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CalcivoTools

Standard Deviation Calculator : Sample (s) vs Population (σ) Standard Deviation

The Calcivo Standard Deviation Calculator measures statistical dispersion in any numerical dataset. It clearly distinguishes between Sample Standard Deviation (s, using Bessel's correction n - 1) and Population Standard Deviation (σ, using N).

Interactive Calculator

100% Client-Side Engine
Sample Standard Deviation (s, N - 1)
5.2372
Used when analyzing a sample from a larger population.
Population Standard Deviation (σ, N)
4.899
Used when analyzing the entire dataset/population.
Mean (x̄)
18
Sample Variance (s²)
27.4286
Population Variance (σ²)
24
Sum of Squares (SS)
192
Step-by-Step Variance and Deviation
1. Calculate Mean (μ or x̄): Σx / N = 144 / 8 = 18
2. Calculate Deviations from Mean (x - x̄) and square each result
3. Sum of Squared Deviations (SS): Σ(x - x̄)² = 192
4. Population Variance (σ²): SS / N = 192 / 8 = 24
5. Population Standard Deviation (σ): √(σ²) = √(24) = 4.899
6. Sample Variance (s²): SS / (n - 1) = 192 / (8 - 1) = 27.4286
7. Sample Standard Deviation (s): √(s²) = √(27.4286) = 5.2372

Step-by-Step Examples

Analyzing 8 Numbers

Values: 10, 12, 23, 23, 16, 23, 21, 16.

Inputs: Values: 10, 12, 23, 23, 16, 23, 21, 16
Result: Sample s = 5.2372, Population σ = 4.8989 (Mean = 18.00)

Mean = 18.00. Sum of squared differences SS = 192.00. s = √(192 / 7) = 5.2372. σ = √(192 / 8) = 4.8989.

How to Use This Tool

  1. Enter dataset values separated by commas, spaces, or tabs.
  2. Click Calculate.
  3. Review both sample and population standard deviations alongside step-by-step deviation tables.

Frequently Asked Questions

Use Sample Standard Deviation (divided by n - 1) when your data is a sample taken from a larger population. Use Population Standard Deviation (divided by N) when your data represents the entire population.

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