Statistics · Unit 5: Sampling Distributions · 14 min read · Updated 2026-05-11
Biased and Unbiased Estimators — AP Statistics
AP Statistics · Unit 5: Sampling Distributions · 14 min read
1. Core Concepts: Biased vs Unbiased Estimators★★☆☆☆⏱ 3 min
An estimator is a sample statistic used to estimate an unknown population parameter. Common examples include the sample proportion $\hat{p}$ to estimate population proportion $p$, and the sample mean $\bar{x}$ to estimate population mean $\mu$. Bias describes how far the center of the estimator's sampling distribution is from the true population parameter.
A common misconception is that bias refers to error in a single estimate. In reality, bias is a property of the entire sampling distribution's center, not spread or error from one sample. If the average of the estimator across infinitely many random samples equals the true parameter, the estimator is unbiased; otherwise it is biased.
2. Formal Definition and Bias Calculation★★★☆☆⏱ 4 min
By definition, an estimator is unbiased when $E(\hat{\theta}) = \theta$, meaning the mean of its sampling distribution is exactly at the true parameter value. A positively biased estimator overestimates the true parameter on average, while a negatively biased estimator underestimates it on average.
3. Common Biased and Unbiased Estimators★★☆☆☆⏱ 3 min
The AP exam expects you to quickly identify the bias status of the most common estimators, without requiring a full expected value calculation for every question:
**Unbiased estimators**: Sample mean $\bar{x}$ for $\mu$, sample proportion $\hat{p}$ for $p$, sample variance $s^2$ (divided by $n-1$) for $\sigma^2$
**Biased estimators**: Sample range for population range, sample standard deviation $s$ for $\sigma$, sample maximum for population maximum, sample variance divided by $n$ for $\sigma^2$
Sample standard deviation is biased even though sample variance is unbiased because the square root function is non-linear, so $E(\sqrt{s^2}) \neq \sqrt{E(s^2)}$. Sample variance with denominator $n$ has a small negative bias, meaning it underestimates true population variance on average.
4. Bias from Flawed Sampling Methods★★★☆☆⏱ 4 min
Bias can come from two sources: inherent bias in the estimator even with random sampling, or bias from a non-random or flawed sampling method. A biased sampling method almost always produces a biased estimator, even if the estimator would be unbiased for a simple random sample.
Common Pitfalls
Why: Expectation does not preserve non-linear transformations like square root, so even if $s^2$ is unbiased, $s$ is not.
Why: Students confuse the property of the sampling distribution's center with the outcome of a single sample.
Why: Students confuse bias (center of the sampling distribution) with variability (spread of the sampling distribution).
Why: Students mix up the order of subtraction in the formal definition of bias.
Why: Students memorize convenience sampling is biased, but AP FRQs require context-specific description of bias direction for full credit.