Standard error tracks uncertainty in sample estimates
Standard error is the standard deviation of a statistic's sampling distribution, often the wobble of the sample mean. It equals population sigma over square root of n, so four times the data halves the error on the mean.
Standard error (SE) measures how much a sample statistic—most famously the sample mean—would vary across repeated draws from the same population. The sampling distribution of means has variance equal to population variance divided by sample size: as n grows, sample means cluster tighter around the true population mean. Thus SE of the mean equals σ/√n, describing dispersion of means rather than individual data points.
Population σ is rarely known, so practitioners estimate SE by replacing σ with the sample standard deviation s, giving σ̄x ≈ s/√n. Confusion is common: σ describes individual observations, σ̄x describes the sample mean's variability, and σ̂x̄ is the estimated SE actually computed. Using s underestimates population SD when n is small—about 25 percent low at n = 2, only 5 percent at n = 6—prompting correction formulas for tiny samples.
The derivation uses independence: for n independent observations with mean μ and variance σ², the sum T has variance nσ², so the sample mean's variance is σ²/n and its standard deviation is σ/√n. Halving the error on a mean estimate requires quadrupling sample size; shrinking it tenfold needs a hundred times as many observations. In regression, "standard error" can also mean the square root of the reduced chi-squared statistic.
When sample size n is large relative to population variance, x̄ stays close to μ. For correlated data, variance calculations follow the Markov chain central limit theorem instead. If sample size itself is random—when observations are accepted by some criterion—additional variation enters because n becomes a random variable alongside the statistic being measured.
Source: Standard error
