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Margin of Error Calculator

Calculate MOE with Step-by-Step Derivation

Sample Size (n)
Confidence Level
Sample Proportion (p̂)

Margin of Error Formula

For Proportions: MOE = z* × √(p̂(1-p̂)/n)
For Means: MOE = z* × (σ / √n)
z* = critical value (1.645 for 90%, 1.96 for 95%, 2.576 for 99%)

Margin of error measures the precision of a sample estimate compared to the true population value.

For proportions, p̂=0.5 gives maximum MOE. Use p̂=0.5 when unknown for conservative estimate.

What is Margin of Error?

Margin of error (MOE) is a measure of the amount of random sampling error in a survey's results. A larger margin of error indicates less confidence that the sample results accurately reflect the true population parameters.

Confidence Level

Higher confidence = larger MOE

Sample Size

Larger n = smaller MOE

Proportion

p̂=0.5 maximizes MOE

Population Size

Only matters if n > 5% of population

💡 Example: n=500, p̂=0.5, 95% confidence. MOE = 1.96 × √(0.5×0.5/500) = 1.96 × 0.02236 = 0.0438 = 4.38%.

Applications

Surveys Polls Market Research Statistics Quality Control

Frequently Asked Questions

What is margin of error?
Margin of error (MOE) is the amount of random sampling error in survey results. It indicates how much the sample estimate might differ from the true population value.
What is a good margin of error?
MOE under 5% is generally considered good for most surveys. Lower MOE requires larger sample size. Acceptable MOE depends on research requirements.
How sample size affects MOE?
Larger sample size reduces MOE. MOE is inversely proportional to the square root of sample size (n). Doubling n reduces MOE by about 30%.
Why use 95% confidence level?
95% is standard convention in statistics. It means we are 95% confident the true value lies within the confidence interval (sample estimate ± MOE).

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