Why is the SE of a proportion largest at p̂ = 0.5?

The formula SE = √(p̂(1−p̂)/n) contains the product p̂(1−p̂), which is maximized at p̂ = 0.5 (where it equals 0.25). At extreme proportions like 0.01 or 0.99, the product is much smaller (0.0099), giving a smaller SE. This is why polls assume p̂ = 0.5 for worst-case sample size calculations.

Why do we add the variances instead of subtracting?

When two random variables are independent, the variance of their difference equals the SUM of their variances: Var(X₁ − X₂) = Var(X₁) + Var(X₂). This is because variability in either group contributes to uncertainty in the difference, regardless of direction.

Should I use pooled or unpooled SE for comparing proportions?

For hypothesis testing (H₀: p₁ = p₂), use the pooled SE where p̂ = (x₁+x₂)/(n₁+n₂). For confidence intervals for the difference, use the unpooled SE (each proportion's own variance). This calculator uses the unpooled formula, which is appropriate for CIs and general-purpose SE estimation.

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Standard Error Calculator

Calculate standard error for means, proportions, and differences with step-by-step solutions and Python code.

Enter Data

Sample Std Dev (s)

Sample Size (n)

Sample Data:

Enter your data and click Calculate SE.

Quick Formulas:

SE of Mean: s / √n
SE of Proportion: √(p̂(1−p̂)/n)
SE of Diff. Means: √(s₁²/n₁ + s₂²/n₂)

Frequently Asked Questions

What is the difference between standard deviation and standard error?

Standard deviation (SD) measures the spread of individual observations in your sample. Standard error (SE) measures the precision of the sample mean as an estimate of the population mean. SE = SD / √n, so SE is always smaller than SD and decreases with larger samples.

Why does standard error decrease with larger sample size?

Larger samples provide more information about the population, making the sample mean a more precise estimate. Mathematically, SE = s/√n, so doubling n reduces SE by a factor of √2 (≈41%). To halve the SE, you need 4× the sample size.

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CI for means and proportions

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Mean, median, std deviation