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**Sampling methods** are techniques used to select a subset of individuals (a sample) from a population to represent the entire group. The goal is to gather data that can be used to make inferences about the population without surveying every member.

In **random sampling**, every individual in the population has an equal chance of being selected.

**Example**: Drawing names from a hat where every name has an equal probability of being picked. The probability of selecting an individual from a population of size \(N\) in a random sample of size \(n\) is:

\[P(\text{selection}) = \frac{1}{N}\]

This method is simple and unbiased, but it can be difficult to implement for very large populations.

In **systematic sampling**, members of the population are selected at regular intervals from a list or ordered group. For example, every \(k\)-th individual is chosen.

**Example**: Selecting every 10th person from a list of registered voters.

To calculate the sampling interval \(k\), use:

\[k = \frac{N}{n}\]

where \(N\) is the population size and \(n\) is the desired sample size.

In **stratified sampling**, the population is divided into distinct subgroups (strata) based on a characteristic such as age, gender, or income. A sample is then drawn from each stratum, either proportionally or equally.

**Example**: Dividing a population into age groups and selecting a sample from each group to ensure representation of all age categories.

The sample size from each stratum can be calculated using:

\[\text{Sample size from stratum} = \frac{\text{Stratum size}}{N} \times n\]

This ensures representation from all strata, making the sample more reflective of the population.

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