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Free online standard deviation calculator

Standard deviation measures how spread out values are around the mean. This calculator gives population σ and sample s, plus variance, mean, and count.

Population σ
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Sample s
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Mean
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Population variance σ²
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Sample variance s²
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Count
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About this calculator

How to use

  1. Type or paste your numbers into the text area — commas, spaces, semicolons, and new lines all work as separators.
  2. Results update as you type.
  3. Use “Share with my numbers” to share the data set via URL.

Formulas

Population standard deviation:

σ = √( Σ(xᵢ − μ)² / n )

Sample standard deviation:

s = √( Σ(xᵢ − x̄)² / (n − 1) )

Where:

  • μ (or x̄) = mean of all values
  • n = count of values
  • Σ(xᵢ − μ)² = sum of squared deviations from the mean

Worked example

Data: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8)

  1. Mean = 40 / 8 = 5
  2. Squared deviations: (2−5)²=9, (4−5)²=1 × 3, (5−5)²=0 × 2, (7−5)²=4, (9−5)²=16
  3. Sum of squared deviations = 9 + 3 + 0 + 4 + 16 = 32
  4. Population variance σ² = 32 / 8 = 4 → σ = 2
  5. Sample variance s² = 32 / 7 ≈ 4.5714 → s ≈ 2.1381

Notes

  • Sample standard deviation requires at least 2 values; with only 1 value the result is 0.
  • Results are rounded to six decimal places.
  • Share URLs encode your data in the d query parameter.

Frequently asked

What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides the sum of squared differences by n. Sample standard deviation (s) divides by n − 1, which corrects for the bias introduced when estimating from a subset of a larger population.
When should I use sample vs. population standard deviation?
Use population (σ) when you have data for every member of the group. Use sample (s) when your data is a subset and you want to estimate the spread of the full population.
What is variance?
Variance is the square of the standard deviation. It measures the average squared distance from the mean. Population variance = σ², sample variance = s².
Why does the sample standard deviation divide by n − 1?
Dividing by n − 1 (Bessel's correction) gives an unbiased estimate of the population variance. Without it, a sample tends to underestimate the true spread.
What does a high standard deviation mean?
A high standard deviation means values are spread widely around the mean. A low standard deviation means values cluster closely around the mean.
How do I enter my numbers?
Type or paste values separated by commas, spaces, semicolons, or new lines. Any mix of delimiters works.