6%, not the 7% a simple average gives.

**When should I use the geometric mean?**

Whenever the quantities multiply: growth rates, investment returns, inflation, and ratios. If the answer to "what happens over several periods" is found by multiplying rather than adding, the arithmetic mean overstates it.

**Does removing outliers make the mean valid?**

Only if the outlier is genuinely an error, such as a typo or a broken sensor. Removing a real observation because it is inconvenient changes what you are describing. The founder's $285,000 is not an error; it is a fact about that payroll, and the honest response is to use the median rather than delete the person.

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_Source: [Do The Calculation](https://dothecalculation.com/blog/math/mean-vs-median-vs-mode). Quote freely with attribution and a link to this page._