AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/3/3-1)
Unit 3 · Topic 3.1
3.1 Estimators
An estimator is a statistic used to estimate a parameter, like p̂ for p. A good estimator is unbiased: its sampling distribution is centered on the true value, so it doesn't systematically run high or low. This topic also separates bias from variability, two ideas students often blur.
Key terms
- estimator
- point estimate
- unbiased estimator
- biased estimator
Estimators and point estimates
An estimator is a rule for estimating a parameter from sample data. The sample proportion p̂ estimates the population proportion p, and the sample mean x̄ estimates the population mean μ.
The single number you get from one sample is a point estimate. If 132 of 400 randomly chosen adults say they read news daily, the point estimate is p̂ = 132/400 = 0.33. A point estimate is your best single guess, but it's almost never exactly right.
Unbiased estimators
An estimator is unbiased if, averaged over all possible random samples, it equals the parameter. In picture form, its sampling distribution is centered exactly at the true value. p̂ and x̄ are both unbiased when the sample is random.
An estimator is biased if its sampling distribution is centered above or below the parameter. The sample range, for example, almost always comes out smaller than the population range, because a sample rarely includes the population's very smallest and very largest values. It's a biased estimator that underestimates.
Bias vs. variability
Bias is about where the sampling distribution is centered. Variability is about how spread out it is. They are separate:
- Low bias, high variability: estimates scatter widely but center on the truth. A small random sample.
- Low bias, low variability: estimates cluster tightly around the truth. A large random sample, the goal.
- High bias, low variability: estimates cluster tightly around the wrong value. A large but badly designed sample, like a big online poll.
- High bias, high variability: scattered and off-center.
What fixes what
Increasing the sample size reduces variability. It does not reduce bias. Bias comes from how the data are collected (topic 1.12) or from the choice of estimator, and it has to be fixed there.
Think of a dartboard. Bias is aiming at the wrong spot. Variability is a shaky hand. Practice (a bigger sample) steadies your hand, but if you're aiming at the wrong spot, you'll just hit the wrong spot more consistently.
Estimators in this course
To judge bias from a simulation, compare the center of the simulated values (their mean) with the parameter used to generate them. If the center sits noticeably above or below the parameter, the estimator is biased.
| Parameter | Estimator | Unbiased with random sampling? |
|---|---|---|
| Population proportion p | Sample proportion p̂ | Yes |
| Population mean μ | Sample mean x̄ | Yes |
| Difference p₁ − p₂ | p̂₁ − p̂₂ | Yes |
| Difference μ₁ − μ₂ | x̄₁ − x̄₂ | Yes |
| Population range | Sample range | No, it tends to underestimate |
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Judge estimators from their sampling distributions
Simulated sampling distributions for three estimators of a parameter equal to 50 are described: Estimator A is centered at 50 with standard deviation 8. Estimator B is centered at 50 with standard deviation 3. Estimator C is centered at 46 with standard deviation 1. Which are unbiased, and which would you choose?
Show the solutionHide the solution
- Step 1: Unbiased means centered at the parameter, 50. A and B are unbiased; C is biased (it tends to underestimate by about 4).
- Step 2: Between A and B, B has much less variability, so its estimates tend to be closer to 50.
- Step 3: C is very consistent, but consistently wrong: most of its estimates are near 46.
Answer: A and B are unbiased; C is biased low. Choose B: it's unbiased and has the smallest variability of the unbiased options.
- Example 2Calculator allowed
Trap: a bigger sample doesn't remove bias
A website asks visitors to vote on whether the school day should start later and gets 12,000 votes, 81% yes. A student says, "With n = 12,000, p̂ = 0.81 must be very close to the true proportion of all students." Respond.
Show the solutionHide the solution
- Step 1: The votes are a voluntary response sample, so the estimator is biased: students who care strongly are more likely to vote.
- Step 2: A huge sample reduces variability, so 0.81 is a very consistent estimate of what this kind of poll produces.
- Step 3: But it doesn't remove bias, so 0.81 may be consistently far from the true proportion of all students.
Answer: The large sample only reduces variability. The voluntary response design makes p̂ biased, so 0.81 could be far from the true proportion.
Common mistakes
- Saying a larger sample reduces bias. It reduces variability only.
- Calling an estimator unbiased because one estimate happened to equal the parameter. Unbiased is about the center of the sampling distribution, over all samples.
- Mixing up the estimator (p̂, a rule) with the parameter (p, a fixed value).
- Preferring a biased estimator just because it has small variability.
On the exam
- Expect graphs of simulated sampling distributions for several estimators. Check each one's center against the parameter for bias, then compare spreads.
- When asked to justify that an estimator is unbiased, refer to the center (mean) of its sampling distribution equaling the parameter.
Connected topics
Videos
Check yourself
3 questions on 3.1 Estimators. Pick an answer to see if you got it, and why.
Four statistics are proposed for estimating a population parameter whose true value is 50. Simulated sampling distributions give the following centers and standard deviations. Which statistic is the best estimator?
A statistic is an unbiased estimator of a parameter. What does this mean?
A researcher simulates the sample range of 5 observations many times to estimate the range of a large population. The simulated values almost always fall below the population range. What does this show?
0 of 3 answered