What does this calculator do?

Compute statistical results from your inputs.

Enter values in the input fields and view results.

What are the formulas?

See the educational content section for formulas.

For statistical analysis and computation.

Check the related calculators section below.

Results follow standard statistical methods.

4 more

STATISTICSSampling DistributionsStatistics Calculator

📊

Standard Error Calculator

Free standard error calculator. SE of mean, proportion, difference of means, difference of proportio

Run CalculatorExplore data analysis and statistical calculations

Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

STATISTICSSampling Distributions

Standard Error — SE of Mean, Proportion, Diff of Means, Diff of Proportions, Correlation, Regression

Compute SE for any statistic. 95% confidence intervals. t or z for small samples. Step-by-step breakdown with interactive visualization.

σ_x̄ = σ/√n →Confidence Intervals →

Real-World Scenarios — Click to Load

SE Mode

Sample SD (s)

Sample size (n)

Sample mean X̄ (for CI)

se_results.sh

CALCULATED

$ standard_error --mode="mean"

Standard Error

2.190890

95% CI Lower

70.7059

95% CI Upper

79.2941

Using

z (1.96)

Standard Error Result

SE of Mean

SE = 2.190890

95% CI: [70.706, 79.294]z

numbervibe.com/calculators/statistics/standard-error-calculator

Sampling Distribution with 95% CI

SE vs Sample Size (n)

SE Formulas by Statistic Type

Calculation Breakdown

COMPUTATION

Standard Error

2.190890

SE = s/√n = 12/√30

Critical value

z* = 1.96

z (n≥30)

95% CI Lower

70.7059

X̄ − 1.96 × SE

95% CI Upper

79.2941

X̄ + 1.96 × SE

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

• SE of mean: SE = s/√n — precision of sample mean as estimate of μ
• SE of proportion: SE = √(p̂(1−p̂)/n) — max when p̂=0.5
• SE of diff of means: SE = √(s₁²/n₁ + s₂²/n₂) — for two independent groups
• SE of diff of proportions: SE = √(p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂)
• SE of correlation: SE_r = √((1−r²)/(n−2)) — depends on r
• SE of regression: SE_b = √(MSE/SSX) — for testing slope = 0
• 95% CI: statistic ± 1.96×SE (or t×SE for n<30 with mean)

Did You Know?

📐Standard error measures the precision of an estimate. Smaller SE = more precise estimate.Source: NIST

📉For the mean, SE = s/√n. Doubling n reduces SE by about 29%.Source: Khan Academy

🗳️Poll margins of error use SE of proportion. n=1000, p=0.5 → ME ≈ ±3% at 95%.Source: Gallup

💊Two-sample t-tests use SE of difference of means. Equal n optimizes power for given total N.Source: FDA

📈SE of correlation depends on r. When r≈0, SE_r ≈ 1/√(n−2). When r≈±1, SE_r ≈ 0.Source: Wolfram

🧪For small samples (n<30), use t-distribution: CI = X̄ ± t* × SE instead of z.Source: OpenIntro

How Each SE Formula Works

SE of Mean

SE = s/√n. The standard deviation of the sampling distribution of X̄. Use for confidence intervals and hypothesis tests about μ.

SE of Proportion

SE = √(p̂(1−p̂)/n). For binary outcomes. Max when p̂=0.5. Normality: np≥10, n(1−p)≥10.

SE of Difference of Means

SE = √(s₁²/n₁ + s₂²/n₂). For comparing two independent groups. Assumes independence.

SE of Difference of Proportions

SE = √(p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂). For A/B tests, conversion rates, etc.

SE of Correlation

SE_r = √((1−r²)/(n−2)). For testing H₀: ρ=0 or constructing CI for ρ. Assumes bivariate normal.

SE of Median

SE ≈ 1.2533 × s/√n for normal populations. The factor 1.2533 ≈ π/2.5 relates median to mean SD.

SE of Regression Coefficient

SE_b = √(MSE/SSX). MSE = mean squared error, SSX = Σ(xᵢ−x̄)². For testing slope = 0.

Expert Tips

Small samples

Use t-distribution for n<30 when estimating mean. t* > 1.96, so CI is wider.

Proportions

When p is near 0 or 1, need larger n for normal approximation. Use exact binomial if np<10.

Two groups

For equal total N, equal n₁=n₂ gives smallest SE of difference (best power).

Correlation

SE of r is not symmetric: it depends on r. Use Fisher z-transformation for precise CI.

Formulas Summary

Statistic	Standard Error
Mean	s/√n
Proportion	√(p̂(1−p̂)/n)
Diff of means	√(s₁²/n₁ + s₂²/n₂)
Diff of proportions	√(p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂)
Correlation	√((1−r²)/(n−2))
Median	1.2533 × s/√n
Regression slope	√(MSE/SSX)

Frequently Asked Questions

What is the difference between standard deviation and standard error?

SD measures spread of data. SE measures precision of an estimate (e.g., sample mean). SE = SD/√n for the mean.

When do I use t instead of z for the confidence interval?

Use t when n<30 and σ is unknown (use s). For n≥30, z and t are very similar.

How do I interpret the 95% CI?

We are 95% confident the true parameter lies in the interval. The interval width is 2×1.96×SE (or 2×t*×SE).

Why does SE of proportion use p̂(1−p̂)?

The variance of a binomial proportion is p(1−p)/n. We estimate p with p̂. Max variance at p=0.5.

Can I use these formulas for non-random samples?

No. SE formulas assume random sampling. Convenience or voluntary samples can produce biased estimates and invalid inference.

SE at a Glance

s/√n

Mean

√(pq/n)

Proportion

±1.96×SE

95% CI

t* for n<30

Small sample

Official Data Sources

NIST Engineering Statistics Handbook ↗

Definitive reference for standard error formulas

Updated: 2024

Wolfram MathWorld - Standard Error ↗

Mathematical formulation of SE

Updated: 2024

Khan Academy - Sampling Distributions ↗

Interactive SE lessons

Updated: 2025

OpenIntro Statistics ↗

Free open-source statistics textbook (4th ed.)

Updated: 2024

Rice Virtual Lab in Statistics ↗

Simulations and applets for sampling

Updated: 2024

Disclaimer: This calculator provides standard errors for educational purposes. Verify assumptions (random sampling, normality when applicable) for each statistic. For small samples, use t-distribution for means. For proportions with np<10, consider exact methods.

👈 START HERE

⬅️Jump in and explore the concept!

Standard Error Calculator

Why This Statistical Analysis Matters

Standard Error — SE of Mean, Proportion, Diff of Means, Diff of Proportions, Correlation, Regression

Real-World Scenarios — Click to Load

SE Mode

Sampling Distribution with 95% CI

SE vs Sample Size (n)

SE Formulas by Statistic Type

Calculation Breakdown

Key Takeaways

Did You Know?

How Each SE Formula Works

SE of Mean

SE of Proportion

SE of Difference of Means

SE of Difference of Proportions

SE of Correlation

SE of Median

SE of Regression Coefficient

Expert Tips

Small samples

Proportions

Two groups

Correlation

Formulas Summary

Frequently Asked Questions

What is the difference between standard deviation and standard error?

When do I use t instead of z for the confidence interval?

How do I interpret the 95% CI?

Why does SE of proportion use p̂(1−p̂)?

Can I use these formulas for non-random samples?

SE at a Glance

Official Data Sources

Related Statistics Calculators

Related Calculators

Standard Deviation of Sample Mean Calculator

Sampling Distribution of the Sample Proportion Calculator

Sampling Error Calculator

Relative Standard Error Calculator

Sample Size Calculator

Normal Probability Calculator for Sampling Distributions

We Value Your Privacy