When is a statistic not a statistic? I have a fairly involved ``statistic'' that involves transformation of a set of samples $\mathbf{x}$ by a massive machine-learned matrix and subsequent nonlinear processing into said statistic $\mathbf{x}_s$. However, it still is reflective of the choice of $\mathbf{x}$.
I am trying to perform a bootstrapped test on the differences between two populations by using two samples, $\mathbf{x}_1$ and $\mathbf{x}_2$. This test calls for using the standard deviation of the samples; however, this data is a set of transformed three-hot vector inputs and one-hot vector selections.
While the inputs and outputs are linked, is it still fair to call $\mathbf{x}_s$ a statistic of $\mathbf{x}$, given how much processing is involved?
 A: This answer is a theoretical supplement to Tim's more practical answer (+1).
There is an axiomatic and mathematical side of statistics. Random variables are measurable functions on the outcome space $\Omega$ of a probability space $(\Omega, \mathcal{F}, P)$. Data are modelled to be instances of random variables, that is subsets of the image of the random variable. In this theoretical context, a statistic is any function of one or more random variables; a functional. Functions of data are then thought of as instances of a statistic. It sometimes creates confusion that the word "statistic" is often used either as a function of random variables or as a function of data, with the former being a model of the latter.
I am not suggesting you think in these terms all the time, or even most of the time depending on what it is you are into doing, but you might like to know that such a perspective exists.

If you are ever curious about this theoretical perspective applied to bootstrapping, see Chapter 17 of Measure Theory and Probability Theory by Athreya and Lahiri 2006.
A: The statistic is defined as

A statistic is a function $T (X^n )$ of the data.

(Larry Wasserman All of Statistics, p. 137)

A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose.

(Wikipedia)


Definition 5.2.1 Let $X_1,\dots,X_n$ be a random sample
of size $n$ from a population and let $T(X_1,\dots,X_n)$ be
a real-valued or vector-valued function whose domain includes
the sample space of $(X_1,\dots,X_n)$. Then the random variable
or random vector $Y = T(X_1,\dots,X_n)$ is called a statistic.
The probability distribution of statistic $Y$ is called the
sampling distribution of $Y$.

The definition of a statistic is very broad, with the only restriction
being that a statistic cannot be a function of a parameter. [...]

(Casella and Berger Statistical inference, p. 211)
TL;DR If you calculate something from the data, "for statistical purposes", it's a statistic.
As noticed by @SextusEmpiricus in the comments, another restriction that may not be instantly obvious to everyone is that $T$ is a function in a mathematical sense, that is, a mapping

a function from a set $X$ to a set $Y$ assigns to each element of $X$ exactly one element of $Y$

So something generating the results at random wouldn't fit the definition.
