@bobthechemist The definitions are really the same. The first one can be written as FullSimplify[(K/(K - 1)) (1 - Sum[v[i]^2, {i, K}]/(Sum[v[i]^2, {i, K}] +
2 Sum[cov[i, j], {i, 2, K}, {j, 1, i - 1}])) /. {Sum[v[i]^2, {i, K}] -> K vbar,
2 Sum[cov[i, j], {i, 2, K}, {j, 1, i - 1}] -> K (K - 1) cbar}]