🎲 Sample Standard Deviation

Sample Standard Deviation#

An estimate of population spread from a sample, using n - 1 in the denominator.

Important

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What it is#

The sample standard deviation \(s\) measures how spread out the values in a sample are around the sample mean \(\bar{x}\). It is the square root of the sample variance — roughly, the average distance of a data point from the mean.

The formula and Bessel’s correction#

\[s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}.\]

The denominator is \(n - 1\), not \(n\) — Bessel’s correction. Dividing by \(n\) would underestimate the true spread, because the deviations are taken from the sample mean (which is itself fitted to the data); using \(n - 1\) makes \(s^2\) an unbiased estimator of the population variance \(\sigma^2\).

Worked example#

For \(\{5, 7, 9\}\): the mean is \(\bar{x} = 7\); deviations are \(-2, 0, +2\); squared, \(4, 0, 4\), summing to 8; divide by \(n - 1 = 2\) to get 4; the square root is \(s = 2\).

Reading it#

A small \(s\) means points cluster near the mean (low variability); a large \(s\) means they’re spread out; \(s = 0\) means every value is identical. The population standard deviation \(\sigma\) uses \(N\) in the denominator (the whole population); the sample version uses \(n - 1\) (an estimate of \(\sigma\)).

Where it shows up#

\(s\) is the raw material of inference: it feeds the standard error (\(SE = s/\sqrt{n}\)), the t-test, ANOVA and regression, and the confidence intervals built around sample means.

Theme: Probability & Statistics Foundations · All terminology

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Mind map — connected ideas

Sample Mean · Standard Error (SE) · True Mean (Population Mean) · Frequentist · Bootstrap Confidence Intervals (CIs) · Margin of Error (MoE)

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More in Probability & Statistics Foundations

Beta Distribution · Confidence Level · Correlation · Critical Value · Cumulative Distribution Function (CDF) · Frequentist · IID (Independent and Identically Distributed) · Likelihood · Margin of Error (MoE) · Mean · Median · Normal Distribution · Outlier · Population Proportion