Margin of Error
ME = z* × (σ / √n)
The margin of error quantifies the maximum expected difference between the sample estimate and the true population parameter at a given confidence level. It is the half-width of a confidence interval and determines the precision of the estimate. Smaller margins of error require larger sample sizes.
Variables
The half-width of the confidence interval
The z-value for the desired confidence level (e.g., 1.96 for 95%)
The population standard deviation (or s when σ is unknown)
The number of observations in the sample
Example Calculation
Scenario
A political poll surveys 1000 voters. The population standard deviation of support is estimated at 0.5 (proportion scale). What is the margin of error at 95% confidence?
Given Data
Calculation
ME = 1.96 × (0.5 / √1000) = 1.96 × (0.5 / 31.62) = 1.96 × 0.01581
Result
ME = 0.031 or about 3.1%
Interpretation
The poll result is accurate to within plus or minus 3.1 percentage points at the 95% confidence level. If 52% of surveyed voters support a candidate, the true support is likely between 48.9% and 55.1%.
When to Use This Formula
- ✓Determining how precise a sample estimate is
- ✓Computing the required sample size for a desired level of precision
- ✓Reporting uncertainty in polls, surveys, and research studies
- ✓Comparing precision across different surveys or studies
Common Mistakes
- ✗Confusing margin of error with standard error (ME = z* times the standard error)
- ✗Thinking that doubling the sample size halves the margin of error (it only reduces it by a factor of √2)
- ✗Using the margin of error formula for proportions without the appropriate standard error formula
- ✗Ignoring that margin of error only accounts for sampling error, not bias or measurement error
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Common questions about this formula
Rearrange the formula: n = (z* × σ / ME)². For example, to achieve a margin of error of 2% at 95% confidence with σ = 0.5: n = (1.96 × 0.5 / 0.02)² = 49² = 2401. Always round up to ensure the margin of error is at most the desired value.
Larger samples provide more information about the population, reducing the variability of the sample mean. The standard error (σ/√n) decreases as n increases because the square root function grows, causing the estimate to be more precise.