We provide more sample-efficient versions of some basic routines in quantum data analysis, along with simpler proofs. Particularly, we give a quantum "Threshold Search" algorithm that requires only $O((\log^2 m)/\epsilon^2)$ samples of a $d$-dimensional state $\rho$. That is, given observables $0 \le A_1, A_2, ..., A_m \le 1$ such that $\mathrm{tr}(\rho A_i) \ge 1/2$ for at least one $i$, the algorithm finds $j$ with $\mathrm{tr}(\rho A_j) \ge 1/2-\epsilon$. As a consequence, we obtain a Shadow Tomography algorith…