  The large imbalance in the neutron and proton densities in very neutron rich
systems increases the nuclear symmetry energy so that it governs many aspects
of neutron stars and their mergers. Extracting the density dependence of the
symmetry energy therefore constitutes an important scientific objective. Many
analyses have been limited to extracting values for the symmetry energy, $S_0$,
and its ``derivative'', $L$, at saturation density $\rho_0 \approx 2.6 \times
10^{14}~\mathrm{g/cm^3}$ $\approx 0.16~\mathrm{nucleons/fm^{3}}$, resulting in
constraints that appear contradictory. We show that most experimental
observables actually probe the symmetry energy at densities far from $\rho_0$,
making the extracted values of $S_0$ or $L$ imprecise. By focusing on the
densities these observables actually probe, we obtain a detailed picture of the
density dependence of the symmetry energy from $0.25\rho_0$ to $1.5\rho_0$.
From this experimentally derived density functional, we extract
$L_{01}=53.1\pm6.1 MeV$ at $\rho \approx 0.10~\mathrm{fm^{-3}}$, a neutron skin
thickness for $^{208}Pb$ of $R_{np} =$ $0.23\pm0.04$ fm, a symmetry pressure at
saturation density of $P_0=3.2\pm1.2 MeV/fm^3$ and suggests a radius for a 1.4
solar mass neutron star of $13.1\pm0.6$ km.
