A single average can hide almost everything interesting about a dataset, which is exactly why real statistics work never stops at the mean.
Free math tool
Statistics Calculator
Mean, median, mode, range, and both population and sample variance/standard deviation, with a live chart of your data.
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Population vs sample, in one sentence
If your numbers are literally everyone or everything you're asking about, use the population figures. If your numbers are a smaller group standing in for a bigger one you can't fully measure, use the sample figures, they divide by one less than the count specifically to correct for a sample's tendency to look less spread out than the true population actually is.
Statistics calculator FAQ
When should I use population vs sample standard deviation?
Use population if your numbers ARE the entire group you care about, every student's score in one specific class. Use sample if your numbers are a subset used to estimate a larger group, a survey of 200 people used to describe a whole city. Sample standard deviation divides by (n − 1) instead of n specifically to correct for the fact that a sample tends to underestimate the true spread.
What does it mean when there is more than one mode?
The mode is whichever value (or values) appear most often. If two or more values are tied for the highest frequency, the data is called multimodal and this tool lists all of them. If every value appears exactly once, there is no meaningful mode, and it says so rather than picking one arbitrarily.
Why is the median sometimes a better summary than the mean?
The mean is pulled toward extreme values, a single very high or very low number can drag it a long way from where most of the data actually sits. The median, the middle value when everything is sorted, ignores how extreme the outliers are and only cares about their rank. Income and house-price data almost always gets reported as a median for exactly this reason.
How do I read the range and standard deviation together?
Range only uses the two most extreme values (max − min) and says nothing about everything in between. Standard deviation accounts for every value's distance from the mean, so it's a far more reliable sense of how spread out the data really is, two datasets can share the same range with very different standard deviations.
Is my data private?
Yes. Every calculation runs in your browser. Nothing you enter is sent to a server.
Free statistics calculator by ANUPRESS
Population or sample: the one question that changes your formula
If your numbers are literally the entire group you care about, every student in one specific class, use the population figures. If your numbers are a smaller group standing in for something bigger you can’t fully measure, a poll of 500 people describing an entire country, use the sample figures instead. Sample variance divides by one less than the count specifically because a sample almost always looks slightly less spread out than the true population it’s drawn from, dividing by a smaller number corrects that bias.
Why the median survives outliers and the mean doesn’t
Add one enormous outlier to a small dataset and the mean lurches toward it, sometimes landing somewhere that doesn’t represent the data at all. The median, the middle value once everything is sorted, barely moves, because it only cares about rank, not magnitude. That’s why household income, house prices, and most real-world “typical value” reporting leans on the median: a handful of billionaires or mansion sales shouldn’t be allowed to define what “typical” means for everyone else.
Range vs standard deviation: two very different kinds of spread
Range only looks at the two most extreme values in the whole dataset and ignores everything else, two datasets can share an identical range while one is tightly clustered and the other is wildly inconsistent in between. Standard deviation accounts for every single value’s distance from the mean, which is why it’s the number statisticians actually trust when they need to describe how consistent or volatile a dataset really is.