Given PDF and CDF of a distribution, how does one find an approximation of $(\operatorname{Var}(X_i))$ using a normal approximation of $(X_i)$?

According to "Mathematica Laboratories for Mathematical Statistics" by Jenny Baglivo (Theorem 9.1 on page 120), it is said that:

$\operatorname{Var}(X_i) \approx p(1-p) / (n+2)f(x)^2$ where $f$ is PDF, $x$ is $p^\text{th}$ quantile of the $X$ distribution, and $p = i/(n+1)$

I have $0$ ideas on how we get to this result. I am absolutely drawing at a blank. Help is very much appreciated!

  • $\begingroup$ Welcome to CV, blamp. The community treats questions about homework and self-study a little differently. I recommend editing your questions (use the "edit" link in the lower left) to include the [self-study] tag. Cool? Cool. $\endgroup$
    – Alexis
    Jan 30, 2018 at 22:43