Most climate impact studies rely on changes in means of meteorological variables, such as temperature, to estimate potential climate impacts, including effects on agricultural production. However, extreme meteorological events, say, a short period of abnormally high temperatures, can have a significant harmful effect on crop growth and final yield. The characteristics of daily temperature time series, specifically mean, variance and autocorrelation, are analyzed to determine possible ranges of probabilities of certain extreme temperature events [e.g., runs of consecutive daily maximum temperatures of at least 95°F (35°C)] with changes in mean temperature of the time series. The extreme temperature events considered are motivated primarily by agricultural concerns, particularly, the effects of high temperatures on corn yields in the U.S. Corn Belt. However, runs of high temperatures can also affect, for example, energy demand or morbidity and mortality of animals and humans. The relationships between changes in mean temperature and the corresponding changes in the probabilities of these extreme temperature events are quite nonlinear, with relatively small changes in mean temperature sometimes resulting in relatively large changes in event probabilities. In particular, the likelihood of occurrence of a run of five consecutive daily maximum temperatures of at least 95°F under a 3°F (1.7°C) increase in the mean (holding the variance and autocorrelation constant) is about three times greater than that under the current climate at Des Moines, Moreover, by allowing either the variance or the autocorrelation as well as the mean to change, this likelihood of a run event varies over a relatively wide range of values. These changes in the probabilities of extreme events need to be taken into consideration in order to obtain realistic estimates of the impact of climate changes such as increases in mean temperature that may arise from increases in atmospheric carbon dioxide concentration.