Standard Deviation Calculator — Sample & Population

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Enter details
ResultUpdates as you type
Sample Std Dev (s)5.237229
Population Std Dev (σ)4.898979
Sample Variance (s²)27.428571
Population Variance (σ²)24
Mean (μ)18
Count (N)8
Sum (∑x)144

Calculate sample standard deviation s, population standard deviation σ, variance, and mean for any dataset with step-by-step math.

Calculate sample standard deviation (s), population standard deviation (σ), variance, mean, and sum for any numerical dataset.

Understanding Standard Deviation and Variance

Standard deviation is a fundamental measure of statistical dispersion. It quantifies how spread out numbers in a data set are from their arithmetic mean.

  • Low Standard Deviation: Indicates that data points tend to be close to the mean.
  • High Standard Deviation: Indicates that data points are spread out over a wider range of values.

Sample vs. Population

  • Use Sample Standard Deviation (s) when working with a sample subset selected from a larger group.
  • Use Population Standard Deviation (σ) when you have data for every member of the entire population.

How to use

  1. Paste or type your numbers separated by commas, spaces, or newlines.
  2. View sample standard deviation s and population standard deviation σ.
  3. Check the sample and population variance, mean, count, and sum.

Formula

Sample Standard Deviations = √[ ∑(x_i - x̄)² / (N - 1) ]Uses N-1 (Bessel's correction) to estimate population variability from a sample.
Population Standard Deviationσ = √[ ∑(x_i - μ)² / N ]Calculated using total population count N.

Worked examples

Standard Deviation Calculation

For dataset [10, 12, 23, 23, 16, 23, 21, 16]: Mean = 18. Sample Std Dev s ≈ 5.237. Population Std Dev σ ≈ 4.899.

Frequently asked questions

Why is Bessel's correction (N-1) used for sample standard deviation?

Dividing by N-1 corrects the bias when estimating a population standard deviation from a sample dataset.

What is standard deviation?

Standard deviation measures the amount of variation or dispersion of a set of values relative to their mean.

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