1. Pooled Two-Proportion z-Test (Default H₀: Δ = 0)
\[ P_1 = \frac{x_1}{n_1}, \quad P_2 = \frac{x_2}{n_2}, \quad \bar{P} = \frac{x_1 + x_2}{n_1 + n_2}, \quad z_{\text{pooled}} = \frac{P_2 - P_1}{\sqrt{\bar{P}(1 - \bar{P})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \]
💬 Read it in plain English: When testing whether two groups have the exact same true conversion rate, we pool their total conversions over total visitors to get P̄, then divide the observed rate gap (P₂ − P₁) by the pooled standard error.
x1, x2
Observed number of successes (conversions, clicks) in Control (1) and Variant (2).
n1, n2
Total visitors or impressions assigned to Control (1) and Variant (2).
P1, P2
Sample conversion rates x₁/n₁ and x₂/n₂.
P̄
Combined conversion rate (x₁ + x₂)/(n₁ + n₂) assuming both groups share one rate.
zpooled
Standard-normal test statistic for H₀: P₂ − P₁ = 0 (its square equals Pearson’s χ²).
2. Non-Zero Benchmark z-Test (Δ₀ ≠ 0) & Unpooled Confidence Interval
\[ \Delta = P_2 - P_1, \qquad z_{\text{unpooled}} = \frac{\Delta - \Delta_0}{\sqrt{\frac{P_1(1 - P_1)}{n_1} + \frac{P_2(1 - P_2)}{n_2}}}, \qquad \text{CI}_{1-\alpha} = \Delta \pm z_{1-\alpha/2}\sqrt{\frac{P_1(1 - P_1)}{n_1} + \frac{P_2(1 - P_2)}{n_2}} \]
💬 Read it in plain English: For confidence intervals and non-zero hurdle tests (Δ₀ ≠ 0), each group keeps its own variance P_i(1 − P_i)/n_i so the uncertainty interval reflects the actual conversion rates observed in each arm.
Δ
Difference in sample conversion rates: P₂ − P₁.
Δ0
Hypothesized proportion difference under H₀ (non-zero uses unpooled SE).
SEunpooled
Standard error √[P₁(1−P₁)/n₁ + P₂(1−P₂)/n₂] used for confidence intervals and Δ₀ ≠ 0.
z1−α/2
Standard-normal critical value (e.g., 1.96 for a 95% two-sided interval).