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Standard Deviation Calculator

Calculators · Added 15 August 2026

Standard deviation measures how spread out a set of numbers is. This calculator shows the whole working — the mean, each squared deviation summed, the divisor and the square root — so the result can be checked by hand, and it makes you choose explicitly between the sample and population formulas.

Separate with commas, spaces or new lines.

Is this a sample or the whole population?

How to use the standard deviation calculator

  1. 1Paste your data, separated by commas, spaces or new lines.
  2. 2Choose sample or population. Sample is right unless you genuinely have every member of the group.
  3. 3Press Calculate.
  4. 4Follow the working panel: mean, sum of squared deviations, divisor, variance, then the square root.
  5. 5The spread panel adds quartiles and the interquartile range, which describe the shape rather than just the width.

Examples

The textbook dataset

Input
2, 4, 4, 4, 5, 5, 7, 9 as a population
Result
Standard deviation exactly 2, variance 4

This set is used in most textbooks precisely because it comes out to a whole number.

The same data as a sample

Input
The identical numbers, treated as a sample
Result
Standard deviation 2.1381 — higher, because it divides by 7 rather than 8

A 7% difference on eight values. On three or four values the gap is far larger, which is why the choice matters.

Comparing spread across different scales

Input
A dataset with mean 1000 and standard deviation 50
Result
Coefficient of variation 5%

Standard deviation alone cannot compare datasets with different means. The coefficient of variation can.

About the standard deviation calculator

What the formula is actually doing

Every step of the standard deviation calculation exists for a reason, and the sequence is easier to remember once the reasons are clear. Start with each value's distance from the mean. Those distances sum to exactly zero by construction — that is what a mean is — so they have to be made positive before they can be added up meaningfully.

Squaring is how that is done, and it is not the only option. Taking absolute values would also work and gives a perfectly respectable measure called the mean absolute deviation. Squaring wins because it is differentiable, which makes it tractable in the wider mathematics built on top of it, and because it weights large deviations more heavily — a single point far from the mean matters more than several points slightly off.

Summing the squares, dividing to get an average, and then taking the square root to undo the squaring gives a number in the original units. That final square root is why standard deviation rather than variance is what appears in reports: 'about 2 centimetres of spread' is a sentence, '4 square centimetres of spread' is not.

Reading spread alongside the average

An average without a spread is half a description. Two production lines both averaging 100 grams per unit are not equivalent if one has a standard deviation of 1 gram and the other 15 — the first is under control and the second is producing a lot of underweight and overweight units around a respectable-looking mean.

This is why the coefficient of variation is worth knowing: it expresses the standard deviation as a percentage of the mean, which makes spread comparable across datasets measured on completely different scales. A 50-gram deviation is enormous on a 100-gram product and trivial on a 100-kilogram one.

The quartiles do a related job differently. Where standard deviation summarises spread in one symmetric number, the interquartile range describes the middle half of the data and is unaffected by extremes. When a dataset has outliers, comparing the two is informative: a standard deviation much larger than the interquartile range suggests the extremes are doing most of the work, and that a median-based summary would describe the data more honestly.

Frequently asked questions

Sample or population — which do I need?
Sample, almost always. You use the population formula only when your data is the entire group you want to describe: every student in one specific class, every item in one specific batch. If your numbers are a subset standing in for something larger, it is a sample. When in doubt, sample is the safer choice because it produces the slightly larger, more conservative estimate.
Why divide by n − 1 for a sample?
Because a sample's own mean sits closer to its own data points than the true population mean does, which makes the squared deviations systematically too small. Dividing by n − 1 rather than n compensates for that bias. It is called Bessel's correction, and its effect shrinks as the sample grows — noticeable at n = 5, negligible at n = 500.
What is variance, and why not just use it?
Variance is the standard deviation squared — the average squared deviation from the mean. It is the more fundamental quantity mathematically and the less useful one practically, because its units are squared. A variance of 4 on a set of heights in centimetres is 4 square centimetres, which means nothing intuitively. Taking the square root puts the answer back into centimetres, which is why standard deviation is what gets reported.
What does the standard deviation actually tell me?
Roughly how far a typical value sits from the mean. For data that is approximately normally distributed, about 68% of values fall within one standard deviation of the mean and about 95% within two — the basis of most quality-control limits. For data that is not normal, it still measures spread, but those percentages do not hold.
Why do my quartiles differ from another calculator?
Because there are several defensible definitions and they disagree on small datasets. This tool uses linear interpolation between order statistics, the same method spreadsheets use for PERCENTILE and QUARTILE. Others use nearest-rank or exclude the median when splitting. On large datasets the differences vanish; on eight values they can be visible.

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