The speed of convergence of the distribution of the normalized maximum, of a sample of independent and identically distributed random variables, to its asymptotic distribution is considered in this article. Assuming that the cumulative distribution function of the random variables is known, the error committed replacing the actual distribution of the normalized maximum by its asympotic distribution is studied. Instead of using the arithmetical scale of probabilities, we measure the difference between the actual and asympotic distribution in terms of the double-log scale used for building the probability plotting paper for the the latter. We demonstrate that the difference between the double-log values corresponding to two probabilities in the upper tail is almost exactly equal to the logarithm of the distribution may not be uniform in this double-log scale and that the difference between the actual and asymptotic distributions, on the probebility plotting paper, may be a logarithmic, power, or even exponential function in the upper tail when the latter distribution is of Fisher-Tippett type I, but that difference is at most logarithmic in the upper tail for type II and III distributions. This fact is exploited to obtain transformed variables that converge tothe asymptotic distribution faster than the original variable on the probabilites plotting paper