3.1 Point Estimates and Unbiased Estimators
An Estimate Is Not the Unknown Parameter
A population proportion p is fixed for a defined population. A random sample’s proportion changes from sample to sample. The statistic is a point estimator of the corresponding population parameter; its observed numerical value is a point estimate. For example, if a random sample of 10 customers contains 3 tea purchasers, estimates the true population tea-purchase proportion. This does not establish that the unknown p equals 0.30.
A sample mean similarly estimates a population mean. If four randomly sampled waiting times are 4,5,7 and 8 minutes, their mean is 6 minutes, which is a point estimate of the population mean waiting time. It is an observed sample calculation, not a supplied theoretical population parameter.
Worked Example: Examine Every Possible Sample
Suppose a toy population has four labeled customers: Y₁ and Y₂ buy tea; N₁ and N₂ do not. The population tea proportion is p=2/4=.50. Select two customers by simple random sampling without replacement. There are six equally likely unordered samples.
| Sample | Sample tea proportion |
|---|---|
| Y₁,Y₂ | 1.00 |
| Y₁,N₁ | 0.50 |
| Y₁,N₂ | 0.50 |
| Y₂,N₁ | 0.50 |
| Y₂,N₂ | 0.50 |
| N₁,N₂ | 0.00 |
The estimator’s long-run average across these equally likely samples is
An estimator is unbiased when its average value does not underestimate or overestimate the population parameter. Here individual samples can give 0 or 1, yet their sampling-distribution center is the true 0.50. Unbiasedness does not mean every estimate is accurate or that the estimator has no spread.
Consider the altered estimator . Its average is 0.25, systematically below p=.50; it is biased downward in this population. The justification compares an estimator’s average over sampling with the population parameter. One unusually high or low sample is not enough to establish bias.
A Simulation Can Help, with a Limit
Repeated random samples can approximate an estimator’s distribution, allowing a comparison of its observed center with a known parameter. A small run may have a center above or below an unbiased estimator’s true center through chance. Exact enumeration, as above, or an appropriate theoretical argument can establish the average; a limited simulation provides evidence rather than a guarantee.
Keep three quantities distinct: the fixed parameter, the rule that produces an estimate from any sample, and the observed estimate from one sample. When reporting a point estimate, name the population parameter it targets and retain the observational units. When judging bias, ask about the average behavior of the rule over repeated random sampling—not whether today’s estimate happens to equal the truth.
Try It Yourself
The following original practice applies the ideas above to a fresh setting. Show the reasoning requested in each part. Answers and task-specific rubrics are on the separate solutions page.
Original generated practice for the Fall 2026 course.
Glossary and deeper reading · 8 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Population proportion
A population proportion is the fraction of the entire population with a specified characteristic. It is a parameter, commonly written p, that a sample proportion can estimate. For example, the fraction of all enrolled students who walk to school is the population proportion of walkers.
Read more: Population proportion in the captured Statistics reading →
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Population
A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.
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Mean
The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.
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Point estimate
A point estimate is the specific numerical value obtained from an estimator for the observed sample. It gives one estimate without expressing uncertainty by itself. For example, a sample mean of 14 minutes is a point estimate of the population’s mean travel time.
Read more: Point estimate in the captured Statistics reading →
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Random sampling
Random sampling uses a specified chance process to select individuals from a population. It supports population inference when the sampling frame and data collection are appropriate, but does not guarantee perfect balance or eliminate every source of bias. For example, a random-number generator can select student identifiers from an enrollment list.
Read more: Random sampling in the captured Statistics reading →
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Unbiased estimator
An unbiased estimator has a sampling-distribution mean equal to the parameter being estimated. Individual estimates can still differ substantially from the parameter, so unbiased does not mean perfectly accurate. For example, a properly sampled mean can be unbiased even when small samples produce highly variable estimates.
Read more: Unbiased estimator in the captured Statistics reading →
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Population parameter
A population parameter is a numerical feature of the population or probability model being studied. It is fixed for that defined population, although its value may be unknown. For example, the true proportion of all enrolled students who walk to school is a parameter.
Read more: Population parameter in the captured Statistics reading →
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Sample
A sample is the subset of a population actually studied. How it is selected matters for the conclusions that can be drawn about the population. For example, 60 selected students may be a sample from a school’s full enrollment.