AP Statistics Unbiased Estimator Practice: Bias vs. Sampling Variability
An unbiased estimator is one whose sampling distribution is centered on the parameter. The sample proportion p̂ is an unbiased estimator of p, so one sample landing above or below p is ordinary sampling variability, not bias. Bias comes from a sampling method that pushes results in one direction, such as a convenience sample.
Try the question
The true proportion of voters who support a measure is p = 0.50. A pollster takes one random sample of 200 voters and gets a sample proportion of 0.44.
Which statement is best?
- A. The sample proportion is a biased estimator, because the result of 0.44 does not equal the true value of 0.50.
- B. The sampling method must be flawed, because a properly done random sample should give a result of exactly 0.50.
- C. The sample proportion is biased low, and taking a larger random sample of 400 voters would remove that bias.
- D. This is consistent with an unbiased estimator, because unbiased means the center of all possible sample proportions is p, not that every sample hits p.
Answer key: why each choice is right or wrong
✕ A — Incorrect
Tempting because the estimate missed the target — but bias is about the long-run center, not one miss. Fix: judge bias by the mean of the sampling distribution.
✕ B — Incorrect
Tempting because random sampling feels like it should reproduce the population — but random samples vary. Fix: expect sample-to-sample variability around p.
✕ C — Incorrect
Tempting because larger samples help — but larger n reduces variability, not bias, and a single 0.44 does not show bias. Fix: bias comes from the method, spread comes from n.
✓ D — Correct
This earns the point because bias describes the sampling distribution over many samples. One sample falling 0.06 below p is ordinary sampling variability and says nothing against unbiasedness.
The core idea
p̂ is the point estimator of the population proportion p. Unbiasedness is a property of p̂'s sampling distribution across every possible sample of size n, not of any one sample's result. The mean of that distribution equals p (μp̂ = p), even though almost every individual p̂ lands somewhat above or below p by chance.
What would make p̂ biased is a sampling method that pulls results in one direction — for example, a convenience sample that over-represents one group — not ordinary sample-to-sample variability.
How points are earned and lost
Earned: name p̂ as the estimator of the parameter p, explain unbiasedness as the estimator's average behavior across repeated sampling, and keep sampling variability (one result differing from p) separate from bias (a systematic direction to that difference).
Lost: calling one sample result biased just because it does not equal the parameter.
The answer move
Name the statistic as an estimator of a population parameter. Then explain unbiasedness as no systematic over- or underestimation before you write your final sentence.
Weak answer vs. point-earning answer
A citywide poll of n = 50 randomly selected voters finds p̂ = 0.58 support for a ballot measure. The true population support is later revealed to be p = 0.52. Was this poll's estimator biased?
✕ Weak answer
Since the poll's result of 0.58 doesn't match the true value of 0.52, the estimator p̂ must be biased.
✓ Point-earning answer
p̂ is an unbiased estimator of p regardless of this one result, because unbiasedness describes the mean of p̂'s sampling distribution across every possible random sample of size 50, which equals p = 0.52 (μp̂ = p). This sample's p̂ = 0.58 differing from p by 0.06 reflects ordinary sampling variability, not bias; bias would require the sampling method itself to systematically favor one outcome.
Before your next bias question
Ask whether the issue is one sample's distance from p (not bias) or the sampling method's systematic direction (bias).
Review the topic behind this question: Topic 3.1: Estimators. More practice: Two-way table practice, Sampling distribution of the sample mean practice and AP Statistics practice questions.
Frequently asked questions
What is a point estimator?
What does unbiased mean in AP Statistics?
Does a larger sample reduce bias?
If my sample result misses the true value, is the estimator biased?
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