A sample statistic used to guess a population parameter is called an estimator. Good estimators are unbiased (right on average) and have low variability (consistent from sample to sample). Simulate six estimators and see which is which.
An estimator is a rule that turns sample data into a guess for a parameter. Pick an estimator, take many samples, and look at the distribution of its estimates. Its center tells you about bias; its spread tells you about variability.
Each dot is one of the six estimators at the current sample size; the highlighted one is selected (hover a gray dot to see its name). Lower left is the goal: low bias and low variability.
A parameter describes the population (p, μ, σ) and is usually unknown. A statistic describes a sample (p̂, x̄, s) and is used as a point estimate of the parameter.
The sampling distribution is the distribution of a statistic's values over all possible samples of the same size. Simulation approximates it by taking many samples.
An estimator is unbiased when the mean of its sampling distribution equals the parameter. p̂, x̄, and s² (with n − 1) are unbiased. Bad sampling methods cause bias.
Variability is the spread of the sampling distribution. Larger random samples give less variability. The population size barely matters if the sample is under 10% of it.
A biased method with a huge sample gives precise but wrong answers. An unbiased method with a tiny sample is right on average but often far off. You want both low.
Bias is aiming off-center; variability is how scattered the darts are. Low bias + low variability means tight shots centered on the bullseye.
How common estimators behave when the data come from a random sample.
| Estimator | Estimates | Unbiased? | Variability |
|---|---|---|---|
| p̂ = X ÷ n | Population proportion p | Yes | σ = √(p(1 − p) ÷ n) |
| x̄ | Population mean μ | Yes | σ ÷ √n |
| s² (divide by n − 1) | Population variance σ² | Yes | Shrinks as n grows |
| Variance dividing by n | σ² | No, too small | Shrinks as n grows |
| Sample maximum | Population maximum | No, too small | Shrinks as n grows |
| p̂ from a convenience or voluntary sample | p | Usually not | Shrinks as n grows, but around the wrong value |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 3.1. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 3 terminology guide.