Sample vs Statistic

A sample is the set of individuals you actually measure, drawn from the larger population you want to describe. Its size is written $n$. A number computed from a sample is a statistic, such as the sample mean $\bar{x}$ (x-bar) or the sample proportion $\hat{p}$ (p-hat), while the matching fact about the population is a parameter, written $\mu$ (mu) or $p$. The sample is the group of individuals, not the number you compute from them.

Poll 1,000 randomly selected voters and suppose 520 say yes. Then $\hat{p} = 520/1000 = 0.52$, an estimate of the unknown $p$. A different random 1,000 would not land on 0.52 again: the standard error is $\sqrt{\frac{0.52 \times 0.48}{1000}} = 0.0158$, so a repeat poll coming back at 0.50 or 0.54 would be ordinary rather than a sign that something went wrong.

Notice what is absent from that calculation. The population size never enters it, which is where the usual objection fails. "A poll of 1,000 cannot speak for 250 million people" sounds like arithmetic, but precision is governed by $n$, not by the fraction of the population sampled. Those 1,000 voters give the same 0.0158 whether the electorate is 250 million or 250 thousand, provided the sample is a small share of it.

What the formula does assume is that chance chose the sample. None of the above survives a sample that was not random: a self-selected sample of 250,000 has a standard error you can still compute and a bias you cannot, which is the point of does a bigger sample fix bias.

Measuring every member of the population instead is a census, and it returns parameters rather than estimates. Sampling exists because most populations are too large, too expensive, or too scattered for that, and because a careful sample often produces better data than a rushed attempt at everyone.

A statistic is a number computed from sample data alone. Because its value depends on which individuals were drawn, it changes from one sample to the next, and that movement is what separates it from a parameter, which stays put. Statistics take Roman letters or hats: $\bar{x}$ (x-bar) for the sample mean, $s$ for the sample standard deviation, $\hat{p}$ (p-hat) for the sample proportion, $r$ for the sample correlation. Each is paired with the parameter it estimates, $\mu$, $\sigma$, $p$, and $\rho$ in turn.

Suppose 60 percent of a large population would vote yes, so $p = 0.60$. Draw a random sample of 80 and find 52 yes votes, giving $\hat{p} = \frac{52}{80} = 0.65$. Draw a second sample of 80 and find 44, giving $\hat{p} = \frac{44}{80} = 0.55$. Two different statistics, one unchanged parameter, and no mistake in either sample. That spread is sampling variability, and the sampling distribution of $\hat{p}$ describes it.

"The sample proportion is 0.65, so the population proportion is 0.65" is the error to name out loud. A statistic is an estimate and it is almost never exactly right, so the honest version attaches an interval or a standard error to that 0.65. The written form of the same mistake is putting $p = 0.65$ where $\hat{p} = 0.65$ belongs. That collapse wrecks everything downstream, because a test compares an observed $\hat{p}$ against a hypothesized $p$, and no comparison is left once both symbols name one quantity.

A statistic has two lives. Before the sample is drawn it is a random variable with a distribution of its own, which is what makes phrases like the mean and standard deviation of $\hat{p}$ meaningful. Once the sample is in hand it is a single number.

Two boundaries. A value computed from an entire population is not a statistic, however it was calculated, and a statistic from a badly chosen sample is still a statistic, just a poor estimator. Topic 3.1 grades estimators on exactly those two axes: whether they are centered on the parameter, and how much they vary.