You want to know the average height of every adult in the country, or the exact share of voters who back a candidate. You cannot measure everyone — so you take one sample and you are left holding a single fistful of data. From that, you must produce your best single guess at the unknown truth. That guess is a point estimate.

A point estimate is exactly that: one number, computed from the sample, offered as our best single answer for a population parameter. The sample mean is your best guess for the population mean; the sample proportion is your best guess for the population proportion. It is honest, it is useful — but it is quietly incomplete, because a single number carries no hint of how uncertain it is. (Fixing that gap is the whole point of a confidence interval, coming up next.)

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.

Two things make an estimator good

Not all guesses are equal. Statisticians ask two questions of any estimator, and a good one answers yes to both:

• Is it unbiased? — does it hit the target on average? An unbiased estimator has no systematic lean; over many samples its estimates centre exactly on the true parameter, neither too high nor too low as a rule.
• Is it precise (low variance)? — do its estimates cluster tightly, or scatter all over the place? Precision is measured by the standard error: a smaller standard error means the estimate wobbles less from sample to sample.

Picture darts. An unbiased but imprecise shooter scatters darts all round the bullseye — centred right, but wildly spread. A biased but precise shooter lands a tight cluster, but off in the corner. The estimator you want is both: centred on the truth and tightly clustered. Bigger samples buy you precision — the standard error \sigma/\sqrt{n} shrinks as n grows — but they cannot fix a bias.

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. It is precisely because the mean of every possible sample averages out to \mu that we trust \bar{x} as our point estimate at all.

Worked example 1 — a point estimate of a mean

A biologist weighs eight randomly caught trout (in grams): 420,\ 455,\ 390,\ 510,\ 470,\ 430,\ 485,\ 440. What is the point estimate of the mean weight of all trout in the lake?

Add them and divide by n = 8:

So the point estimate of \mu is 450 g. That single number is our best guess for the whole lake — but it is almost certainly not the exact truth, and on its own it says nothing about how far off it might be.

Worked example 2 — a point estimate of a proportion

A pollster asks 800 randomly chosen voters whether they support a candidate; 344 say yes. The point estimate of the population support p is the sample proportion:

Now here is the danger. A headline that reads simply "43% support" is missing the most important half of the story. With a sample of 800, the true figure could comfortably be 40% or 46% — an election-deciding difference. The bare point estimate 0.43 means very little until it is paired with a margin of error, which turns it into a confidence interval like "43% ± 3%." The point estimate is the centre of the dartboard; the margin of error tells you how big the board is.

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.