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Statistics · Descriptive Statistics Formula verified

Standard Deviation Calculator

Type or paste a list of numbers to find the standard deviation, variance, and each value's z-score — choose population or sample depending on what your list represents.

Numbers

Separate numbers with commas, spaces, or new lines. Use Sample if this list is drawn from a larger group.

Population standard deviation
12.3153
6 numbers
Variance
151.6667
Mean
18
Coefficient of variation
68.4%

Step-by-step proofCheck by hand

Population vs. sample standard deviation

Both start the same way: find the mean, find how far each value is from the mean, square those differences, and average them — that average is the variance, and its square root is the standard deviation. The only difference is what you divide by. Population divides by the count of numbers (n), treating your list as the entire group. Sample divides by one fewer (n − 1), which corrects for the fact that a smaller sample tends to underestimate the true spread of the larger population it's drawn from.

The formulas

population: σ = √(Σ(x−mean)² / n)  ·  sample: s = √(Σ(x−mean)² / (n−1))

Worked example

For 4, 8, 15, 16, 23, 42 (mean 18): the population standard deviation is 12.3153, and the sample standard deviation — dividing by 5 instead of 6 — is the slightly larger 13.4907.

Frequently asked questions

Should I use population or sample standard deviation?
Use population if the list you typed is the entire group you care about — every student in a class, every day in a month. Use sample if your list is a smaller sample drawn from a larger group you're trying to estimate — a subset of customers standing in for all customers. Sample standard deviation divides by one fewer than the count, which corrects for the extra uncertainty that comes from only seeing part of the whole group.
What does a z-score tell you?
A z-score says how many standard deviations a particular value sits from the mean, and in which direction. A z-score of 0 means the value equals the mean; positive means above it, negative means below it. A z-score of 2 is unusually far from the mean for most real-world data, while a z-score under 1 is unremarkable.
Why is sample standard deviation always a bit larger than population?
Dividing by a smaller number (n − 1, instead of n) always produces a larger result from the same sum of squared differences. This built-in inflation is intentional — a small sample tends to underestimate how spread out the full population really is, and dividing by n − 1 corrects for that bias.
Sources and method
  • Standard population and sample variance/standard deviation formulas, and z-scores — general published statistics.
Last reviewed: 19 Sep 2026 Sources last verified: 19 Sep 2026 Results use the assumptions explained on this page. Report an error How we check calculations

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