# AB test sample size calculation by hand

Evan Miller has created a well-known online AB test sample size calculator. For the sake of being able to program and modify this formula, I would like to know how to calculate sample size Evan Miller-style by hand.

Personally, I'll calculate such a metric by working backwards from how we calculate a 95% confidence interval with the z-test of proportions around the difference in conversion between the two variations ($$\hat{d}$$) by setting it zero.

I'll define/assume:

• $$\alpha$$ = .05, $$\beta$$ = .2
• a 50/50 split between the control and experiment, i.e. $$n\_exp$$ = $$n\_control$$
• the control conversion rate, i.e. the base rate before the experiment = $$c$$
• $$p$$ = pooled conversion rate = (number of exp conversions + number of control conversions/ (n_control + n_experiment)) -> in this context -> $$(nc+n(c+\hat{d}))/2n$$ = $$(2c+\hat{d})/2$$

Now time to solve for $$n$$ ...

$$\hat{d} + Z_{(1+\alpha)/2} * StandardError = 0$$ $$\hat{d} + 1.96 * StandardError = 0$$ $$\hat{d} + 1.96 * \sqrt{p(1-p)(\frac{1}{n\_exp} + \frac{1}{n\_control})} = 0$$ $$\hat{d} + 1.96 * \sqrt{p(1-p)(\frac{2}{n})} = 0$$ $$\sqrt{p(1-p)(\frac{2}{n})} =\frac{-\hat{d}}{1.96}$$

with more simplifying we get to:

$$\frac{(1.96^2) 2p(1-p)}{\hat{d}^2} = n$$ $$\frac{(1.96^2) (2c+2c\hat{d}-2c^2+\frac{3}{2}\hat{d}^2)}{\hat{d}^2} = n$$

At the moment though, my calculation doesn't incorporate power (1-$$\beta$$) but Evan Miller's does.

What should I think about as next steps to incorporate power into my sample size calculation?

(Feel free to also point out other errors in my calculation or assumptions!)