Statistics Calculator: Mean, Median, Mode & Std Dev
Paste a list of numbers and instantly get the mean, median, mode, range, variance, and standard deviation, both population and sample.
How this calculator computes descriptive statistics, and why it reports two different standard deviations
Paste in a list of numbers and this tool returns the full descriptive picture at once: mean, median, mode, range, and both population and sample versions of variance and standard deviation. That last part, computing two different spreads instead of one, is not redundant. Population and sample statistics answer genuinely different questions, and picking the wrong one is one of the most common quiet errors in applied statistics.
Every calculation runs in your browser on the array of numbers you type or paste in. Nothing is uploaded or stored.
The core measures, in the order they’re computed
That n−1 in the sample formula is called Bessel’s correction, named after Friedrich Bessel. When your data is only a sample drawn from a larger population, using the mean of that sample to measure spread systematically underestimates the true population variance, because the sample mean is, by construction, the point that minimizes squared distance to your own sample. Dividing by n−1 instead of n corrects that downward bias, producing an unbiased estimate of the population’s actual variance.
| Statistic | Formula | Use when |
|---|---|---|
| Population variance | Σ(x−μ)² / n | Your data is the entire group |
| Sample variance | Σ(x−x̄)² / (n−1) | Your data estimates a larger group |
| Population std dev (σ) | √(population variance) | Entire group known |
| Sample std dev (s) | √(sample variance) | Estimating from a subset |
Multimodal detection
If two or more values tie for highest frequency, the tool lists all of them and labels the result multimodal, rather than arbitrarily picking just one and hiding the tie.
Flexible input parsing
Numbers can be separated by commas, spaces, semicolons, or line breaks in any combination, and the parser strips and validates each token before running the math.
- Bessel’s correction explains the n−1 divisor and the bias it corrects for in sample variance.
- Variance covers the general mathematical definition behind both formulas used here.
- Standard deviation details the square-root relationship to variance and its interpretation as a spread measure.
Summarising a dataset
Summarizing a batch of quality control measurements from a production line, checking the spread of exam scores across a class, analyzing a small sample of customer survey ratings to estimate how the full customer base likely feels, comparing two data sets by their mean and standard deviation before running a more formal statistical test, and verifying textbook or homework statistics problems step by step. Whenever you have a list of numbers and need to understand not just their center but how spread out they are, and whether that data represents everything you care about or just a sample of it, this calculator handles both cases correctly.
FAQ: Statistics Calculator
The mean is the sum of all values divided by how many there are (the arithmetic average). The median is the middle value when the data is sorted (or the average of the two middle values if there’s an even count), which is less affected by extreme outliers than the mean. The mode is the value that appears most frequently, and a data set can have one mode, multiple modes (multimodal), or no mode at all if every value is unique.
Use the median when your data contains outliers or is heavily skewed, since a few extreme values can pull the mean far from what feels ‘typical,’ while the median stays anchored to the actual middle of the sorted data. A classic example is household income, where a small number of very high earners inflate the mean well above what most households actually earn, making the median a more representative figure.
Population standard deviation divides the sum of squared differences by N (the total count) and is used when your data represents every member of the group you’re studying. Sample standard deviation divides by N−1 instead, a correction (called Bessel’s correction) that compensates for the tendency of a sample to underestimate the true variability of the full population it was drawn from. Since most real-world data collection involves sampling rather than measuring an entire population, the sample version is more commonly the correct choice.
When you calculate variance using a sample’s own mean (rather than the true population mean, which is usually unknown), the result systematically underestimates the true population variance, because the sample mean is, by definition, the value that minimizes the sum of squared distances within that particular sample. Dividing by N−1 instead of N corrects for this bias, producing a more accurate estimate of the population’s actual variance.
A large standard deviation means your data points are spread out widely around the mean, indicating high variability, while a small standard deviation means values cluster tightly around the mean. For example, two classes could have the same average test score, but one with a small standard deviation had most students scoring close to that average, while one with a large standard deviation had scores spread from very low to very high.
If every value in your data set appears exactly once, there’s no value that occurs more often than any other, so the data set has no mode, this is a normal and valid outcome, not an error. This is especially common in continuous, precisely measured data (like exact temperatures or prices) where exact repeated values are less likely than in categorical or rounded data.
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