How is the median calculated with an even number of values?

Sort the values, then average the two middle numbers. With an odd count, the median is simply the single middle value after sorting.

Statistics Calculator — Mean, Median, Mode & Standard Deviation

Q: What's the difference between population and sample standard deviation?

Population standard deviation divides by n (the count) and is used when your data is the entire group you care about. Sample standard deviation divides by n−1 and is used when your data is a sample used to estimate a larger population — dividing by the smaller number (n−1) corrects for the fact that a sample tends to underestimate the true population variance.

Q: What if there are multiple modes, or no mode at all?

If two or more values are tied for the most frequent, all of them are shown (a dataset can be bimodal or multimodal). If every value appears exactly once, there is no mode, and this is shown as such rather than picking an arbitrary value.

Q: What input formats does this accept?

Numbers separated by commas, spaces, or line breaks — or any mix of the three. Non-numeric text is ignored automatically.

What's the difference between population and sample standard deviation?

Population standard deviation treats your data as the entire group you care about, and divides the sum of squared differences by n (the count): σ = √(Σ(x − mean)² ÷ n). Sample standard deviation treats your data as a sample drawn from a larger population you're trying to estimate, and divides by n−1 instead: s = √(Σ(x − mean)² ÷ (n−1)). Dividing by the smaller number (n−1) — known as Bessel's correction — makes the sample figure slightly larger, correcting for the fact that a sample's own mean is closer to its own data points than the true population mean would be, which would otherwise make sample variance systematically underestimate the truth.

Which one should I use?

If you have every value that exists (every student in one specific class, every day in a specific month), use population statistics. If your numbers are a sample meant to represent a bigger group you haven't fully measured (a survey sample, a handful of test batches, a subset of transactions), use sample statistics. When in doubt, sample statistics is the more common and conservative choice in research and quality-control contexts.

What if there are multiple modes, or no mode at all?

If two or more values tie for the highest frequency, this calculator lists all of them — a dataset can genuinely be bimodal or multimodal, and picking just one would be misleading. If every value in the list appears exactly once, there's no meaningful mode, and that's shown explicitly rather than defaulting to an arbitrary number.

How is the median calculated?

Sort all values from smallest to largest. With an odd count, the median is the single middle value. With an even count, it's the average of the two middle values. Unlike the mean, the median isn't pulled around by extreme outliers — a useful cross-check whenever the mean and median differ substantially, which usually signals a skewed distribution.

What does a large standard deviation actually tell you?

Standard deviation measures how spread out values are around the mean — a small standard deviation means most values cluster close to the average, while a large one means values are widely scattered. Two datasets can have the identical mean but very different standard deviations, which is exactly why reporting an average alone (without any measure of spread) can be misleading.

What input formats does this accept?

Numbers separated by commas, spaces, line breaks, or any mix — paste directly from a spreadsheet column or a comma-separated list and it parses correctly either way. Non-numeric text is simply ignored.

Worked example: the list 4, 8, 6, 5, 3, 9, 8, 7, 2, 8 has count 10, sum 60, mean 6.0, sorted order 2,3,4,5,6,7,8,8,8,9 giving a median of (6+7)/2 = 6.5, a mode of 8 (appears three times), a range of 9−2=7, a population standard deviation of about 2.28, and a slightly larger sample standard deviation of about 2.40.