Summary  
This chapter demonstrates how to implement four fundamental sampling techniques—simple random, stratified, cluster, and systematic sampling—in Python using the random module, covering selection of unique elements, proportional subgroup sampling, random cluster selection, and fixed-interval sampling.  

General domain of usage  
Statistical research and data collection

import random

N = 30   # population size
n = 5    # sample size

sample_srs = random.sample(range(1, N+1), n)
print(f"Simple Random Sample: {sample_srs}")

* `random.sample(range(1, N+1), n)` randomly selects **n unique values** from the population;
* Works **without replacement** (no repeats);
* Every member of the population has an equal chance of being chosen.

N_males = 18
N_females = 12
N_total = N_males + N_females
n_total = 10

n_males = round((N_males / N_total) * n_total)
n_females = round((N_females / N_total) * n_total)

print(f"Stratified Sample Size -> Males: {n_males}, Females: {n_females}")

* Population is divided into **subgroups (strata)**;
* Sample is drawn **proportionally** from each subgroup;
* Ensures representation of key groups.

import random

clusters = 5
students_per_cluster = 25

selected_cluster = random.randint(1, clusters)
print(f"Selected cluster (classroom): {selected_cluster} containing {students_per_cluster} students")

* Population divided into **clusters** (e.g., classrooms);
* One or more clusters are selected **randomly**;
* Everyone in chosen cluster(s) is surveyed;
* Efficient when listing every individual is impractical.

import random

N = 1000
n = 100

k = N // n                # Sampling interval
start = random.randint(1, k)  # Random start

sample_systematic = list(range(start, N+1, k))
print(f"Sampling interval k = {k}")
print(f"Random start = {start}")
print(f"First 10 samples: {sample_systematic[:10]}")

* Interval $$k = \frac{N}{n}$$;
* Start point chosen randomly between 1 and $$k$$;
* Select every $$k$$-th element from ordered population.

## Summary of Methods

* **Simple Random:** equal chance for all, no repeats;
* **Stratified:** ensures subgroup representation;
* **Cluster:** randomly selects whole groups;
* **Systematic:** selects at fixed intervals after random start.

What function is used for simple random sampling without replacement?

This course distills the essentials of Python's math module, focusing on trigonometric functions, logarithmic operations, and mathematical constants. Learners will gain hands-on experience using Python to perform these standard mathematical computations, ideal for beginners seeking practical skills with the math module.

Implementing Sampling to Python

Simple Random Sampling

Stratified Sampling

Cluster Sampling

Systematic Sampling

Summary of Methods