Explanation
What mean, median and mode tell you
A dataset can have several different centres. The mean balances every observation; the median locates the middle after sorting; the mode identifies the most frequent value. Their disagreement can reveal skew or outliers.
Choose the centre you need
Enter between one and 100 values separated by semicolons, all in the same unit. Choose the mean for an arithmetic balance point, the median when a few very large or small values should not dominate, or the mode for the most common value. Every value has equal weight.
x̄ = Σxᵢ ÷ n
| x̄ | mean |
|---|---|
| xᵢ | a value in the list |
| n | number of values |
M = middle of x₍₁₎ ≤ … ≤ x₍ₙ₎
| M | median |
|---|---|
| x₍₁₎ … x₍ₙ₎ | values sorted from smallest to largest |
| n | number of values |
Mo = the value that occurs most often
| Mo | mode |
|---|---|
| xᵢ | a value in the list |
Mean of 2, 4 and 6
| Values, separated by semicolons | 2;4;6 |
|---|---|
| Result | 4 |
Sum 12 ÷ 3 values = 4
Median of 1, 2, 8 and 9
| Values, separated by semicolons | 1;2;8;9 |
|---|---|
| Result | 5 |
4 values (sum 20), sorted; the middle value is 5
Mode of 2, 4, 4 and 6
| Values, separated by semicolons | 2;4;4;6 |
|---|---|
| Result | 4 |
4 values (sum 16); the most frequent value is 4
Diagram of this result
A dot plot stacks repeated values, with markers for the mean, median and mode.
Each value is a dot on a number line; equal values stack. Labelled markers show the mean, median and mode, and the selected one is highlighted. The exact numbers are stated in the answer and the steps.
Common mistakes
Do not average averages unless their group sizes are equal. Sort before finding the median, and use the mean of the two central values when the count is even. A mode is not the largest value. Lists with tied frequencies or no repeated value have no single mode and are rejected.
Assumptions and limits
This is unweighted descriptive arithmetic. Leave out missing values deliberately. The mode needs exactly one most frequent value occurring at least twice; ties are not resolved by picking one. Values use up to 30 significant digits and 50 internal digits; the answer is rounded to two decimals only for display.
Sources
OpenStax Introductory Statistics 2e, descriptive statistics: https://openstax.org/details/books/introductory-statistics-2e
Why do mean and median differ?
Large or small outliers move the mean because every value contributes to its sum. The median depends on order and the middle positions, so it often moves less.