Sample size calculation with different standard deviations when calculating the required sample size with a given power and p threshold, usually the calculator only requires two estimated population mean and standard deviation. is it always valid to assume the two populations' standard deviations are the same? what if they are not the same?
An example of sample size/power calculator here: https://www.stat.ubc.ca/~rollin/stats/ssize/n2.html
 A: For $\mu_1 = 100, \mu_2 = 110, \sigma = 10, \alpha = 0.05$ and desired power $0.8,$ your online calculator for a pooled 2-sample t test gave required sample sizes $n_1 = n_2 = 16.$
The following simulation in R can be used to see if this result is valid.
With 100,000 iterations we can expect about two place accuracy, so
agreement is pretty good.
set.seed(2020)
pv = replicate(10^5, 
 t.test(rnorm(16,100,10),rnorm(16,110,10),var.eq=T)$p.val)
mean(pv <= 0.05)
[1] 0.7812

If you want to know the power of a Welch t test (in which population SDs
may differ) with parameters $\mu_1 = 100, \mu_2 = 110,$ $\sigma_1 = 10, \sigma_2=15,$ $n_1 = 12, n_2 = 20, \alpha = 0.05,$ then you can make obvious changes in the simulation to obtain power about 59%.
set.seed(1124)
pv = replicate(10^5, 
 t.test(rnorm(12,100,10),rnorm(20,110,15))$p.val)
mean(pv <= 0.05)
[1] 0.58547

So, in order to accommodate the different SDs (one larger than above), you might
consider increasing sample sizes to $n_1 = 25, n_2=35,$ assuming it is
easier or cheaper to sample from the second population. Then the power
is about 86%.
set.seed(1234)
pv = replicate(10^5, 
 t.test(rnorm(25,100,10),rnorm(35,110,15))$p.val)
mean(pv <= 0.05)
[1]  0.85936

Roughly speaking, balanced designs with $n_1=n_2,$ give
the best power for a given total sample size $n_1+n_2.$ Unless there is
a good reason for using an unbalanced design, it is best to avoid them.
Notes: For pooled tests, there are fairly simple formulas involving
noncentral t distributions, which make simulation unnecessary. If
population SDs do not differ by much, you might use formulas for a
pooled test as a starting place for designing an experiment for a Welch test. I have not seen formulas for Welch
tests; only simulation results.
