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Let's say we have a country with 3000 people, divided into 3 cities containing 1000 people. We want to know the proportion of women in the whole country so we decide to take a sample of 30 people, 10 from each town. Suppose each town has the same sample mean and variance.

Using the formula for the variance of stratified random sampling:

$$s^2 = \sum_{h=1}^{3} (\frac {N_h}{N})^2 \frac {s_h^2}{n_h}=\sum_{h=1}^{3} (\frac {100}{300})^2 \frac {s_h^2}{10}=\frac{s_h^2} {30} $$

But we know that when we have 3 equal samples os variance ##s_h^2## the resulting sample has ##s_h^2/3##

So what was my mistake?