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!)


Have a look at this lecture.


It explains how to calculate sample size for any given level of power in detail. The power is the probability of not making a type 2 error, so it is introduced at the point where you decide on the effect size you want to detect.

  • $\begingroup$ Wanna add more details to this answer? $\endgroup$ – zthomas.nc Mar 1 at 17:18

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