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Standard Deviation Calculator (Population & Sample)

Calculate Mean, Variance, and Standard Deviation (σ vs s). Features a "Show Your Work" step-by-step table ideal for students and researchers.

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About

Statistical analysis hinges on understanding the spread of data. The choice between Population and Sample standard deviation is critical; using the wrong one distorts the Margin of Error. This tool handles both scenarios. It calculates the Population deviation (σ) when you have the entire data set, and the Sample deviation (s) when analyzing a subset to infer properties of the whole. The step-by-step breakdown ensures transparency, showing the squared differences and variance calculations required for academic verification.

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Formulas

The calculation involves three primary steps. First, calculate the Mean. Second, determine the Squared Difference for each data point:

d2 = (xi − μ)2

Finally, the Variance is the average of these squared differences. For a Sample, we divide by N-1 (Bessel's Correction) to unbiasedly estimate the population variance.

Reference Data

MetricSymbolFormulaUse Case
Meanx or μ∑ xNAverage of the data set.
Sample SDs√∑(x - x)2N - 1Use when data is a subset (Survey, Experiment).
Population SDσ√∑(x - μ)2NUse when you have ALL data (Census, Logs).

Frequently Asked Questions

This is known as Bessel's Correction. When working with a sample, the calculated variance tends to underestimate the true population variance. Dividing by N-1 instead of N corrects this bias.
The tool accepts numbers separated by commas, spaces, newlines, or tabs. It automatically filters out text and empty spaces.
Variance is simply the Standard Deviation squared. It measures the average degree to which each point differs from the mean.