  We study theoretically and experimentally the quantification of non-Gaussian
distributions via non-destructive measurements. Using the theory of cumulants,
their unbiased estimators, and the uncertainties of these estimators, we
describe a quantification which is simultaneously efficient, unbiased by
measurement noise, and suitable for hypothesis tests, e.g., to detect
non-classical states. The theory is applied to cold $^{87}$Rb spin ensembles
prepared in non-gaussian states by optical pumping and measured by
non-destructive Faraday rotation probing. We find an optimal use of measurement
resources under realistic conditions, e.g., in atomic ensemble quantum
memories.
