Two families of kurtosis measures are defined as K1(b) = E[ab-|z|] and K2(b) = E[a(1 - |z|b)] where z denotes the standardized variable and a is a normalizing constant chosen such that the kurtosis is equal to 3 for normal distributions. K2(b) is an extension of Stavig's robust kurtosis. As with Pearson's measure of kurtosis β2 = E[z4], both measures are expected values of continuous functions of z that are even, convex or linear and strictly monotonic in ℜ- and in ℜ+. In contrast to β2, our proposed kurtosis measures give more importance to the central part of the distribution instead of the tails. Tests of normality based on these new measures are more sensitive with respect to the peak of the distribution. K1(b) and K2(b) satisfy Van Zwet's ordering and correlate highly with other kurtosis measures such as L-kurtosis and quantile kurtosis.