Calculating Sample Size For Unknown Population

Yamane Formula:

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

Z-Score (Z):

standard deviations

Prevalence (p):

proportion (0-1)

Margin of Error (e):

proportion (0-1)

Population (N):

individuals

Sample Size (n):

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1. What is the Yamane Formula?

The Yamane formula is a simplified equation for calculating sample size from a given population. It provides an approximation that is particularly useful when the population size is unknown or very large, using a simplified version of the Cochran formula with finite population correction.

2. How Does the Calculator Work?

The calculator uses the Yamane formula approximation:

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

Where:

\( n \) — Required sample size
\( Z \) — Z-score (standard normal deviate)
\( p \) — Estimated prevalence or proportion
\( e \) — Margin of error
\( N \) — Population size

Explanation: The formula calculates the minimum sample size needed to achieve a desired level of precision in statistical estimates, accounting for population size through finite population correction.

3. Importance of Sample Size Calculation

Details: Proper sample size calculation ensures research studies have sufficient statistical power to detect meaningful effects, prevents wasted resources on underpowered studies, and provides reliable and generalizable results.

4. Using the Calculator

Tips: Enter Z-score based on confidence level (1.96 for 95%, 2.576 for 99%), estimated prevalence (use 0.5 for maximum variability), margin of error (typically 0.05 for 5%), and total population size.

5. Frequently Asked Questions (FAQ)

Q1: What Z-score should I use?
A: Common Z-scores are 1.645 (90% confidence), 1.96 (95% confidence), and 2.576 (99% confidence). Choose based on your desired confidence level.

Q2: What if I don't know the prevalence?
A: Use 0.5 (50%) as this provides the most conservative estimate and maximum sample size requirement.

Q3: How do I choose the margin of error?
A: Typical values are 0.05 (5%) or 0.03 (3%). Smaller margins require larger sample sizes but provide more precise estimates.

Q4: When is finite population correction important?
A: When your sample size exceeds 5% of the total population, finite population correction becomes significant for accurate calculations.

Q5: Can this be used for very large populations?
A: Yes, for populations over 100,000, the sample size becomes relatively stable and approaches the infinite population formula result.