Concepts · 3 min read

Mean vs Median: When Averages Lie

Published October 3, 2026Written and reviewed by Mateuss M.

In short: The mean is sensitive to extreme values; the median is not. Five similar numbers give Mean 11.6 · Median 12, but add one outlier and it becomes Mean 25.5 · Median 12.

A small data set, then one outlier

Imagine five quiz scores near 12. The mean and median agree. Now add a single score of 95.

Before the outlier

AnswerMean 11.6 · Median 12

Mode: 12
Range: 3
Sample SD: 1.1402
Population SD: 1.0198

Step-by-step working 7 steps

1
Sort the data
```
10, 11, 12, 12, 13
```
5 values.
2
Mean = sum ÷ count
```
Sum = 58
Mean = 58 ÷ 5 = 11.6
```
3
Median = middle value
```
Value 3 of 5 = 12
```
4
Mode = most frequent value
```
12 (appears 2 times)
```
5
Range = largest − smallest
```
13 − 10 = 3
```

Squared deviations from the mean

(10 − 11.6)² = 2.56
(12 − 11.6)² = 0.16
(11 − 11.6)² = 0.36
(13 − 11.6)² = 1.96
(12 − 11.6)² = 0.16
Sum of squares = 5.2

Sum these for every value to get the total used below.

7
Variance and standard deviation
```
Population: σ² = 5.2 ÷ 5 = 1.04,  σ = 1.0198
Sample:     s² = 5.2 ÷ 4 = 1.3,  s = 1.1402
```
Use the sample version (÷ n−1) when your data is a sample from a larger group; use the population version when it is the whole group.

After adding 95

AnswerMean 25.5 · Median 12

Mode: 12
Range: 85
Sample SD: 34.0632
Population SD: 31.0953

Step-by-step working 7 steps

1
Sort the data
```
10, 11, 12, 12, 13, 95
```
6 values.
2
Mean = sum ÷ count
```
Sum = 153
Mean = 153 ÷ 6 = 25.5
```

Median = middle value

Average of values 3 and 4: (12 + 12) ÷ 2 = 12

4
Mode = most frequent value
```
12 (appears 2 times)
```
5
Range = largest − smallest
```
95 − 10 = 85
```

Squared deviations from the mean

(10 − 25.5)² = 240.25
(12 − 25.5)² = 182.25
(11 − 25.5)² = 210.25
(13 − 25.5)² = 156.25
(12 − 25.5)² = 182.25
(95 − 25.5)² = 4830.25
Sum of squares = 5801.5

Sum these for every value to get the total used below.

7
Variance and standard deviation
```
Population: σ² = 5801.5 ÷ 6 = 966.9167,  σ = 31.0953
Sample:     s² = 5801.5 ÷ 5 = 1160.3,  s = 34.0632
```
Use the sample version (÷ n−1) when your data is a sample from a larger group; use the population version when it is the whole group.

Why the mean moves and the median does not

The mean uses every value’s size, so one huge number pulls it up. The median only cares about position — the middle of the sorted list — so an extreme value at the end shifts it by at most one place.

A simple rule

Roughly symmetric data with no outliers → report the mean.
Skewed data or obvious outliers (incomes, house prices, response times) → report the median, and mention the mean.
When it matters, report both and the spread.

Frequently asked questions

What is an outlier?

A value far from the rest of the data. A common rule of thumb flags values more than 1.5 interquartile ranges beyond the quartiles.

Should I delete outliers?

Only if you can show they are errors. Otherwise keep them and choose a measure that is robust to them.

Written and reviewed by Mateuss M.

Mateuss M. writes and reviews mathematical content for CalcSolver, focusing on online calculators, formulas, equations, and practical math tools. He reviews calculator functionality, calculation methods, formulas, examples, and explanations to help ensure that each tool is clear, useful, and easy to understand.

Keep reading

Concepts

Mean vs Median: When Averages Lie

A small data set, then one outlier

Why the mean moves and the median does not

A simple rule

Frequently asked questions

Written and reviewed by Mateuss M.

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