Calculators

Standard Error Calculator

Standard error of the mean, of a proportion, and of the difference between two independent means — with a sampling-distribution diagram and step-by-step working.

1 · Standard Error of the Mean

Given a sample standard deviation s and sample size n, computes SE(x̄) = s / √n.

2 · Standard Error of a Proportion

For a survey proportion p̂ (between 0 and 1) with sample size n, SE(p̂) = √(p(1 − p) / n).

3 · Standard Error of the Difference Between Two Means

For two independent samples, SE(x̄₁ − x̄₂) = √(s₁²/n₁ + s₂²/n₂). This is the SE used in a two-sample (Welch's) t-test.

Standard error explained, step by step

Standard error measures the precision of an estimate, not the spread of the raw data. Each card below covers one of the three SE forms this calculator supports.

1. SE vs SD — precision of the estimate vs spread of the data

Standard deviation describes how spread out individual data points are around the mean. Standard error describes how spread out the estimate itself would be across repeated samples. SD does not shrink with n; SE shrinks as 1 / √n. Report SE (or a CI) when you're talking about how precise the mean is.

SE = s / √n  ·  s
= √(Σ(xᵢ − x̄)² / (n − 1))
Example
Given
s = 3.16
n = 5
Substitute
SE = 3.16 / √5
Answer
SE ≈ 1.414

2. SE of a mean — s / √n

For a single quantitative variable, plug the sample standard deviation and sample size into s / √n. Quadrupling n halves SE, matching the √n rule you see everywhere in sampling theory.

SE(x̄) = s / √n
Example
Given
s = 15
n = 25
Substitute
SE = 15 / √25
Answer
SE = 3.0

3. SE of a proportion — √(p(1 − p) / n)

For a yes/no outcome the SE uses the sample proportion, not a separate s. The variance p(1 − p) peaks at p = 0.5, so the widest SE for a given n occurs when the outcome is a coin-flip.

SE(p̂) = √( p(1 − p) / n )
Example
Given
p = 0.60
n = 1000
Substitute
√(0.24 / 1000)
Answer
SE ≈ 0.01549

4. SE of a difference — variances add, not SEs

For two independent means, the sampling variances add. That's why you square each SE, sum, then take the square root. This is the Welch (unequal-variance) form and is the safer default. It's exactly what a two-sample t-test uses in the denominator.

SE(x̄₁ − x̄₂) = √( s₁²/n₁ + s₂²/n₂ )
Example
Given
s₁=10
n₁=50
s₂=12
n₂=60
Substitute
√(2.000 + 2.400)
Answer
SE ≈ 2.098

Features of this calculator

  • Three SE tools in one page — mean, proportion, and difference of two means.
  • Sampling-distribution diagram with the ±1 SE region shaded for every result.
  • Show / hide step-by-step working powered by the site's shared solution component.
  • Uses exact formulas — no rounding until the final display.
  • Direct links to the Margin of Error and Confidence Interval calculators to turn SE into a full interval.

Frequently asked questions

+Does standard error assume the data is normal?
No — the formulas above only need the sample statistic to have an approximately normal sampling distribution, which is guaranteed for large enough n by the Central Limit Theorem even when the raw data is skewed. For small n with strongly non-normal data, prefer a bootstrap SE or a distribution-specific method.
+Is the '±1 SE' band on the diagram a confidence interval?
It covers roughly 68% of the sampling distribution — useful as a visual scale, but you'd normally report a 95% interval, which is roughly ±2 SE (±1.96 SE, exactly).
+What if my two samples have very different sizes or variances?
The two-sample formula √(s₁²/n₁ + s₂²/n₂) already accounts for that — it does NOT assume equal variances. That's the Welch (unequal-variance) form and is the safer default.
+Can I use this for the difference of two proportions?
The general principle is the same — SE(p̂₁ − p̂₂) = √(p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂) — but this page focuses on the three most-used forms. Support for the difference-of-proportions form can be added on request.
+Why divide by n − 1 for s but by n for SE?
You don't divide SE by n on its own — SE = s / √n, and s itself already uses n − 1 in its variance formula. The n − 1 is Bessel's correction inside the standard-deviation step; it doesn't reappear when you form the standard error.

Related calculators

Standard Deviation CalculatorConfidence Interval CalculatorMargin of Error CalculatorSample Size Calculator