Sample Size From Population

Sample Size Formula for Finite Population:

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

Z-Score:

(dimensionless)

Proportion:

(decimal)

Precision:

(decimal)

Population Size:

(integer)

Required Sample Size:

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1. What is Sample Size Calculation?

The sample size calculation determines the number of participants needed in a study to detect an effect of a given size with a certain degree of confidence. It's crucial for study design to ensure results are statistically significant and resources are used efficiently.

2. How Does the Calculator Work?

The calculator uses the finite population correction formula:

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

Where:

\( n \) — Required sample size
\( Z \) — Z-score for desired confidence level
\( p \) — Estimated proportion (0.5 for maximum variability)
\( d \) — Margin of error (precision)
\( N \) — Total population size

Explanation: The formula adjusts the sample size based on population size, with the finite population correction factor reducing the required sample when the sample represents a significant fraction of the population.

3. Importance of Sample Size

Details: Proper sample size ensures study validity, prevents wasting resources on overly large studies, and ensures adequate power to detect meaningful effects.

4. Using the Calculator

Tips: Common Z-scores: 1.645 (90% CI), 1.96 (95% CI), 2.576 (99% CI). Use p=0.5 for most conservative estimate. Precision (d) is typically 0.05 (5%) for social sciences.

5. Frequently Asked Questions (FAQ)

Q1: What Z-score should I use?
A: 1.96 for 95% confidence (most common), 1.645 for 90%, 2.576 for 99% confidence levels.

Q2: Why use p=0.5?
A: This gives the most conservative (largest) sample size as it maximizes the p*(1-p) term in the formula.

Q3: When is finite population correction needed?
A: When your sample size exceeds 5% of the total population (N). For infinite populations, use the simpler formula without denominator.

Q4: How does precision affect sample size?
A: Smaller precision (d) values require larger samples. Halving the precision quadruples the required sample size.

Q5: What if my population is very large?
A: For populations >100,000, the finite correction becomes negligible and you can use the infinite population formula.