Free Mean, Median, Mode Calculator
Paste or type a list of numbers and instantly see the mean, median, mode, range, and standard deviation. It handles commas, spaces, and line breaks, and lists every mode when your data has more than one.
Enter numbers separated by commas, spaces, or new lines.
Use sample statistics (÷ n−1) when your data is a sample of a larger group, and population statistics (÷ n) when it is the entire group.
Quick answer
The mean is the sum of all values divided by how many there are; the median is the middle value of the sorted list (the average of the two middle values when the count is even); and the mode is the value that appears most often. For the list 10, 20, 20, 30, 40 the mean is 24, the median is 20, and the mode is 20.
Formula & method
Mean (arithmetic average)
mean = (x₁ + x₂ + … + xₙ) / n
- xᵢ — each value in the list
- n — count of values
Add every value and divide by how many values there are.
Median (middle value)
median = middle value of the sorted list; if n is even, median = (x at n/2 + x at n/2 + 1) / 2
Sort the numbers first. With an odd count the median is the single center value; with an even count it is the average of the two center values.
Mode (most frequent)
mode = value(s) with the highest frequency
A list can have one mode, several modes (bimodal/multimodal), or no mode at all when every value is unique.
Variance and standard deviation
population σ² = Σ(xᵢ − mean)² / n; sample s² = Σ(xᵢ − mean)² / (n − 1); σ = √σ², s = √s²
- σ² — population variance (divide by n)
- s² — sample variance (divide by n − 1)
Use the sample versions (÷ n − 1) when your data is a sample of a larger group; use the population versions (÷ n) when the data is the entire group.
Examples
- Input
- 10, 20, 20, 30, 40
- Result
- Mean = 24, Median = 20, Mode = 20, Range = 30, Sample SD ≈ 11.401754
- Why
- Sum = 10 + 20 + 20 + 30 + 40 = 120, so mean = 120 ÷ 5 = 24. Sorted, the middle (3rd) value is 20, so median = 20. The value 20 appears twice and every other value once, so the mode is 20. Range = 40 − 10 = 30. Squared deviations from the mean sum to 520, giving population variance 520 ÷ 5 = 104 (σ ≈ 10.198039) and sample variance 520 ÷ 4 = 130 (s ≈ 11.401754).
- Input
- 4, 8, 15, 16, 23, 42
- Result
- Mean = 18, Median = 15.5, Mode = none, Range = 38
- Why
- Sum = 4 + 8 + 15 + 16 + 23 + 42 = 108, so mean = 108 ÷ 6 = 18. With 6 values (even count) the median is the average of the two middle values: (15 + 16) ÷ 2 = 15.5. Every value appears exactly once, so there is no mode. Range = 42 − 4 = 38.
- Input
- 5, 5, 9, 9, 10
- Result
- Mean = 7.6, Median = 9, Mode = 5 and 9, Range = 5
- Why
- Sum = 5 + 5 + 9 + 9 + 10 = 38, so mean = 38 ÷ 5 = 7.6. The middle (3rd) value of the sorted list is 9, so median = 9. Both 5 and 9 appear twice, the highest frequency, so the data is bimodal with modes 5 and 9. Range = 10 − 5 = 5.
When to use this tool
- Summarizing test scores, survey responses, prices, or measurements with a single representative number.
- Checking homework or stats coursework on central tendency and spread without doing the arithmetic by hand.
- Deciding whether the mean or median better describes skewed data such as incomes or response times.
- Quickly getting variance and standard deviation for a column of numbers pasted from a spreadsheet.
Common mistakes
- Forgetting to sort the list before reading off the median — the middle value only works on the sorted data, not the order you typed it.
- Reporting a single mode when the data is bimodal or multimodal. If two or more values tie for the highest frequency, all of them are modes.
- Confusing population standard deviation (÷ n) with sample standard deviation (÷ n − 1). Most real-world data is a sample, so the sample version is usually correct.
- Letting one extreme outlier distort your conclusion. The mean is pulled toward outliers, while the median stays robust — report both when the data is skewed.
Frequently asked questions
What is the difference between mean, median, and mode?
The mean is the arithmetic average (sum ÷ count), the median is the middle value of the sorted list, and the mode is the value that occurs most often. They can all differ: for 1, 2, 2, 9 the mean is 3.5, the median is 2, and the mode is 2. The mean reacts to every value, the median only to the center, and the mode only to frequency.
Can a data set have more than one mode?
Yes. A list with two values tied for the highest frequency is bimodal, and three or more is multimodal — this calculator lists every mode. If all values are unique, there is no mode at all, which we show as "none".
How do I find the median when there is an even number of values?
Sort the numbers, then take the two values in the middle and average them. For 4, 8, 15, 16, 23, 42 the two middle values are 15 and 16, so the median is (15 + 16) ÷ 2 = 15.5.
Should I use the mean or the median?
Use the mean for roughly symmetric data with no extreme outliers. Use the median for skewed data — like incomes, house prices, or response times — because one very large or very small value barely moves the median but can heavily distort the mean.
What is the difference between sample and population standard deviation?
Population standard deviation divides the summed squared deviations by n and is used when your numbers are the entire group. Sample standard deviation divides by n − 1 (Bessel's correction) and is used when your numbers are a sample drawn from a larger group, which is the more common case.
What separators can I use between numbers?
Commas, spaces, tabs, and new lines all work, so you can paste a row like "10, 20, 30" or a whole column copied from a spreadsheet. Any text that is not a valid number is ignored, so stray labels or units won't break the result.
Sources & references
External references open in a new tab. We are independent and not affiliated with these organizations.
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Provided “as is” for general information only — results may be inaccurate, so verify before you rely on them. No warranty; use at your own risk.
Built and reviewed by HIFreeTools against the formula shown above and any authoritative references cited on this page. See our methodology and editorial standards.
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