Experiments on biodiversity have shown that productivity is often a decelerating monotonic function of biodiversity. A property of nonlinear functions, known as Jensen's inequality, predicts negative effects of the variance of predictor variables on the mean of response variables. One implication of this relationship is that an increase in spatial variability of biodiversity can cause dramatic decreases in the mean productivity of the system. Here I quantify these effects by conducting a meta-analysis of experimental data on biodiversity–productivity relationships in grasslands and using the empirically derived estimates of parameters to simulate various scenarios of levels of spatial variance and mean values of biodiversity. Jensen's inequality was estimated independently using Monte Carlo simulations and quadratic approximations. The median values of Jensen's inequality estimated with the first method ranged from 3.2 to 26.7%, whilst values obtained with the second method ranged from 5.0 to 45.0%. Meta-analyses conducted separately for each combination of simulated values of mean and spatial variance of biodiversity indicated that effect sizes were significantly larger than zero in all cases. Because patterns of biodiversity are becoming increasingly variable under intense anthropogenic pressure, the impact of loss of biodiversity on productivity may be larger than current estimates indicate.