Heavy tailed models assign much higher probabilities to extreme discharge values than the normal distribution. Traditionally, these models have been applied to the entire temporal domain of a given set of river discharge data, regardless of the potential over- or under-estimation of tail probabilities resulting from the natural variability of the controlling mechanisms. To prove the hypothesis that such indiscriminate application of models to the river discharge leads to bias in the estimates, we utilized a method for dividing a given daily river discharge data set into three annual seasons: dry, transition, and wet. The data sets were acquired from the Global Runoff Data Centre as well as the USGS Surface Water Information site, including rivers in the US and Europe. We then applied some heavy tailed probability models to the discharge values associated with each season individually. We found that for many rivers the discharge from dry, transition, and wet seasons could each be described with a particular heavy tailed probability distribution. The distinction is most pronounced in large rivers such as the Umpqua in Oregon and the Rhine in the Netherlands. Although there are several exceptions, typically, the discharge recorded during the wet season tends to follow the lognormal distribution, while the discharge recorded during the dry and transition seasons tends to follow the power-law distribution. Further implications will be discussed.