Calculate Sample Size From Power

Sample Size Formula:

\[ n = \frac{(Z_{\alpha/2} + Z_{\beta})^2 \times \sigma^2}{\delta^2} \]

Required Sample Size (n):

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1. What is Sample Size Calculation?

The sample size calculation determines how many participants are needed in a study to detect an effect of a given size with a specified degree of confidence and power. It's a crucial step in study design to ensure adequate statistical power while avoiding unnecessary resource expenditure.

2. How Does the Calculator Work?

The calculator uses the standard sample size formula for comparing means:

\[ n = \frac{(Z_{\alpha/2} + Z_{\beta})^2 \times \sigma^2}{\delta^2} \]

Where:

\( Z_{\alpha/2} \) — Critical value for desired confidence level (1.96 for 95% CI)
\( Z_{\beta} \) — Critical value for desired power (0.84 for 80% power)
\( \sigma \) — Standard deviation of the outcome variable
\( \delta \) — Minimum detectable effect size

Explanation: The formula accounts for the trade-off between Type I (α) and Type II (β) errors, variability in the data (σ), and the smallest effect size you want to detect (δ).

3. Importance of Sample Size Calculation

Details: Proper sample size calculation ensures your study has adequate power to detect meaningful effects while avoiding wasted resources on excessively large samples. Underpowered studies may miss important effects, while overpowered studies may detect trivial effects.

4. Using the Calculator

Tips: Enter Z-scores for your desired confidence level and power (common values pre-filled), the expected standard deviation of your outcome, and the minimum effect size you want to detect. All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What are typical values for Z_α/2 and Z_β?
A: For 95% confidence (α=0.05), Z_α/2=1.96. For 80% power (β=0.2), Z_β=0.84. For 90% power, Z_β=1.28.

Q2: How do I estimate standard deviation?
A: Use pilot data, published studies, or reasonable assumptions. For proportions, σ=√[p(1-p)] where p is the expected proportion.

Q3: What if I need to calculate for two groups?
A: Multiply the result by 2 for equal-sized groups. For unequal allocation, adjust accordingly (e.g., multiply by (r+1)/r where r is allocation ratio).

Q4: What's a reasonable effect size?
A: This depends on your field and what's clinically meaningful. Small effects typically require larger samples.

Q5: Are there other formulas for different study designs?
A: Yes, different formulas exist for proportions, survival analysis, etc. This calculator is for comparing means.