Although the relationship between strength and physical performance in older adults is probably non-linear, few empirical studies have demonstrated that this is so. In a population-based sample of adults aged 60-96 years (n = 409), leg strength was measured in four muscle groups (knee extensor, knee flexor, ankle plantar flexor, ankle dorsiflexor) of both legs using an isokinetic dynamometer. A leg strength score was calculated as the sum of the four strength measurements in the right leg. Usual gait speed was measured over a 15.2 metre course. With a linear model, leg strength explained 17% of the variance in gait speed. Non-linear models (quadratic and inverse) explained significantly more variance (22%). The nature of the non-linear relationship was that, in stronger subjects, there was no association between strength and gait speed, while in weaker subjects, there was an association. Body weight and age also explained significant amounts of variance in gait speed, while sex and height did not. The results supported the hypothesis of a non-linear relationship between leg strength and gait speed that is similar for older men and women. This finding represents a mechanism for how small changes in physiological capacity may have substantial effects on performance in frail adults, while large changes in capacity have little or no effect in healthy adults.