Calculate the standard error of the sample mean with optional finite population correction for small populations.
Last updated: March 2026
Known population variability
Number of observations
Leave blank for infinite population
The Standard Deviation of the Sample Mean (σ_x̄), also called Standard Error of the Mean, measures how much sample means vary from the true population mean. When you repeatedly draw samples from a population, each sample has a different mean. This metric quantifies that variation.
The formula is: σ_x̄ = σ / √n, where σ is the population SD and n is the sample size. For large populations, this formula works well. However, for small populations (finite populations), we apply a Finite Population Correction (FPC): σ_x̄ = (σ / √n) × √((N-n)/(N-1)). This correction reduces the standard error because we're sampling without replacement from a limited population, making the sample more representative.
The FPC matters when the sample size is a significant fraction of the population (typically when n > 0.05N). For large populations, FPC ≈ 1 and the correction becomes negligible.
Surveying a Small School District
σ (population SD) measures how spread out individual values are. σ_x̄ (SE of mean) measures how spread out sample means are. σ_x̄ is always smaller than σ by a factor of √n — sample means are more stable than individual values.
Use FPC when sampling without replacement from a small/finite population. A practical rule: apply FPC if n {'>'} 0.05N (sample {'>'} 5% of population). For large populations or sampling with replacement, FPC ≈ 1 and can be ignored.
Larger samples provide better estimates of the population mean. Mathematically, σ_x̄ includes √n in the denominator, so as n increases, SE decreases. Doubling n reduces SE by about 30%; quadrupling n cuts SE in half.
FPC = 0.95 means the corrected SE is 95% of the uncorrected SE. You're saving only 5% of error by the population adjustment, indicating the finite population effect is small. This occurs when n is a small fraction of N.
Theoretically, only if σ = 0 (no variation in population) or n = ∞ (infinite sample). In practice, SE is never zero because real populations always have variation, and infinite samples are impossible.
A 95% CI for the population mean = sample mean ± 1.96 × σ_x̄ . The standard error directly scales the width of confidence intervals. Smaller SE means narrower, more precise confidence intervals.
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