Mean, Median, Mode Calculator

Last updated: 2026-05-19

Overview

Mean, median, and mode are the three standard measures of central tendency — single numbers that summarize where the middle of a dataset lies. The range is the simplest measure of spread, describing how far the data extends. Together these four statistics give a compact summary of any list of numbers. This calculator computes all four from comma-separated input (for example: 12, 15, 11, 19, 14).

The four measures

Mean (arithmetic average)

The mean is the sum of all values divided by the count of values:

$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

For the dataset 4, 8, 15, 16, 23, 42:

$\bar{x} = \frac{4 + 8 + 15 + 16 + 23 + 42}{6} = \frac{108}{6} = 18$

The mean uses every data point, so a single extreme value can raise or lower it significantly. This makes it well suited to symmetric, unskewed distributions but misleading when outliers are present.

Median

The median is the middle value of a sorted dataset. If the count is even, it is the average of the two middle values:

• Odd n: median = the value at position ⌊n/2⌋ + 1 (1-indexed) after sorting.
• Even n: median = average of values at positions n/2 and n/2 + 1 after sorting.

For 4, 8, 15, 16, 23, 42 (n = 6, even):

$M_d = \frac{15 + 16}{2} = 15.5$

Because it relies on position rather than magnitude, the median is resistant to outliers. A household-income distribution with a billionaire included will have a mean far above what most people earn, but the median stays close to the typical household.

Mode

The mode is the value that appears most often. A dataset can have:

• No mode — all values appear exactly once (this calculator displays "no mode").
• One mode (unimodal) — one value appears more than all others.
• Two modes (bimodal) — two values tie for most frequent.
• Multiple modes (multimodal) — three or more values share the highest frequency.

For 2, 3, 3, 5, 7, 7, both 3 and 7 appear twice, so the modes are 3, 7.

The mode is the only measure of central tendency that applies to categorical (non-numeric) data. In a class poll on favorite colors, the mode is the only meaningful average.

Range

The range is the simplest measure of spread:

$R = x_{\max} - x_{\min}$

For 4, 8, 15, 16, 23, 42: range = 42 − 4 = 38.

The range describes how wide the data is, but like the mean it is sensitive to outliers. A single extreme value can make the range large while most of the data stays tightly clustered. For a more robust view of spread, standard deviation is preferable (available in the Descriptive Statistics Calculator).

Worked example: test scores

Suppose seven students scored 55, 62, 70, 70, 78, 84, 95 on an exam.

Measure	Calculation	Result
Count	7 values	7
Mean	(55+62+70+70+78+84+95) ÷ 7 = 514 ÷ 7	73.43
Median	middle value of sorted list (position 4)	70
Mode	70 appears twice; all others once	70
Range	95 − 55	40

Interpretation: The mean (73.43) and median (70) are close, suggesting the distribution is roughly symmetric. The mode at 70 confirms that score is the most common. The range of 40 indicates moderate spread across the class.

Choosing a measure

Situation	Best measure
Symmetric data, no outliers	Mean
Skewed data or outliers present	Median
Categorical data, or the most common value	Mode
Overall extent of the data	Range (or standard deviation)

Why the formulas work

The mean minimizes the sum of squared deviations — it is the value c that makes Σ(xᵢ − c)² as small as possible. This optimality property underlies least-squares regression and much of statistics.

The median minimizes the sum of absolute deviations — it is the value c that makes Σ|xᵢ − c| as small as possible. This robustness to outliers is why it is preferred in skewed distributions.

Application

The choice of measure has real consequences when summarizing skewed data. U.S. household income is reported using the median ($74,580 in 2022) rather than the mean because a small number of very high earners inflate the mean far above what a typical household earns. Reporting that "the average household earns $100,000+" would be technically true but misleading, since most households earn considerably less. The same reasoning applies to house prices, salaries, and any distribution with a long tail.

Frequently Asked Questions (FAQ)

When should I use mean vs median?

The mean is appropriate when the data has no extreme outliers — it uses every value and works best for symmetric distributions. The median is appropriate when the data is skewed or contains outliers (e.g. household incomes, house prices) because it represents the "typical" value without being dragged by extremes.

For example, in a dataset of salaries where one person earns ten times more than everyone else, the median better reflects what most employees earn.

Can a dataset have more than one mode?

Yes. A dataset is called bimodal if it has two modes, and multimodal if it has more than two. For example, the dataset 2, 3, 3, 5, 7, 7 has modes 3 and 7 — both appear twice. This calculator lists all modes separated by commas.

What happens when all values are unique?

When every value in the dataset appears exactly once, there is no value that occurs more frequently than the others. By convention, such a dataset has no mode. This calculator displays "no mode" in that case. The mean, median, and range are still computed normally.

How is this different from a descriptive statistics calculator?

This calculator focuses on the four most commonly taught summary statistics — mean, median, mode, and range — for any list of comma-separated numbers. A full descriptive statistics calculator also computes variance, standard deviation, and may be limited to a fixed number of inputs. This one suits quick central-tendency lookups; the descriptive statistics calculator is the right tool when spread measures like standard deviation are needed.