Difference between pooled variance equations

Question

I'm currently doing A-level Further Maths A2. I've seen two different equations to calculate the estimate of pooled variance.

I do not know when to use which and what makes each significantly different. And I could not find any good resource to help me either.

So far, most of the people I asked said I should just use the bottom equation and disregard the other one. But they don't tell me a good reason why, other than that they're "more or less the same" and the bottom one is better.

Edit:

http://www.stat.yale.edu/Courses/1997-98/101/meancomp.htm This article helped me out. The first formula is used for unknown means and known sd of two samples which may not be the same (which is often the case in biology.) The second formula is used for when it is assumed that the two samples have the same sd, (e.g. samples come from the same population) Only the second one is considered a pooled estimate.

Answer 1

The first formula is not more or less the same with the second one. Just substitute a few values, you'll see the difference. Or, you might set $n_1=n_2$, choose them very large etc. you will obtain very different results.

For the second one, the intuition is very simple. For the sake of simplicity assume we have $n_i$ instead of $n_i-1$, which comes due to Bessel's correction. What you do is just weighted averaging based on how many samples each set has, i.e. $s_1^2\frac{n_1}{n_1+n_2}+s_2^2\frac{n_2}{n_1+n_2}$.