# Independent Samples - Quantum Column Proportions Test

Where $p_1$ and $p_2$ are the two proportions and $e_1$ and $e_2$ are the effective sample sizes and $\pi_1$ and $\pi_2$ are the weighted sample sizes:

$t=\frac{p_1-p_2}{\sqrt{p_{12}(1-p_{12})(e^{-1}_1+e^{-1}_2)(\frac{e_1+e_2}{e_1+e_2-1})}}$, where $p_{12}=\frac{\pi_1 p_1 + \pi_2 p_2}{\pi_1+\pi_2}$

where $p \approx 2 \Pr(t_{e_1+e_2-1} \ge |t|)$.

Note that if wishing to replicate results from IBM SPSS Data Collection products (including Survey Reporter and Quantum) additional settings need to be modified (see How to Replicate Survey Reporter Significance Tests).