Tariff indices such as trade-weighted means, variances, and coefficients of variation are commonly used despite their lack of a theoretical foundation. This paper provides a theoretical framework for the index-number problem for tariffs. This paper shows that the Anderson and Neary (1993) trade-restrictiveness index is a function of “marginal trade-weighted moments” of the tariff schedule, higher mean and generalized variance both implying a higher TRI. In the CES case, the TRI is increasing in the trade-weighted mean and in the variance of the tariff schedule. For this special case the mean and the variance are partially legitimized.