🎲 Standard Error (SE)

Standard Error (SE)#

The standard deviation of a statistic’s sampling distribution — how much an estimate varies across samples.

Important

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What it is#

The standard error (SE) measures how much a sample statistic — a mean or a proportion — would vary across many random samples from the same population. It is the uncertainty of the estimate: how far the sample value is likely to fall from the true population value.

The formulas#

For a mean (population SD \(\sigma\), or sample SD \(s\) when \(\sigma\) is unknown):

\[SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \quad\left(\text{or } \frac{s}{\sqrt{n}}\right).\]

For a proportion:

\[SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}.\]

Key properties#

Bigger samples shrink it (\(SE \propto 1/\sqrt{n}\), so estimates get more precise), more spread inflates it (larger \(\sigma\)), and it is the bridge to the Central Limit Theorem: for large \(n\) the sample mean is approximately normal with the SE as its spread.

Examples#

With \(\mu = 100, \sigma = 20, n = 25\), \(SE = 20/\sqrt{25} = 4\) — sample means vary about ±4 around the truth. For a proportion, 520 “yes” out of 1,000 gives \(\hat{p} = 0.52\) and \(SE = \sqrt{0.52 \times 0.48 / 1000} \approx 0.016\), a ±1.6% margin.

SE vs standard deviation#

They are easy to confuse: the standard deviation describes the spread of individual data points, while the standard error describes the spread of the estimate. SD is variability in the data; SE is variability in the statistic — and SE shrinks with \(n\) while SD does not.

Where it shows up#

The SE is the denominator of inference: confidence intervals (\(\text{estimate} \pm z_{\alpha/2}\, SE\)), test statistics (the z- and t-tests), and A/B comparisons of conversion rates all run on it.

Theme: Probability & Statistics Foundations · All terminology

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Mind map — connected ideas

True Conversion Rate · Bootstrap Confidence Intervals (CIs) · Frequentist · Conversion Rate (CR) · True Mean (Population Mean) · A/B Testing

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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