Point Estimate vs Parameter

A point estimate is the value a statistic takes in the one sample you collected, offered as the estimate of a parameter you cannot observe. Each parameter has a standard partner: $\bar{x}$ (x-bar) estimates $\mu$ (mu), $\hat{p}$ (p-hat) estimates $p$, and $s$ estimates $\sigma$ (sigma). The word point is doing work in that phrase. It signals a single value with no width attached to it.

If 63 of 200 randomly sampled voters approve, the point estimate is $\hat{p} = \frac{63}{200} = 0.315$. That number alone says nothing about how far off it might be. Attach a margin of error and the picture changes: the standard error is $\sqrt{\frac{0.315(0.685)}{200}} = 0.0328$, so with $z^* = 1.96$ the margin is 0.064 and the 95% interval runs from 0.251 to 0.379. An estimate that looked exact carries a span of nearly 13 points around it.

The misreading is a missing word. "Thirty-one and a half percent of voters approve" states a sample statistic as though it were the population parameter. It is one draw from a sampling distribution centered at $p$, and the chance it equals $p$ exactly is essentially zero. Notation carries the distinction, so writing $p$ where you computed $\hat{p}$ claims something you did not compute.

An unbiased point estimate is still allowed to be far off. Unbiasedness is a property of the procedure across all samples, not a promise about this one, so knowing that $\bar{x}$ is unbiased does not license treating $\bar{x}$ as $\mu$. For a difference, the point estimate is the difference of the two point estimates, $\hat{p}_1 - \hat{p}_2$ or $\bar{x}_1 - \bar{x}_2$, and it sits at the center of the matching interval.

Topic 3.1 is Estimators, where point estimates are judged on bias and on variability.

A parameter is a number attached to a population. It is fixed: it does not change when you draw a different sample, and it does not change because you failed to measure it. What is missing is your knowledge of the value, not the value. The symbols are mostly Greek: $\mu$ (mu) for the population mean, $\sigma$ (sigma) for the standard deviation, $\rho$ (rho) for the correlation. The proportion is the exception, since the population value is the Roman letter $p$ and the sample value is $\hat{p}$ (p-hat).

A company has exactly five regional offices, staffed by 12, 15, 18, 20, and 25 people. If the population is those five offices, then $\mu = \frac{90}{5} = 18$ employees, and the population standard deviation divides the squared deviations by $N = 5$, giving $\sigma = \sqrt{\frac{98}{5}} = 4.43$. Both are parameters and both are known, which is unusual and entirely legitimate.

Two sentences to unlearn: "a parameter is a number you cannot know" and "if I calculated it, it must be a statistic." The label comes from which group the number describes, not from whether the value is available and not from the group's size. Treat the same five offices as a random sample of 60 offices nationwide and the same data give $\bar{x} = 18$ and $s = \sqrt{\frac{98}{4}} = 4.95$, dividing by $n - 1 = 4$. Same numbers in, different label out, and a different divisor for the spread.

Which one you hold depends on how the population was defined, and that comes from the question asked rather than from the spreadsheet in front of you. The mean of 500 rows is a parameter when those 500 are the whole group you want to describe and a statistic when they stand in for a larger one.

Inference is written about parameters: a confidence interval brackets one, and a hypothesis states a value for one, which is why $H_0: p = 0.5$ is well formed and $H_0: \hat{p} = 0.5$ is not. The sample value is already in front of you, so there is nothing left to hypothesize.