Calculate standard error and margin of error for sample estimates with optional finite population correction.
Last updated: March 2026
Sampling error is the difference between a sample statistic and the true population parameter it estimates. It arises naturally whenever you sample from a population—no sample is perfect. Unlike bias (which is systematic and preventable), sampling error is random and always present to some degree.
The margin of error (MOE) quantifies sampling error: it specifies the range around your sample estimate where the true population value likely falls, given a chosen confidence level (typically 95%). For example, a poll reporting "45% support ±3% at 95% confidence" means they're 95% confident the true population support is between 42% and 48%.
Standard error measures the typical variation of sample statistics. Smaller standard errors (larger samples, less variability) produce tighter confidence intervals and more reliable estimates. The finite population correction applies when sampling significantly from a finite population, reducing standard error slightly.
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MOE is inversely proportional to √n. Doubling sample size reduces MOE by ~29%. Quadrupling sample size halves MOE. Larger samples = more precise estimates.
Use it when your sample size exceeds ~5% of the population. For large populations (millions), FPC has minimal impact. It slightly reduces SE for finite populations.
Proportion SE depends on p×(1-p); peak variability at p=0.5. Mean SE depends on population std dev. Use proportions for yes/no questions, means for continuous measurements.
95% is the statistical standard. 90% gives narrower MOE (less certainty), 99% gives wider MOE (more certainty). Choose based on your risk tolerance and precision needs.
Limited options: higher confidence increases MOE, lower confidence decreases it. For proportions, MOE is lowest near 0% or 100% (more certain outcomes). Generally, larger samples are needed.
Sampling error and bias are different. Sampling error is random; bias is systematic. Larger samples reduce sampling error but NOT bias. You must address bias through study design.
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