Formula for Calculating Sample Size

Sample Size Formulas:

\[ n = \frac{Z^2 \times \sigma^2}{d^2} \text{ (for means)} \] \[ n = \frac{Z^2 \times p \times (1-p)}{d^2} \text{ (for proportions)} \]

Calculation Type:

Z-Score:

Standard Deviation (σ):

Proportion (p):

Margin of Error (d):

Required Sample Size (n):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Sample Size Calculation?

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 helps ensure studies are adequately powered without wasting resources.

2. How Does the Calculator Work?

The calculator uses two standard formulas:

\[ n = \frac{Z^2 \times \sigma^2}{d^2} \text{ (for means)} \] \[ n = \frac{Z^2 \times p \times (1-p)}{d^2} \text{ (for proportions)} \]

Where:

\( n \) — Required sample size
\( Z \) — Z-score for desired confidence level (1.96 for 95% CI)
\( \sigma \) — Standard deviation of the population (for means)
\( p \) — Estimated proportion (for proportions)
\( d \) — Margin of error (precision)

Explanation: The formulas account for the variability in data and desired precision to determine how many samples are needed.

3. Importance of Sample Size

Details: Proper sample size ensures study validity, prevents underpowered studies, and optimizes resource use. Too small samples may miss effects, while too large wastes resources.

4. Using the Calculator

Tips: Select calculation type (means or proportions), enter Z-score (1.96 for 95% CI), standard deviation or proportion, and desired margin of error. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What Z-score should I use?
A: Common values are 1.96 (95% CI) or 2.576 (99% CI). The exact value depends on your desired confidence level.

Q2: How to estimate standard deviation?
A: Use pilot data, previous studies, or literature. For proportions, 0.5 gives the most conservative estimate.

Q3: What's a reasonable margin of error?
A: Typically 5% (0.05) for proportions. For means, it depends on the measurement scale and expected variability.

Q4: Does this work for small populations?
A: For populations <10,000, use finite population correction: \( n_{adj} = \frac{n}{1 + \frac{n-1}{N}} \) where N is population size.

Q5: What about different study designs?
A: More complex designs (e.g., cluster RCTs) require additional adjustments to the sample size calculation.