SAMPLE SIZE DETERMINATION
🔹 Definition
Sample size determination is the process of calculating the minimum number of subjects or observations required in a study to:
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Detect a true effect or difference,
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Achieve statistical significance, and
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Ensure adequate power and precision of results.
It prevents both underpowered studies (too few participants → false negatives) and wasteful studies (too many participants → unnecessary cost/time).
⚖️ Key Factors Affecting Sample Size
| Factor | Meaning | Effect |
|---|---|---|
| Confidence Level (Z) | Certainty we want (usually 95%) | Higher confidence → Larger sample |
| Margin of Error (E) | Acceptable level of error | Smaller error → Larger sample |
| Population Variability (SD or p) | How much the data vary | Higher variability → Larger sample |
| Power (1−β) | Ability to detect a true effect (usually 80% or 90%) | Higher power → Larger sample |
| Expected Effect Size (Δ) | Minimum difference considered clinically important | Smaller difference → Larger sample |
🧩 1. For Estimating a Mean
n=(Z×SDE)2n = \left( \frac{Z \times SD}{E} \right)^2n=(EZ×SD)2
Where:
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n = required sample size
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Z = Z-score for desired confidence (1.96 for 95%)
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SD = estimated standard deviation
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E = margin of error (precision)
Example 1 (Mean)
A study aims to estimate mean fasting blood sugar.
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SD = 15 mg/dL
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Desired precision (E) = ±5 mg/dL
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Confidence = 95% (Z = 1.96)
n=(1.96×155)2=(5.88)2=34.6n = \left( \frac{1.96 \times 15}{5} \right)^2 = (5.88)^2 = 34.6n=(51.96×15)2=(5.88)2=34.6
✅ Minimum sample size = 35 participants
🧩 2. For Estimating a Proportion
n=Z2×p(1−p)E2n = \frac{Z^2 \times p(1-p)}{E^2}n=E2Z2×p(1−p)
Where:
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p = estimated proportion (from pilot or literature)
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E = acceptable margin of error
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Z = Z-score for desired confidence level
Example 2 (Proportion)
Prevalence of hypertension estimated at 30% (p = 0.3).
Margin of error = 5% (E = 0.05).
Confidence = 95% (Z = 1.96).
n=(1.96)2×0.3(1−0.3)(0.05)2=3.8416×0.210.0025=323.1n = \frac{(1.96)^2 \times 0.3(1 - 0.3)}{(0.05)^2} = \frac{3.8416 \times 0.21}{0.0025} = 323.1n=(0.05)2(1.96)2×0.3(1−0.3)=0.00253.8416×0.21=323.1
✅ Minimum sample size = 324 participants
🧩 3. For Comparing Two Groups (Means)
n=2×(Zα/2+ZβΔ/SD)2n = 2 \times \left( \frac{Z_{\alpha/2} + Z_{\beta}}{\Delta/SD} \right)^2n=2×(Δ/SDZα/2+Zβ)2
Where:
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Δ = expected difference between group means
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Zα/2 = 1.96 for 95% confidence
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Zβ = 0.84 for 80% power
Example 3 (Two Means)
Comparing two antihypertensive drugs:
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Expected difference = 10 mmHg
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SD = 15 mmHg
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α = 0.05, β = 0.20 (power 80%)
n=2×(1.96+0.8410/15)2=2×(2.8×1.5)2=2×(4.2)2=2×17.64=35.28n = 2 \times \left( \frac{1.96 + 0.84}{10/15} \right)^2 = 2 \times (2.8 \times 1.5)^2 = 2 \times (4.2)^2 = 2 \times 17.64 = 35.28n=2×(10/151.96+0.84)2=2×(2.8×1.5)2=2×(4.2)2=2×17.64=35.28
✅ n ≈ 36 per group
📊 4. Adjustments
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For Non-response or Attrition:
nadjusted=n(1−expected dropout rate)n_{adjusted} = \frac{n}{(1 - \text{expected dropout rate})}nadjusted=(1−expected dropout rate)n
e.g., If dropout = 10%, then multiply by 1.11. -
For Finite Population (N 10,000):
nadj=n1+n−1Nn_{adj} = \frac{n}{1 + \frac{n - 1}{N}}nadj=1+Nn−1n
💡 Practical Notes for Medical Research
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Use pilot studies or previous literature to estimate SD or p.
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For unknown proportions, assume p = 0.5 (gives largest sample).
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Always justify sample size in proposals (Ethics & IRB requirement).
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Many use software like OpenEpi, Raosoft, or Epi Info for precise calculations.
