Populations, samples and sampling distributions

What you’ll learn

• Distinguish a population, sampling frame and sampling unit.
• Compare a census with a sample survey.
• Construct a simple sampling distribution.

Understand the idea

The population is the collection about which a conclusion is required. A sampling unit is one selectable member, and a sampling frame is the list or mechanism used to select units. Missing members or duplicate entries in the frame can create bias before any numerical analysis begins.

A census attempts to measure every population member; a sample measures only some. A sample can be faster, cheaper and essential when measurement destroys the item, while a census avoids sampling variation if genuinely complete. Both can still suffer from inaccurate measurement or non-response.

A statistic is calculated from the sample without containing an unknown population parameter. It changes from sample to sample. Its sampling distribution lists the values it can take and their probabilities over repeated samples selected by the stated method. Treat ordered samples correctly when enumerating equally likely outcomes.

Key formulae

$$\bar X=\frac1n\sum_{i=1}^{n}X_i$$

$$P(T=t)=\sum_{\text{samples giving }t}P(\text{sample})$$

$$\sum_tP(T=t)=1$$

Worked example

A population contains the values 1 and 3, equally likely on each draw. Draw twice independently with replacement. Find the sampling distribution of the sample mean.

1. The four ordered samples are $(1,1)$, $(1,3)$, $(3,1)$ and $(3,3)$, each with probability $1/4$.
2. Their means are respectively 1, 2, 2 and 3.
3. Combine samples with the same mean: $P(\bar X=1)=1/4$, $P(\bar X=2)=1/2$, $P(\bar X=3)=1/4$.
4. The probabilities sum to one and $E(\bar X)=2$, matching the population mean.

Answer: Means $1,2,3$ have probabilities $1/4,1/2,1/4$.