Sample Size Derivation & MDE Budget

The A/B Test Minimum Sample Size Formula & MDE Derivation establishes the exact sample requirements per variant under significance $\alpha$ and power $1 - \beta$; for continuous mean metrics, equating critical $Z$-score thresholds under $H_0$ and $H_1$ yields the closed-form sample size formula: $n = \frac{(z_{\alpha/2} + z_\beta)^2 \cdot 2\sigma^2}{\delta^2}$ (where $\delta = |\mu_T - \mu_C|$ is target effect size, $\sigma^2$ metric variance); for binomial ratio metrics (CTR/CVR with baseline probability $p$), substituting $\sigma^2 = \bar{p}(1-\bar{p})$ derives: $n = \frac{(z_{\alpha/2} + z_\beta)^2 \cdot 2 \bar{p}(1-\bar{p})}{\delta^2}$; inverted under fixed traffic constraints, it computes the Minimum Detectable Effect: $\text{MDE} = (z_{\alpha/2} + z_\beta) \sqrt{\frac{2\sigma^2}{n}}$.

Pre-experiment traffic budgeting, duration scheduling, and statistical sensitivity audits.

Underpowered tests miss real wins due to noisy variance, while overpowered tests waste valuable user traffic and delay product deployment cycles.

Derive the step-by-step mathematical proof of sample size $n = \frac{(z_{\alpha/2} + z_\beta)^2 \cdot 2\sigma^2}{\delta^2}$ from normal distribution quantile critical boundaries?

Explain why halving the target effect size $\delta$ quadruples ($4\times$) required sample size due to the inverse-square law?

Derive effective sample size $n_{\text{eff}} = \frac{2 n_T n_C}{n_T + n_C}$ under skewed 90/10 traffic splits and quantify statistical power loss?

Why must online experiments run for integer weekly cycles (7 or 14 days) regardless of reaching sample size early to absorb weekly seasonality?

How to resolve sample size explosions on heavy-tailed revenue metrics via Winsorization, log transforms, or CUPED variance reduction?