AP Statistics Unit 5 Sampling Distributions

Topic 5.4: Biased and Unbiased Point Estimates

When Does a Sample Statistic "Hit the Target" on Average?
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:

Key Vocabulary

Point Estimator The sample statistic used to estimate a population parameter (e.g., x estimates μ)
Point Estimate The specific numerical value of the statistic from a particular sample (e.g., x = 2.3)
Unbiased Estimator A statistic whose sampling distribution is centered at the true parameter — on average, it equals the parameter
Biased Estimator A statistic whose sampling distribution is not centered at the true parameter — it systematically over- or under-estimates
Sampling Variability The natural variation in a statistic from sample to sample; can be modeled using probability

Part 1: The Sample Mean — An Unbiased Estimator

[0:00 – 2:40]
Setup — Weimaraner Dogs: [0:19] A breeder has 5 dogs with ages: 0, 2, 5, 8, 10. This is the entire population. Population mean μ = 5. We randomly select samples of 3 dogs.
1. [1:21] One sample gives ages {0, 2, 5} with x = 2.3. The statistic x is called the , and the specific value 2.3 is called the .
2. [1:35] This single sample gave x = 2.3, which is not equal to μ = 5. Does this one "miss" make x a bad estimator? Think about what "on average" means.
3. [1:47] The video lists all possible samples of size 3 and their means. The mean of all those sample means turns out to be , which equals the population mean μ.
4. [2:27] Because the average value of x across all possible samples equals μ, we say x is an estimator of μ.
Key Idea: [2:18] Individual sample means vary around μ — some are too high, some too low. But the sampling distribution is at the population mean. That's what "unbiased" means: no systematic tendency to miss in one direction.

Part 2: The Sample Range — A Biased Estimator

[2:41 – 4:05]
5. [2:41] The population range is 10 − 0 = . The sample {0, 2, 5} has a range of .
6. [3:17] After computing the range for all possible samples of size 3, the mean of all sample ranges is . Since this does not equal the population range (10), the sample range is a estimator.
7. [3:27] On the dot plot of all sample ranges, the values cluster to the of the population range. This means the sample range systematically the population range.
8. [3:27] Why is this bias predictable? A sample of 3 dogs drawn from a population of 5 can never include both the youngest and oldest dog every time. So the sample range can never be than the population range — it can only equal it or fall short.

Part 3: Key Takeaways

[4:05 – 4:32]
[4:07]
  1. A sample statistic is a of the corresponding population parameter.
  2. Estimators exhibit that can be modeled using probability.
  3. An estimator is unbiased if, , it equals the population parameter.

Reflection Questions

[AI-Graded]
R1. The sample mean of {0, 2, 5} was 2.3 — way off from the population mean of 5. Yet we still call x an "unbiased" estimator. Explain what "unbiased" actually promises and what it does not promise about any single sample.
R2. Explain in your own words why the sample range is a biased estimator of the population range. Use the Weimaraner example to describe why the bias goes in one direction (always underestimating, never overestimating). Could a different statistic be biased in the opposite direction — systematically overestimating?

Exit Ticket

A population of 4 students has quiz scores: {60, 70, 80, 90}. The population mean is 75 and the population range is 30. A researcher takes all possible samples of size 2.

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