Mathwords logoMathwords

Sampling Error — Definition, Formula & Examples

Sampling error is the difference between a value you calculate from a sample, such as a sample mean, and the true value for the whole population. It happens by chance, because a sample contains only some members of the population.

Let θ^\hat{\theta} be a statistic computed from a random sample, used to estimate a population parameter θ\theta. The sampling error of that sample is θ^−θ\hat{\theta} - \theta. This error is a random variable that changes from one sample to the next. For an unbiased estimator its expected value is 00. Its typical size is measured by the standard error, the standard deviation of the sampling distribution of θ^\hat{\theta}.

Key Formula

SExˉ=σnSE_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
Where:
  • SExˉSE_{\bar{x}} = standard error of the sample mean, the typical size of the sampling error
  • σ\sigma = population standard deviation
  • nn = sample size

Worked Example

Problem: A college has students whose mean weekly study time is μ=10\mu = 10 hours, with population standard deviation σ=5\sigma = 5 hours. You take a random sample of n=25n = 25 students and get a sample mean of xˉ=11.2\bar{x} = 11.2 hours. Find the sampling error for this sample and the standard error. Then find the standard error for a sample of 100100 students.
Step 1: The sampling error is the sample mean minus the population mean.
xˉ−μ=11.2−10=1.2 hours\bar{x} - \mu = 11.2 - 10 = 1.2 \text{ hours}
Step 2: The standard error tells you how large the sampling error typically is for samples of size 25.
SExˉ=525=55=1 hourSE_{\bar{x}} = \frac{5}{\sqrt{25}} = \frac{5}{5} = 1 \text{ hour}
Step 3: Quadruple the sample size to 100. Because nn sits under a square root, the standard error is cut in half.
SExˉ=5100=510=0.5 hoursSE_{\bar{x}} = \frac{5}{\sqrt{100}} = \frac{5}{10} = 0.5 \text{ hours}
Answer: This sample's sampling error is 1.21.2 hours, a little larger than the typical 11 hour for n=25n = 25. With n=100n = 100, the standard error falls to 0.50.5 hours.

Why It Matters

Margins of error in election polls are built directly from the standard error. In an introductory statistics course, confidence intervals and hypothesis tests both rely on measuring how much sampling error to expect. Researchers use the formula σ/n\sigma/\sqrt{n} to choose a sample size large enough to make that error acceptably small.

Common Mistakes

Mistake: Treating sampling error and sampling bias as the same thing, or believing a bigger sample fixes both.
Correction: Sampling error is random. It averages out across samples, and it shrinks as nn grows. Bias is systematic and comes from a flawed method, such as convenience sampling. A larger biased sample is still wrong in the same direction.

Related Terms