Sampling Techniques

Sampling is the process of selecting a subset (sample) from a larger group (population) so that conclusions drawn from the sample can be generalised to the population.

Key Terminology

Why Sample at All?

1. Cost — surveying 1.4 billion people is impossible; 5,000 well-chosen people is not.
2. Time — results are needed before they become irrelevant.
3. Feasibility — destructive testing (crash tests, food-quality tests) rules out testing everything.
4. Accuracy — a small, well-controlled sample often beats a huge sloppy census, because effort per unit is higher.

A. Probability (Random) Sampling

Every member has a known, non-zero probability of selection — this is what makes statistical inference (Unit 2) valid.

1. Simple Random Sampling (SRS)

Every unit has an equal chance of being selected; selection is by lottery or random number generator.

• With replacement — a selected unit returns to the pool and can be picked again
• Without replacement — more common in practice

import pandas as pd
import numpy as np

np.random.seed(42)
population = pd.DataFrame({
    "student_id": range(1, 101),
    "marks": np.random.randint(35, 100, 100),
    "section": np.random.choice(["A", "B", "C"], 100, p=[0.5, 0.3, 0.2]),
})

# Simple random sample of 10 students
srs = population.sample(n=10, random_state=1)
print(srs["marks"].mean(), "vs population", round(population["marks"].mean(), 2))

2. Systematic Sampling

Select every k-th unit after a random start, where k = N / n is the sampling interval.

N = 1000 students, n = 100 needed
k = 1000 / 100 = 10
Random start r between 1 and 10, say r = 7
Selected: 7, 17, 27, 37, 47, ... , 997

Periodicity trap: if a list of houses repeats "corner house every 10th" and k = 10, every sampled house is a corner house — a systematically biased sample.

k = len(population) // 10
start = np.random.randint(0, k)
systematic = population.iloc[start::k]
print(len(systematic), "students selected, every", k, "th")

3. Stratified Sampling

Divide the population into homogeneous, non-overlapping strata (e.g. by gender, section, income band), then randomly sample within each stratum.

• Proportionate — each stratum contributes in proportion to its share of the population
• Disproportionate — small but important strata are over-sampled

# Proportionate stratified sample: 20% from each section
stratified = (population
              .groupby("section", group_keys=False)
              .apply(lambda g: g.sample(frac=0.2, random_state=1)))
print(stratified["section"].value_counts())
# A    10
# B     6
# C     4     -> proportions preserved

4. Cluster Sampling

Divide the population into naturally occurring clusters (villages, colleges, city blocks), randomly select entire clusters, and survey everyone inside the chosen clusters.

Stratified vs Cluster — The Classic Exam Question

5. Multi-stage Sampling

Sampling applied in stages: e.g. randomly select states → within them districts → within them villages → within them households. Used in national surveys like the NSSO and the Census follow-up surveys.

B. Non-Probability Sampling

Selection probability is unknown; results cannot be statistically generalised, but these methods are cheap and useful for exploratory work.

Sample Size — What Drives It

For estimating a population mean:

        n = (Z × σ / E)²

  Z = z-score for the confidence level (1.96 for 95%)
  σ = population standard deviation (estimated)
  E = margin of error you can tolerate

Example: σ = 15 marks, E = 2 marks, 95% confidence
        n = (1.96 × 15 / 2)² = (14.7)² ≈ 216 students

Larger samples reduce sampling error (proportional to 1/√n) — note the diminishing returns: quadrupling n only halves the error.

Sampling Bias — What to Watch For

A well-drawn sample is the bridge between the data you have and the population you want to talk about — and every inferential technique in Unit 2 silently assumes that bridge is sound.