What Is Sampling Error?
Sampling error is the natural difference between an estimate computed from a sample and the true value in the whole population. Because you observe only part of the population, your sample proportion is almost never exactly equal to the population proportion.
The size of the sampling error shrinks as the sample grows. For a proportion, the standard error is the square root of p̂(1 − p̂)/n, and a common summary is the 95% margin of error, which is about 1.96 times that standard error. When the proportion is unknown, using p̂ = 0.5 gives the largest, most conservative estimate of error.
Formula
Standard error (SE) = √( p̂(1 − p̂) / n )
Margin of error (95%) ≈ 1.96 × SE
where:
p̂ = sample proportion
n = sample size
Applications
- Reporting the margin of error for an opinion poll
- Determining how large a survey sample must be for a target precision
- Interpreting the reliability of acceptance-sampling results
- Adding error bars to survey-based estimates of proportions
Sources
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2021). Introduction to the Practice of Statistics (10th ed.). Macmillan.
- Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (7th ed.). Wiley.