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Estimators

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A parameter ($p$, $\mu$, $\sigma$) is a fixed unknown number describing a population; a statistic ($\hat{p}$, $\bar{x}$, $s$) is computed from a sample, known exactly, and different on the next sample. An estimator is a statistic used to estimate a parameter, and its point estimate is the value it takes on one sample. It is unbiased when the mean of its sampling distribution equals the parameter, which is true of $\hat{p}$ under simple random sampling, and its variability is the spread of that same sampling distribution. The two properties are independent, which is what the four-target picture shows, and only variability responds to sample size.

The failures are all confusions of the two objects. The parameter gets reported as whatever the sample produced, so "the parameter is 0.62" replaces "the estimate is 0.62 and $p$ is unknown". A single sample landing away from the truth gets called bias, when it is ordinary variability. Unbiased gets read as accurate, though an unbiased estimator with a wide sampling distribution can miss badly on any one sample. And a larger sample is offered as the fix for a biased design, when a bigger voluntary-response survey only tightens the distribution around the wrong value.

two columns that never trade places quantity parameter statistic proportion p p-hat mean mu x-bar std deviation sigma s fixed, unknown, one per population known, varying, new one per sample
The hat is the whole distinction. A null hypothesis and a confidence interval both speak about the left column, so a symbol from the right column appearing in either is a sentence that has changed subject.
unbiased, low spread what every method wants unbiased, high spread centered, but any one sample can miss badly biased, low spread: the dangerous one biased and variable
Bias and variability vary independently. A larger sample tightens every cluster and moves none of them, so the bottom-left target only becomes more convincing as the data grow.

The work

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Lesson
Estimators ›

Separates the fixed unknown parameter from the statistic a sample supplies, fixes the notation that carries the difference, defines bias as the centering of a sampling distribution rather than one sample's miss, and sets variability beside it as the property sample size controls.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items on estimators: parameters reported as whatever the sample gave, one sample's miss called bias, unbiased read as accurate, and a larger sample offered as the cure for a biased design. Take it cold to find your habit, or after the lesson to check it is gone.

Not started · 10 items · ~15 min
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Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.

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