Sampling Error vs Bias of an Estimator

Sampling error is not an error in the everyday sense, and the name is what trips students up. Even a flawless simple random sample produces a statistic that misses the parameter, purely because a different random sample would have landed somewhere else. If the true population proportion is $p = 0.50$ and one random sample of 100 gives $\hat{p} = 0.46$ (p-hat, the sample proportion), the sampling error for that sample is $0.46 - 0.50 = -0.04$. Sampling error shrinks as $n$ grows, but errors from bad methods, such as undercoverage or nonresponse, are non-sampling errors and a bigger sample will not touch them.

Bias is about the center of a sampling distribution, not its spread: an estimator is biased when its average value across all samples sits above or below the true parameter. The sample mean is unbiased because $E(\bar{x}) = \mu$ (the expected value of x-bar equals mu, the population mean), so it is right on average even though any one sample misses. The range of a sample, by contrast, is biased low, since a sample rarely contains both the population minimum and the population maximum. Collecting more data does not remove bias that comes from how the sample was chosen.