Simple Random Sampling

Every member of the population has an equal chance of being selected.
This is like drawing names out of a hat.

Where:

• $N$ = population size.

Example 1:

You have a class of 30 students. You want to randomly select 5 for a survey.

Solution: Use a random number generator to select 5 unique numbers between 1 and 30. Each student has a $\tfrac{\raisebox{1pt}{$1$}}{\raisebox{-1pt}{$30$}}$ chance of being selected.

Example 2:

You have a class of 30 students and want to select 5 to participate in a survey.

• Total population: $N=30$;
• Sample size: $n=5$.

What is the probability that Alice and Bob are both selected?

Total number of ways to choose 5 students from 30:

Number of favorable samples containing both Alice and Bob:
Fix Alice and Bob — choose 3 more from the remaining 28:

So the probability is:

Stratified Sampling

The population is divided into meaningful subgroups (strata), and random samples are taken from each.

Where:

• $N_h$ - size of subgroup $h$;
• $N$ - total population size;
• $n$ - total sample size;
• $n_{\raisebox{-1pt}{$h$}}$ - sample size from subgroup $h$.

Example:

A class has 30 students: 18 males and 12 females. You want to sample 10 students proportionally:

• From males: $\tfrac{\raisebox{1pt}{$18$}}{\raisebox{-1pt}{$30$}} \times 10 = 6$;
• From females: $\tfrac{\raisebox{1pt}{$12$}}{\raisebox{-1pt}{$30$}} \times 10 = 4$.

Why it's good: Ensures representation of key subgroups.

Cluster Sampling

The population is split into groups (clusters), and entire clusters are randomly selected.

Where:

• Clusters are pre-existing groups (e.g., classrooms, teams);
• You randomly pick entire clusters, not individuals.

Example 1:

Your school has 5 classrooms. You want a sample of 25 students, but surveying individuals is too time-consuming.

Solution: Randomly select 1 classroom (since each has ~25 students) and survey all.

Example 2:

A university has 20 dorm buildings, each housing 50 students. You randomly select 4 dorms and survey everyone inside.

• Number of clusters: $N=20$;
• Selected clusters: $n=4$;
• Students per dorm: $M=50$;
• Total students sampled: $n \times M = 200$.

What's the probability that a specific student (e.g., Sarah) is included?
It equals the probability that her dorm is selected:

Complex case:
If 10 dorms have 30 students and 10 have 70 students, and you select 4 dorms randomly, what's the expected sample size?

Let:

• $D_{30} = 10$ dorms with 30 students;
• $D_{70} = 10$ dorms with 70 students.

Expected sample size:

So even if clusters differ in size, the expected sample size remains the same if dorm types are balanced.

Systematic Sampling

Select every $k$-th item from a list.

Where:

• $N$ - total population;
• $n$ - sample size desired;
• $k$ - sampling interval.

Example:

A list of 1000 customers. You want a sample of 100. So:

Pick a random start point (e.g., 7), then select every 10th customer: 7, 17, 27, etc.

Why it's good: Easy to implement and systematic.

All Methods Applied to One Problem

Problem Setup:
You're studying cafeteria satisfaction at a school with 300 students across 10 classrooms (30 per room). You want a sample of 30 students.

• Simple random: randomly pick 30 names from the full list;
• Stratified: if 60% are boys and 40% girls, sample 18 boys and 12 girls;
• Cluster: randomly select 1 class (30 students) and survey all;
• Systematic: pick every 10th student from an ordered list.

Summary

• Sampling reduces data collection effort while allowing generalization;
• Random and stratified sampling are best for accuracy;
• Cluster sampling is efficient but works best when clusters are similar;
• Systematic sampling is simple and practical;
• Convenience sampling is risky and should be avoided when possible;
• Always document your sampling method in real-world analysis.