Margin of Error Calculator
Calculate the margin of error for a survey proportion given sample size and confidence level.
How to use this tool
- Enter sample size (n), proportion (p) and z* (confidence level) in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your margin of error and the full breakdown beneath it.
Calculate margin of error: MOE = z* × √(p(1−p)/n). Commonly reported as ±X% in survey results.
Formula
Standard error: SE = √( p × (1 − p) / n )
Margin of error: MOE = z* × SE
As a percentage: MOE% = MOE × 100
How it works
This calculator computes the margin of error for a survey proportion, quantifying the uncertainty around a sample estimate at a given confidence level.
It first calculates the standard error of the proportion, then multiplies by the z* critical value. The result assumes simple random sampling and applies best when the sample is large enough for the normal approximation to hold (typically n × p ≥ 5 and n × (1 − p) ≥ 5).
Worked example
Worked example
- Inputs: sample size n = 1000, proportion p = 0.5, confidence level z* = 1.96 (95%).
- Standard error: SE = √(0.5 × 0.5 / 1000) = √(0.00025) ≈ 0.015811.
- Margin of error: MOE = 1.96 × 0.015811 ≈ 0.031.
- As a percentage: MOE% = 0.031 × 100 = 3.1%.
Margin of error = 0.031 (3.1%)
Key terms
- Standard error (SE)
- The estimated standard deviation of the sampling distribution of a proportion; measures how much the sample proportion is expected to vary from sample to sample.
- Margin of error (MOE)
- Half the width of a confidence interval; the maximum likely difference between the sample proportion and the true population proportion.
- Confidence interval
- A range of values — computed as p ± MOE — that is expected to contain the true population proportion with the stated probability.
- Z* (critical value)
- The standard normal quantile corresponding to the chosen confidence level (1.96 for 95%, 2.576 for 99%).
- Normal approximation
- The use of the normal distribution to approximate the binomial distribution of a proportion, valid when n is large and p is not too close to 0 or 1.
Frequently asked questions
- What is the margin of error?
- The margin of error quantifies the uncertainty in a sample proportion estimate. A MOE of ±3% at 95% confidence means the true proportion is within 3 percentage points of the observed proportion 95% of the time.