Statistics

Standard Deviation Calculator — Mean & Variance

Enter any list of numbers to instantly calculate the mean, variance, and both population and sample standard deviation — with the formula shown.

📊 Enter Your Data Set

Numbers (comma or space separated)

📊 Result

Enter a data set to see mean, variance and standard deviation

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Disclaimer: This tool performs standard mathematical calculations for educational and reference purposes. Always double-check results independently for graded coursework, exams, or safety-critical applications.

What Standard Deviation Measures

Standard deviation tells you how spread out a set of numbers is from the average. A small standard deviation means most values cluster tightly around the mean; a large one means the values are spread widely. It's the square root of variance, which puts the result back into the same units as your original data (variance is squared units, which is harder to interpret directly).

Population vs. Sample Standard Deviation

Use population standard deviation (divides by n) when your data set already represents the entire group you care about — every student in a class, every transaction in a month. Use sample standard deviation (divides by n−1) when your data is a sample drawn from a larger population and you're estimating that population's spread — a survey of 200 customers out of 50,000, for example. The n−1 adjustment (Bessel's correction) corrects for the fact that a sample slightly underestimates the true population variance.

Worked Example

Data set: 2, 4, 4, 4, 5, 5, 7, 9 (8 values).

Step 1 — Mean: (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5
Step 2 — Squared differences from mean: 9, 1, 1, 1, 0, 0, 4, 16 (sum = 32)
Step 3 — Population variance: 32 / 8 = 4, so population SD = √4 = 2
Step 4 — Sample variance: 32 / 7 ≈ 4.571, so sample SD ≈ 2.138

Frequently Asked Questions

What does a standard deviation of 0 mean? ▼

It means every value in your data set is identical — there's no spread at all around the mean.

Why is sample standard deviation always slightly larger than population standard deviation? ▼

Sample SD divides by n−1 instead of n, which produces a slightly larger number. This correction exists because a sample tends to slightly underestimate the true spread of the full population it was drawn from, and dividing by a smaller number compensates for that bias.

Can standard deviation be negative? ▼

No. It's a square root of an average of squared values, so it's always zero or positive. A result close to zero means low variability; a large result relative to the mean means high variability.