Bias of an Estimator vs Variability of an Estimator

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.

Variability describes the spread of an estimator's sampling distribution, so low variability means two different samples would hand you nearly the same answer. It is a separate goal from low bias: an estimator can be centered correctly and still swing wildly, or be very consistent and consistently wrong. Larger samples cut variability, and the square root sets the pace: with $p = 0.5$, $n = 100$ gives $\sqrt{\frac{0.5(0.5)}{100}} = \sqrt{0.0025} = 0.05$, while $n = 400$ gives $\sqrt{\frac{0.25}{400}} = \sqrt{0.000625} = 0.025$. Only a better sampling design fixes bias.