Once you have a sample, the natural next move is to turn it into a single number that stands in for the unknown truth. A point estimate is exactly that: one number, computed from the sample, offered as our best single guess for a population parameter.

Each parameter has its sample-based counterpart:

• the sample mean \bar{x} estimates the population mean \mu;
• the sample standard deviation s estimates the population \sigma;
• the sample proportion \hat{p} estimates the population proportion p.

The hat and the bar are the giveaway: a parameter (\mu, \sigma, p) is a fixed but usually unknown number describing the whole population; an estimate (\bar{x}, s, \hat{p}) is something we actually compute from the data we hold.

Right on average: unbiased

A good estimator is unbiased — it is correct on average over all the samples we might have drawn. For the sample mean,

This does not mean any particular \bar{x} equals \mu. It means that if we repeated the sampling endlessly, the estimates would centre on \mu with no systematic lean to one side. Unbiasedness is a promise about the long run, not about the single sample in your hand.

One estimate is a dot with no error bar

The vertical line is the true \mu; the small dots are estimates from different samples, landing on both sides of it. Slide to draw a fresh sample and watch your estimate jump around \mu — sometimes high, sometimes low, rarely exactly right.

That is the catch with a point estimate: it is a single dot, with no sense of its own error. It never tells you how far off it might plausibly be. To report that uncertainty we need to widen the dot into a range — the idea behind a confidence interval.