What is margin of error?

The margin of error quantifies the maximum expected difference between the sample estimate and the true population value at a given confidence level.

Should I use 50% for p if I don't know the proportion?

Yes. Using p = 0.50 yields the largest variance and thus the most conservative (largest) margin of error or sample size.

When should I apply finite population correction (FPC)?

Use FPC when sampling without replacement from a relatively small population (common guidance: when n/N ≥ 0.05). It reduces the margin of error.

Is the normal approximation valid for small samples?

The normal approximation is reasonable when n·p ≥ 5 and n·(1−p) ≥ 5. Otherwise consider exact (Clopper–Pearson) or Wilson intervals.

How does confidence level affect margin of error?

Higher confidence levels use larger critical values (z), which increases the margin of error or the required sample size.

Can I calculate sample size from a desired margin of error?

Yes. The tool provides a 'Sample size' mode that computes the minimum n for your target margin of error, confidence level, and optional FPC.

Does population size matter?

Population size matters when it is relatively small compared to the sample (use FPC). Otherwise, margin of error depends mainly on n, p, and the confidence level.

Margin of Error Calculator

This professional-grade margin of error calculator helps researchers, marketers, journalists, and students quantify survey uncertainty or determine the minimum sample size required to achieve a target precision. It supports conservative and user-specified proportions and includes an optional finite population correction (FPC) for small populations.

Calculator

Margin of error Sample size

Confidence level *

Estimated proportion (p) *

Sample size (n) *

Target margin of error (E) *

Finite population correction (FPC)

Use finite population size

Population size (N)

Results

Margin of error (E) —

Confidence interval —

Z critical value —

FPC factor 1.000

Data Source and Methodology

Authoritative references:
1) NIST/SEMATECH e-Handbook of Statistical Methods — Proportions and Confidence Intervals (accessed 2025). https://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm
2) NIST/SEMATECH — Finite Population Correction (FPC). https://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm
3) AAPOR (2016). Standard Definitions: Final Dispositions of Case Codes and Outcome Rates for Surveys, 9th ed. aapor.org

Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.

The Formula Explained

For a proportion p with sample size n and confidence level z:

Inline LaTeX: \( \mathrm{MOE} = z \cdot \sqrt{\dfrac{p(1-p)}{n}} \times \mathrm{FPC} \)

Finite Population Correction (optional): \( \mathrm{FPC} = \sqrt{\dfrac{N-n}{N-1}} \)

Required sample size for target margin of error E: \( n_0 = \dfrac{z^2\,p(1-p)}{E^2}, \quad n = \dfrac{n_0}{1 + \dfrac{n_0 - 1}{N}} \) (with FPC)

Confidence interval for proportion: \( \hat{p} \pm \mathrm{MOE} \)

Glossary of Variables

p — Estimated proportion (between 0 and 1). You can input it as a percentage (e.g., 50%).

n — Sample size (number of observations).

N — Population size (if applying finite population correction).

z — Critical value from the standard normal distribution for the chosen confidence level.

E (MOE) — Margin of error in absolute percentage points (e.g., ±3 pp).

FPC — Finite population correction factor that reduces variance when n/N is not negligible.

Confidence Interval — Range \( [p - E,\, p + E] \) within which the true proportion falls with the chosen confidence.

Come Funziona: Un Esempio Passo-Passo

Suppose you poll n = 400 people in a city of N = 20,000 to estimate the share who support a policy. With p = 0.50 and 95% confidence (z ≈ 1.96):

Compute standard error: \( \sqrt{p(1-p)/n} = \sqrt{0.5 \cdot 0.5 / 400} = 0.025 \).
Compute FPC: \( \sqrt{(N-n)/(N-1)} = \sqrt{(20000-400)/(19999)} \approx 0.990 \).
MOE: \( 1.96 \times 0.025 \times 0.990 \approx 0.0485 \) (4.85 pp).
Confidence Interval: 50% ± 4.85% → [45.15%, 54.85%].

If instead you want E = 3 pp at 95% confidence with FPC: first compute \( n_0 = z^2 p(1-p)/E^2 = 1.96^2 \cdot 0.25 / 0.03^2 \approx 1067.1 \). Then adjust with FPC: \( n = n_0 / \left(1 + \frac{n_0 - 1}{N}\right) \approx 1015.6 \), so you need at least 1,016 respondents.

Frequently Asked Questions (FAQ)

Is this calculator suitable for any survey?

It’s designed for simple random samples measuring a single proportion. For complex designs (e.g., clustering, stratification), incorporate the design effect (DEFF) by inflating n or the MOE accordingly.

What if n·p or n·(1−p) is small?

When these are below about 5, the normal approximation may be inaccurate. Consider Wilson, Agresti–Coull, or exact (Clopper–Pearson) intervals.

How does using FPC change results?

FPC reduces the margin of error when sampling a meaningful fraction of a finite population. As N becomes large relative to n, FPC approaches 1 and has negligible effect.

Why is 50% conservative?

Variance of a Bernoulli variable is p(1−p), maximized at p = 0.5. This gives the largest MOE and largest required n, ensuring a conservative plan.

What units does E use here?

E is an absolute percentage point margin of error. “3” means ±3 percentage points (not relative percent).

Can I use this for means?

This tool is for proportions. For means you need an estimate of the standard deviation and a t- or z-based interval, which is different.

Does nonresponse affect MOE?

MOE is a sampling variability concept. Nonresponse and bias are separate issues; they can invalidate inferences even with a small MOE.

Tool limitations: assumes simple random sampling without replacement and the normal approximation for proportions.

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Last reviewed for accuracy on: September 14, 2025.