Parameters, Statistics, and Sampling Variability
Distinguish population parameters (fixed, unknown) from sample statistics (variable, observable). Understand sampling variability: the value of a statistic changes from sample to sample and has its own distribution.
A lesson of AP Statistics, in Sampling Distributions, about 10 minutes.
By the end of this lesson, you will be able to distinguish parameters from statistics, explain sampling variability, and define bias and variability in the context of estimation.
Parameters vs. Statistics
Parameter: a number that describes a characteristic of the population. Fixed (but usually unknown).
| Symbol | Parameter | Description |
|---|---|---|
| μ | Population mean | Mean of all values in the population |
| σ | Population SD | SD of all values in the population |
| p | Population proportion | Fraction of the population with a characteristic |
Statistic: a number computed from sample data. Varies from sample to sample.
| Symbol | Statistic | Description |
|---|---|---|
| $\bar{x}$ | Sample mean | Estimates μ |
| s | Sample SD | Estimates σ |
| $\hat{p}$ | Sample proportion | Estimates p |
Memory trick: Parameter → Population; Statistic → Sample (double P and double S)
Sampling Variability
Take repeated samples of the same size from the same population. Each sample produces a different value of the statistic. This natural variation is called sampling variability.
Example: What fraction of US adults approve of the President? Take 10 different random samples of 1,000 adults. Each gives a slightly different $\hat{p}$.
This is NOT a problem with how we're measuring — it's the fundamental nature of random sampling.
The Sampling Distribution
The sampling distribution of a statistic is the distribution of all possible values of that statistic from all possible samples of the same size from the same population.
Describing the sampling distribution answers: how does the statistic behave in the long run?