  We consider the jellium model of $N$ particles on a line confined in an
external harmonic potential and with a pairwise one-dimensional Coulomb
repulsion of strength $\alpha > 0$. Using a Coulomb gas method, we study the
statistics of $s = (1/N) \sum_{i=1}^N f(x_i)$ where $f(x)$, in principle, is an
arbitrary smooth function. While the mean of $s$ is easy to compute, the
variance is nontrivial due to the long-range Coulomb interactions. In this
paper we demonstrate that the fluctuations around this mean are Gaussian with a
variance ${\rm Var}(s) \approx b/N^3$ for large $N$. We provide an exact
compact formula for the constant $b = 1/(4\alpha) \int_{-2 \alpha}^{2\alpha}
[f'(x)]^2\, dx$. In addition, we also calculate the full large deviation
function characterising the tails of the full distribution ${\cal P}(s,N)$ for
several different examples of $f(x)$. Our analytical predictions are confirmed
by numerical simulations.
