Let $n$ be an even number. We draw values $a_1,\dots,a_n$ independently, uniformly from $[0,1]$. The expected values of the numbers, from largest to smallest, are $\frac{n}{n+1},\frac{n-1}{n+1},\dots,\frac{1}{n+1}$. Let $A$ be the sum of the $n/2$ largest numbers. So $E[A]$ is (approximately) $3n/8$.
How can we bound the probability that $A$ deviates from $E[A]$ by more than a constant multiplicative factor? For example, what is an upper bound for $Pr[A<n/4]$? Normally we would use Chernoff's bound, but the problem here is that the variables are not independent.
I guess $n$ had better be even in order for $A$ to be well-defined.
Conditional on the order statistic $X_{(n/2)} = t$, the top $n/2$ order statistics are uniform on $(t,1)$. Thus the conditional distribution of $A$ given $X_{(n/2)} = t$ is that of $tn/2 + (1-t) B$, where $B$ is the sum of $n/2$ independent $U(0,1)$ random variables. Thus
$$Pr(A < r n \mid X_{(n/2)} = t) = Pr\left(B < \dfrac{rn - nt/2}{1-t} = \dfrac{n(2r-t)}{2(1-t)}\right)$$
With high probability, $X_{(n/2)}$ is close to $1/2$, while if $(2r-t)/(2(1-t)) \ne 1/4$ we can use Chernoff bounds on the right side.