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Standard Deviation Calculator

Calculate sample and population standard deviation, variance, mean, and margins of error.

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How to Use the Standard Deviation Calculator

1Paste or enter comma-separated numbers into the data input box.
2Select whether your data represents an entire Population (N) or a Sample (n - 1).
3Inspect the standard deviation, variance, mean, and sum of squared differences.
4Copy the comprehensive statistical report.

Formula & How It Works

Sample vs. Population Formulas

Sample s = √[ ∑(xᵢ - x̄)² / (n - 1) ]

Sample standard deviation uses n - 1 in the denominator to provide an unbiased estimator of variance, whereas population standard deviation divides by N.

Worked Examples

Sample Dataset

Input: 10, 12, 23, 23, 16, 23, 21, 16

Explanation: Sum of squared deviations is 192. For sample, 192 / 7 = 27.43, and √27.43 = 5.24.

Result: Mean = 18, Sample s = 5.237, Population σ = 4.899

Homogeneous Dataset

Input: 5, 5, 5, 5

Explanation: All values are identical, meaning zero dispersion or variability.

Result: Standard Deviation = 0

Frequently Asked Questions

When should I use sample vs. population standard deviation?▼

Use sample standard deviation when your data is a subset of a broader group. Use population standard deviation if you have measured every single member of the group.

What does a high standard deviation indicate?▼

A high standard deviation indicates that data points are spread out widely across a large range of values relative to the mean.

What is variance?▼

Variance is simply the square of the standard deviation (s² or σ²).

Does this tool calculate the standard error of the mean?▼

Yes, it reports SE = s / √n for statistical significance testing.

Last reviewed & updated: 2026-03-27

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