For the measurement of elasticity, given two observations of a bivariate relationship, the arc elasticity formula has been traditionally used by economists and statisticians. However, no proper statistical justification for this procedure exists in the literature. In this paper measures of elasticity on an arc are derived using widely accepted statistical criteria such as the minimum absolute deviation criterion and the least squares criterion. It is shown that in the linear bivariate case the minimum absolute deviation elasticity is the arc elasticity. However, the formulae according to the least squares criterion and other criteria differ from arc elasticity even in the linear case. A numerical comparison of formulae is also provided and these are assessed on the basis of a 'goodness of fit' statistic developed for this purpose.