Standard Deviation Calculator

What this tool is used for

• Measuring how tightly a set of values clusters around its mean.
• Comparing two datasets that share a mean but not a spread.
• Getting the variance for a downstream statistical calculation.
• Checking whether a single observation is unusual for its set.
• Confirming a figure a spreadsheet produced with an ambiguous function.

Frequently Asked Questions

Population or sample — which one applies to my data?

Population (÷ n) when your numbers are the entire group: every employee, all twelve months of the year that happened. Sample (÷ n−1) when they are a draw from something larger and you are estimating that larger thing's spread. Survey responses, test batches and measurements are almost always samples.

Why does the sample version divide by n − 1?

Because deviations are measured from the SAMPLE mean, which is fitted to the data and sits closer to the points than the true mean does. That makes the raw squared deviations too small, and dividing by n−1 corrects the bias exactly — Bessel's correction, n−1 being the degrees of freedom left.

What does it show for a single value?

A population standard deviation of 0 — one point has no spread — and no sample figure at all, because n−1 is zero and the quantity is undefined. That is not a defensive check bolted on; a sample of one carries no information about the variability of the population it came from.

Why report variance as well as standard deviation?

Because variance is what the maths is built on and standard deviation is what humans read. Variances of independent quantities add; standard deviations do not. But variance is in squared units — squared pounds, squared centimetres — so taking the root returns it to the scale of the original data.

Is a standard deviation of 5 large?

Unanswerable without the mean. 5 around a mean of 10 is enormous spread; 5 around 10,000 is negligible. The scale-free version is the coefficient of variation, σ/μ — and for a roughly normal distribution the useful rule is that about 68% of values fall within one σ of the mean and 95% within two.

Common errors and gotchas

• Using the population formula on a sample, which understates the spread — the denominator differs.
• Reporting a standard deviation for data that is not roughly symmetric, where it describes little.
• Comparing standard deviations across datasets with different units or scales.
• Letting one outlier dominate, since the deviation is squared before averaging.
• Confusing standard deviation with standard error, which shrinks as the sample grows.