Margin of Error Calculator

Compute the margin of error (MOE) and confidence intervals for surveys and mean estimates with customizable confidence levels and Finite Population Correction.

MOE Parameters

Live calculation
Preset Scenarios:

Setting proportion to 50% gives the most conservative (maximum possible) margin of error for any sample size.

Margin of Error (MOE)
±4.99%at 95% confidence
Critical Z-Valuez* = 1.96
Estimate: 50%95% CI: [45.01%, 54.99%]
Sample Size (n)385
FPC Factor1
Base MOE (No FPC)±4.99%
Confidence Level95%

Confidence Level Comparison for Sample n = 385

ConfidenceCritical z*Margin of ErrorRelative Uncertainty
90%1.645±4.19%Narrowest (10% risk)
95%1.96±4.99%Standard Benchmark
99%2.576±6.56%Widest (1% risk)

What Is Margin of Error?

The margin of error (MOE) expresses the maximum expected difference between a sample statistic and the true population parameter at a specified level of statistical confidence.

1. Margin of Error for Proportions (%)
MOE=z* × √[ p(1 − p) / n ](× √[(N − n)/(N − 1)] if FPC)
2. Margin of Error for Population Means
MOE=
z* × σ√n
Step-by-Step Calculation Breakdown (Example: n = 1,000, 95% Confidence, p = 50%)
Step 1: Identify Critical Value (z*) for 95% Confidence
• For 95% confidence (α = 0.05), critical value z* = 1.960
Step 2: Calculate Standard Error of Proportion
• SE = √[ (0.50 × 0.50) ÷ 1000 ] = √[ 0.25 ÷ 1000 ] = √0.00025 = 0.01581 (1.581%)
Step 3: Multiply Critical Value by Standard Error for Final MOE
• MOE = 1.960 × 1.581% = ±3.10%
• 95% Confidence Interval = [50% − 3.10%, 50% + 3.10%] = [46.90%, 53.10%]

Key Drivers of Margin of Error

  • Sample Size (n): Increasing sample size shrinks uncertainty exponentially.
  • Confidence Level (1 − α): Demanding higher confidence widens the interval to ensure coverage.
  • Population Variability (p or σ): For proportions, variability peaks at p = 50%, producing the most conservative (widest) margin of error.

Frequently Asked Questions

What does margin of error represent in polling and research?
Margin of error is the plus-or-minus radius around a sample estimate at a chosen confidence level. For example, if a poll finds 52% support with a ±3% margin of error at 95% confidence, we are 95% confident the true population support lies between 49% and 55%.
How does sample size affect margin of error?
Margin of error decreases in inverse proportion to the square root of sample size (1/√n). To cut your margin of error in half, you must quadruple (4×) your sample size.
Why does higher confidence widen the margin of error?
Higher confidence levels (e.g. 99% vs 90%) require larger critical values (z* = 2.576 vs 1.645) to ensure that more of the sampling distribution is enclosed within the confidence interval.
What is a typical margin of error in national opinion polls?
Most national political and consumer surveys sample approximately 1,000 respondents, yielding a margin of error of roughly ±3.1% at the 95% confidence level for a 50/50 proportion.
When should I apply Finite Population Correction (FPC)?
Apply FPC when your sample represents more than 5% of the total population (n/N > 0.05). FPC accounts for the fact that sampling without replacement from a small population reduces uncertainty.
Can I use this calculator for both percentages and continuous means?
Yes. In Proportion Mode, enter estimated percentage and sample size. In Mean Mode, enter sample standard deviation and sample size.

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