One number is unknown, the other is measured

A parameter is a number describing a population. It has one value, that value does not change when you take a sample, and in a real problem you never learn it. A statistic is a number computed from a sample. It is known exactly once the data are in, and it would come out differently on the next sample.

An estimator is a statistic used to estimate a parameter, and the value it produces on a particular sample is a point estimate. If 62% of 400 surveyed voters support a measure, then $\hat{p} = 0.62$ is the point estimate and $p$, the proportion of all voters who support it, is what is being estimated.

The sentence that gets this backwards is common enough to name: "the parameter is 0.62 because that is what the sample gave". The sample gave the statistic. The parameter was 0.62 or was not, before the survey and after it, and the survey's job is to say what values of $p$ are plausible.

Each parameter has a matching statistic, and the symbols are not interchangeable:

1. Proportion: parameter $p$, statistic $\hat{p}$.
2. Mean: parameter $\mu$, statistic $\bar{x}$.
3. Standard deviation: parameter $\sigma$, statistic $s$.

Greek letters and unadorned symbols belong to the population; hats and Roman letters belong to the sample. Writing $\hat{p}$ where $p$ belongs turns a claim about a population into a claim about the data already in hand, which is never in question. This matters most in two places later: a null hypothesis is always about a parameter, and a confidence interval always estimates a parameter, so a $\hat{p}$ appearing in either one is a signal that the sentence has drifted off its subject.

An estimator is unbiased when the mean of its sampling distribution equals the parameter: across all possible samples of a given size, the estimates center on the target with no systematic tendency to land high or low. For a simple random sample, $\hat{p}$ is unbiased for $p$, since $\mu_{\hat{p}} = p$.

Two readings of that definition fail:

1. A single miss is not bias. Every sample misses by something; that is variability. Bias asks whether the misses cancel over many samples or pile up on one side.
2. Unbiased does not mean accurate. An unbiased estimator with enormous variability can be far off on any given sample, and it is still unbiased.

Bias comes from the data collection or the estimator's construction, not from bad luck. A voluntary-response survey produces a $\hat{p}$ whose sampling distribution is centered away from $p$, and taking a larger voluntary-response sample tightens that distribution around the wrong value. More data cannot fix bias.

The variability of an estimator is the spread of its sampling distribution: how much the estimate jumps from sample to sample. Low variability means repeated samples give similar answers.

Bias and variability are independent properties, which is why the standard picture is four targets: shots centered on the bullseye and tight (unbiased, low variability, the goal), centered but scattered (unbiased, high variability), tight but clustered off-center (biased, low variability, the dangerous one), and scattered off-center (biased and variable).

Only variability responds to sample size. A larger simple random sample narrows the sampling distribution of $\hat{p}$ around $p$, which is the entire mechanism behind every interval and test that follows. The biased-but-tight target is the one to fear precisely because a large sample makes it look authoritative: consistent, repeatable, and wrong.

Comparing two estimators of the same parameter therefore takes both questions. Is each one centered on the target, and if both are, which has the smaller spread. An estimator that is unbiased with lower variability beats an unbiased one with higher variability, and a biased estimator is usually out of the running whatever its spread.