Power statistics of a finite group

Definition

The power statistics of a finite group $G$ of order $n$ can be viewed in the following ways:

The non-cumulative power statistics is a function from the set of divisors of $n$ to the nonnegative integers, which sends $d$ to the number of elements $g\in G$ for which there exists $h\in G$ satisfying $h^{d}=g$ but such that there is no larger divisor $d'$ of $n$ for which there exists $k\in G$ with $k^{d'}=g$ .
The cumulative power statistics is a function from the set of divisors of $n$ to the nonnegative integers, which sends $d$ to the number of elements $g\in G$ for which there exists $h\in G$ satisfying $h^{d}=g$ .

Order statistics of a finite group: This gives information on the number of elements of each order.
Order-cum-power statistics of a finite group: This gives information on the number of elements of a given order that arise as given powers.