Sampling

The $\textbf{population}$ is the whole group being studied. A $\textbf{sample}$ is part of it. A census surveys everyone.

Samples are cheaper and quicker, but give less certain conclusions. Larger samples are generally more reliable.

A sample is $\textbf{representative}$ if it reflects the population’s make-up.

A sample is $\textbf{biased}$ if some members of the population are more likely to be included than others, in a way that affects the results.

Common causes: sampling at a particular time or place, using volunteers, too small a sample, or leading questions.

To reduce bias, use a random method, a larger sample, and ask neutral questions.

In a $\textbf{simple random sample}$, every member of the population has an equal chance of selection — for example, numbering everyone and using a random number generator.

In a $\textbf{stratified sample}$, the population is split into groups (strata), and each group is sampled in proportion to its size.

Number from each group $= \dfrac{\text{group size}}{\text{population size}} \times \text{sample size}$.

Capture a sample, mark them and release them. Later, capture a second sample and count how many are marked.

Assume the proportion marked in the second sample equals the proportion marked in the whole population: $\dfrac{M}{N} = \dfrac{m}{n}$.

So the population estimate is $N = \dfrac{M \times n}{m}$, where $M$ is the number first marked, $n$ the size of the second sample and $m$ the number marked in it.