Chemical kinetics databases for many elementary gas-phase reactions provide the recommended values of the Arrhenius parameters, the temperature range of their validity, and the temperature dependence of the uncertainty of the rate coefficient k. An analytical expression is derived that describes the temperature dependence of the uncertainty of k as a function of the elements of the covariance matrix of the Arrhenius parameters. Based on this analytical expression, the various descriptions of the temperature dependence of the uncertainty of k used in the combustion, and in the IUPAC and JPL atmospheric chemical databases are analyzed in detail. Recommendations are given for an improved representation of the uncertainty information in future chemical kinetics databases using the covariance matrix of the Arrhenius parameters. Utilization of the joint uncertainty of the Arrhenius parameters is needed for a correct uncertainty analysis in varying temperature chemical kinetic systems. A method is suggested for the determination of the covariance matrix and the joint probability density function of the Arrhenius parameters from the present uncertainty information given in the kinetics databases. The method is demonstrated on seven gas kinetic reactions exhibiting different types of uncertainty representation.