Statistics Calculator — Mean, Median, Mode & Standard Deviation
A statistics calculator takes a list of numbers and returns the descriptive measures that summarize it: mean (the arithmetic average), median (the middle value when sorted), mode (the most frequent value), range, variance, and standard deviation. Paste or type your data set separated by commas, spaces, or line breaks — the calculator parses the values, counts them, and computes every statistic at once, so you can check homework or analyze real data without spreadsheet formulas.
The key decision is population versus sample. If your data covers every member of the group you care about, use population variance (divide by N). If it is a sample drawn from a larger group, use sample variance (divide by n − 1, Bessel's correction) so the estimate is not biased low. The calculator shows both, along with the sum, minimum, maximum, and quartiles, giving a complete descriptive picture of the distribution.
How it works
Mean: x̄ = Σx ÷ n. Median: middle value of the sorted list (average of the two middle values if n is even). Population variance: σ² = Σ(x − μ)² ÷ N. Sample variance: s² = Σ(x − x̄)² ÷ (n − 1). Standard deviation: σ = √σ² or s = √s². Range = max − min.
Use cases
- Checking homework answers for mean, median, mode, and standard deviation
- Summarizing survey responses or experiment measurements
- Computing the standard deviation of grades, prices, or lab readings
- Comparing the spread of two data sets using variance and range
- Finding outliers by looking at quartiles, minimum, and maximum
- Deciding between population and sample formulas for a class project
Frequently asked questions
What is the difference between mean, median, and mode?
The mean is the sum of all values divided by how many there are; the median is the middle value once the data is sorted; and the mode is the value that appears most often. In the set 2, 3, 3, 8, 14 the mean is 6, the median is 3, and the mode is 3. The median resists distortion by extreme values, which is why it is preferred for skewed data like incomes.
When should I use sample standard deviation instead of population?
Use the population formula (divide by N) when your data includes every member of the group of interest, such as the grades of all students in one class. Use the sample formula (divide by n − 1) when your data is a subset used to estimate a larger group, such as 100 survey respondents representing a city. Dividing by n − 1, known as Bessel's correction, compensates for the fact that a sample tends to underestimate the true spread.
How do I calculate standard deviation by hand?
First compute the mean, then subtract the mean from each value and square the result. Sum the squared differences and divide by N (population) or n − 1 (sample) to get the variance. The standard deviation is the square root of the variance. For 2, 4, 6 the mean is 4, squared differences are 4, 0, 4, population variance is 8 ÷ 3 ≈ 2.67, and σ ≈ 1.63.
What does the standard deviation actually tell me?
It measures how far values typically sit from the mean, in the same units as the data. A small standard deviation means the values cluster tightly around the average; a large one means they are spread out. For roughly bell-shaped distributions, about 68% of values fall within one standard deviation of the mean and about 95% within two — a useful rule of thumb for spotting unusual values.
Can a data set have more than one mode or no mode at all?
Yes. If two values tie for the highest frequency the set is bimodal, and with more ties it is multimodal — the calculator lists every mode. If every value appears exactly once, there is no mode. Mode is the only measure of central tendency that also works for categorical data, such as the most common answer in a poll.