Every individual has an equal probability of selection.

• Use random number table or computer generator
• Gold standard for unbiased selection
• Requires a complete sampling frame (list of all individuals)

When to use:

• Small, well-defined populations
• Complete sampling frame available

Limitation:

• Impractical for large/geographically dispersed populations
• May miss small subgroups

Select every k-th individual from the frame.

1. Calculate interval: k = N/n
2. Pick random start (1 to k)
3. Select every k-th person thereafter

Example (NFHS):

• 500 households in a village, need 50
• k = 500/50 = 10
• Random start = 3
• Select: 3, 13, 23, 33, …

Advantage: Simpler than SRS, spreads evenly

Risk: Periodicity bias (if list has hidden pattern)

Divide population into strata, sample from each.

1. Define strata (age groups, districts, urban/rural)
2. Sample independently within each stratum
3. Can use proportional or disproportional allocation

Why stratify?

• Ensures representation of all subgroups
• Reduces sampling error (more homogeneous within strata)
• Enables stratum-specific estimates

Example (NFHS-5):

• Strata: Urban vs Rural within each state
• Sample proportionally from each

Randomly select whole clusters, sample everyone within.

1. Divide population into clusters (villages, schools, PHCs)
2. Randomly select clusters
3. Include all individuals in selected clusters

Why use clusters?

• No need for complete individual-level frame
• Cheaper — travel to fewer locations
• Standard for WHO EPI surveys

Disadvantage:

• Higher sampling error (individuals within clusters are similar)
• Need design effect correction for sample size

SE quantifies typical sampling error.

\[SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\]

\[SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\]

Key insight (CLT):

• Sampling error is random and follows a normal distribution
• SE decreases with √n — the √n law
• Doubling precision requires 4× the sample size

\[n = \frac{z^2_{\alpha/2} \times p(1-p)}{E^2}\]

Where:

• \(z_{\alpha/2}\) = 1.96 (for 95% CI)
• \(p\) = expected proportion
• \(E\) = desired margin of error

If p is unknown: Use p = 0.5 (maximizes sample size — conservative)

Worked Example:

Estimate diabetes prevalence in Bhopal.

• Expected: p = 0.10 (10%)
• Precision: E = 0.02 (±2%)
• Confidence: 95% (z = 1.96)

\[n = \frac{1.96^2 \times 0.10 \times 0.90}{0.02^2}\]

\[n = \frac{3.8416 \times 0.09}{0.0004} = \frac{0.346}{0.0004}\]

\[\boxed{n = 864}\]

\[n_{per\ group} = \frac{2(z_{\alpha/2} + z_{\beta})^2 \times \sigma^2}{\delta^2}\]

Where:

• \(z_{\alpha/2}\) = 1.96 (α = 0.05, two-tailed)
• \(z_{\beta}\) = 0.84 (β = 0.20, power = 80%)
• σ = common SD in both groups
• δ = difference to detect

Worked Example:

New antihypertensive vs standard.

• Detect: δ = 10 mmHg difference
• SD: σ = 15 mmHg
• Power: 80%, α = 0.05

\[n = \frac{2(1.96 + 0.84)^2 \times 15^2}{10^2}\]

\[= \frac{2 \times 7.84 \times 225}{100}\]

\[\boxed{n = 36 \text{ per group}}\]

\[n = \frac{2(z_{\alpha/2} + z_{\beta})^2 \times [p_1(1-p_1) + p_2(1-p_2)]}{(p_2 - p_1)^2}\]

Where:

• \(p_1\) = proportion in control group
• \(p_2\) = proportion in treatment group

Worked Example:

TB cure rate: 70% → 85%.

\[n = \frac{2 \times 7.84 \times [0.70 \times 0.30 + 0.85 \times 0.15]}{(0.85 - 0.70)^2}\]

\[= \frac{2 \times 7.84 \times [0.21 + 0.1275]}{0.0225}\]

\[= \frac{2 \times 7.84 \times 0.3375}{0.0225}\]

\[\boxed{n \approx 94 \text{ per group}}\]

Where:

• m = cluster size
• ρ = intracluster correlation (ICC)

\[n_{adjusted} = n_{SRS} \times DE\]

Example:

• SRS sample size: 36 per group
• Cluster size: m = 12 patients per PHC
• ICC: ρ = 0.02

\[DE = 1 + (12 - 1) \times 0.02 = 1.22\]

\[n_{adjusted} = 36 \times 1.22 \approx 44 \text{ per group}\]