The Power Of Sample Size Calculation

Statistical Power Formula:

\[ Power = 1 - \beta = \Phi\left(\frac{|\mu_1 - \mu_2|}{\sigma/\sqrt{n}} - z_{1-\alpha/2}\right) \]

Effect Size (d):

Cohen's d

Sample Size (n):

participants

Alpha Level (α):

Test Type:

Statistical Power:

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1. What Is Statistical Power?

Statistical power is the probability that a test will correctly reject a false null hypothesis. It represents the likelihood of detecting an effect when one truly exists, calculated as 1 - β (where β is the Type II error rate).

2. How Does The Calculator Work?

The calculator uses the standard power calculation formula:

\[ Power = 1 - \beta = \Phi\left(\frac{|\mu_1 - \mu_2|}{\sigma/\sqrt{n}} - z_{1-\alpha/2}\right) \]

Where:

\( \mu_1 - \mu_2 \) — True difference between groups
\( \sigma \) — Standard deviation
\( n \) — Sample size per group
\( \alpha \) — Significance level (Type I error rate)
\( z_{1-\alpha/2} \) — Critical z-value
\( \Phi \) — Standard normal cumulative distribution function

Explanation: Power increases with larger effect sizes, larger sample sizes, and higher alpha levels, but decreases with greater variability.

3. Importance Of Power Analysis

Details: Adequate statistical power is essential for reliable research findings. Underpowered studies may fail to detect real effects, leading to false negative results and wasted resources.

4. Using The Calculator

Tips: Enter effect size (Cohen's d), sample size per group, select alpha level (typically 0.05), and choose test type (one-tailed or two-tailed). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is considered adequate statistical power?
A: Typically 80% or higher is considered adequate, though 90% is preferred for critical research.

Q2: How does effect size impact power?
A: Larger effect sizes require smaller sample sizes to achieve the same power. Small effects require larger samples.

Q3: What is the difference between one-tailed and two-tailed tests?
A: One-tailed tests have higher power for directional hypotheses, while two-tailed tests are more conservative and appropriate for non-directional hypotheses.

Q4: When should I conduct power analysis?
A: Before study design (a priori) to determine required sample size, and sometimes after data collection (post hoc) to interpret results.

Q5: What if my study is underpowered?
A: Consider increasing sample size, using more sensitive measures, or collaborating with other researchers to pool data.