Calculation of Sample Size for Unknown Population

Sample Size Formula for Unknown Population:

\[ n = \frac{Z^2 \times p \times (1-p)}{e^2} \]

Z-Score (Confidence Level):

unitless

Proportion (p):

decimal

Margin of Error (e):

decimal

Population Size (Optional):

count

Required Sample Size (n):

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1. What is Sample Size Calculation?

Sample size calculation is a statistical process used to determine the number of participants needed in a study to detect a significant effect or difference. For unknown populations, we use the standard formula that assumes an infinite population size.

2. How Does the Calculator Work?

The calculator uses the sample size formula for unknown populations:

\[ n = \frac{Z^2 \times p \times (1-p)}{e^2} \]

Where:

\( n \) — Required sample size
\( Z \) — Z-score corresponding to confidence level (1.96 for 95% confidence)
\( p \) — Estimated proportion (0.5 for maximum variance)
\( e \) — Margin of error (decimal form)

Finite Population Correction: If population size is known, apply: \( n_{adjusted} = \frac{n}{1 + \frac{n-1}{N}} \)

3. Importance of Sample Size Determination

Details: Proper sample size calculation ensures studies have adequate power to detect effects, prevents wasting resources on underpowered studies, and provides reliable and valid results.

4. Using the Calculator

Tips: Enter Z-score (typically 1.96 for 95% confidence), proportion (use 0.5 for maximum sample size), margin of error (e.g., 0.05 for ±5%), and optionally population size for finite correction.

5. Frequently Asked Questions (FAQ)

Q1: Why use p=0.5 as default?
A: Using p=0.5 provides the maximum possible sample size, ensuring adequate power regardless of the actual proportion in the population.

Q2: What Z-score should I use?
A: Common Z-scores are 1.645 (90% confidence), 1.96 (95% confidence), and 2.576 (99% confidence).

Q3: When should I use finite population correction?
A: Use finite correction when your sample represents more than 5% of the total population to avoid overestimating required sample size.

Q4: What is an acceptable margin of error?
A: Typically 0.05 (±5%) for general research, but can be smaller (0.01-0.03) for precise studies or larger (0.10) for exploratory research.

Q5: Can this calculator be used for clinical trials?
A: This calculator is suitable for prevalence studies and surveys. Clinical trials often require more complex power calculations based on effect sizes.