Point Estimates and Their Variability

A point estimate is one value computed from your sample data that stands in for an unknown population parameter. For a population proportion, the point estimate is the sample proportion: p-hat = (number with the trait) / (sample size). The trouble is, a different sample would almost certainly give a different p-hat. That's sampling variability, and it's why a lone number like 'p-hat = 0.33' feels incomplete: it says nothing about how far off it might be. This is the whole reason confidence intervals exist later in this unit — they attach a margin of error to the point estimate so you can say how much you trust it.

Unbiasedness and variability are separate properties, and mixing them up causes real confusion. An estimator is unbiased if, averaged over many, many samples, its values center on the true parameter — no systematic pull in one direction. But two unbiased estimators can differ wildly in how spread out their values are from sample to sample. One might cluster tightly around the truth, the other might swing wide, and both still average out correctly in the long run. A good estimator needs both: centered on the truth, and tightly clustered there.

Sample size connects directly here: larger samples shrink the typical spread of p-hat across hypothetical repeated samples, which is why bigger studies are trusted more. But that's a statement about long-run behavior, not about any single sample you actually collected. This sets up the next idea in the unit: quantifying that spread using standard error, then building it into confidence intervals and test statistics.