Sampling error

Sampling error is the gap between a statistic and the parameter it estimates, for one particular sample: $\hat{p} - p$ (p-hat minus p) for a proportion, or $\bar{x} - \mu$ (x-bar minus mu) for a mean. The word error is used in its statistical sense of distance from a target, not in the everyday sense of a blunder. It exists because a sample is only part of a population, so a flawless simple random sample still lands somewhere other than the truth, and it can come out positive or negative.

Suppose the true population proportion is $p = 0.50$ and one random sample of 100 gives $\hat{p} = 0.46$. The sampling error is $0.46 - 0.50 = -0.04$. Judge that against the spread it came from: the standard deviation of $\hat{p}$ here is $\sqrt{0.5(0.5)/100} = 0.05$, so the sample missed by 0.8 standard deviations. Counting exactly, about 48 percent of all samples of 100 miss by at least 0.04. Nothing went wrong in this sample; roughly half of them do this.

The misreading is right there in the name: "the poll had a sampling error, so the pollsters made a mistake." No. The companion version is "we eliminated sampling error by being careful." Care does not touch it. Only a larger $n$ shrinks it, and only a census removes it, which is why every honest poll publishes a number quantifying how big its sampling error is likely to be.

In practice you can never compute it, because computing $\hat{p} - p$ would require knowing $p$, and if you knew $p$ you would not be sampling. What gets reported instead is a margin of error, which bounds the likely size of the sampling error at a stated confidence level.

One boundary. Sampling error averages out to zero across repeated samples when the estimator is unbiased, so it is variation rather than a lean. Bias does not average out, and neither do non-sampling errors such as undercoverage, nonresponse, or a leading question. A bigger sample shrinks the first and leaves the others exactly where they were.