We consider consequences of the first and second moments of the Dirichlet coefficients in one-parameter families of elliptic curves. Assuming standard conjectures (known for rational surfaces), Rosen and Silverman proved a conjecture of Nagao that the first moment is related to the rank; we use this to construct families of moderate rank by having large first moment sums. For non-CM one-parameter families, Michel proved the second moment of the Fourier coefficients is p^2 + O(p^{3/2}). Cohomological arguments show that the lower order terms are of sizes p^3/2, p, p^1/2 and 1. In every case we are able to analyze, the largest lower order term in the second moment expansion that does not average to zero is on average negative. The negative bias in these lower order terms has implications toward the excess rank conjecture and the behavior of zeros near the central point of elliptic curve L-functions.