Margin of Error Calculator

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Statistics

Margin of Error Calculator

Calculate the margin of error for sample estimates. Essential for surveys, polls, and statistical inference.

Input Parameters

Or 0.5 for conservative estimate

Results

Margin of Error (MOE)
±0.0490
±4.90 percentage points
Z-score
1.96
Sample Prop (p̂)
0.5000
95% Confidence Interval
[0.4510, 0.5490]
Interpretation
The true population proportion lies within ±4.90 percentage points of the sample estimate with 95% confidence.

What is Margin of Error?

Margin of error (MOE) is the range of uncertainty around a sample estimate. It quantifies how much the sample result might differ from the true population value.

MOE = z × SE = z × (σ / √n) For proportions: MOE = z × √[p(1−p) / n]

Key Concepts:

  • Confidence interval: Sample estimate ± MOE gives the likely range for the population parameter.
  • Confidence level: Higher confidence (95% vs 90%) produces wider MOE.
  • Sample size: Larger samples yield smaller MOE (proportional to 1/√n).
  • Variability: More variable data (higher σ or p closer to 0.5) yields larger MOE.
  • Uses: Surveys, polls, quality control, clinical trials, market research.

Example: A poll showing "52% approve ±3%" at 95% confidence means we're 95% confident the true approval rate is between 49% and 55%.

How to Calculate Margin of Error

For Proportions:

1

Choose your confidence level (90%, 95%, or 99%) → get z-score (1.645, 1.96, 2.576)

2

Enter sample size (n) and sample proportion (p̂)

3

Calculate: MOE = z × √[p(1−p) / n]

For Means:

1

Choose confidence level → get z-score

2

Enter sample size (n) and standard deviation (σ)

3

Calculate: MOE = z × (σ / √n)

Example Calculations

Proportion Example: Online Poll

Survey: 400 people, 52% favor new policy Confidence level: 95% (z = 1.96) p = 0.52, n = 400 MOE = 1.96 × √(0.52 × 0.48 / 400) = 1.96 × √(0.2496 / 400) = 1.96 × √0.000624 = 1.96 × 0.0249 ≈ ±0.0489 (≈ ±4.89 percentage points) Result: 52% ±4.89% means true support is likely between 47.11% and 56.89%

Mean Example: Height Measurement

Sample: 100 people, mean = 68 inches Standard deviation: 4 inches Confidence level: 95% (z = 1.96) MOE = 1.96 × (4 / √100) = 1.96 × (4 / 10) = 1.96 × 0.4 ≈ ±0.784 inches Result: Average height is 68 ±0.784 inches (95% confident true mean is between 67.216 and 68.784 inches)

Frequently Asked Questions

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