Application of general unit hydrograph model for marathon finish time distributions

The marathon finish time distribution is the probability distribution of the time a group of runners completes a marathon race. The knowledge of this distribution is required to optimize a running process when planning and operating a marathon; it is also required to compare different marathons and evaluate individual running performance. This research demonstrated that the marathon finish time distribution can be described by the general unit hydrograph model from applied hydrology. Specifically, we found that marathon finish time data, in terms of distribution density, often show a positively skewed distribution, which implies that a few runners are superfast, many runners are very slow, and most runners are between the two extremes. This asymmetrical feature is similar to a unit hydrograph that describes rainfall-runoff processes in watersheds. We thus hypothesized that the marathon finish time distribution follows Guo's three-parameter general unit hydrograph model. To test this hypothesis, we fit the general unit hydrograph model to the 2013–2021 New York City (NYC) Marathon finish time distribution data, resulting in determination coefficients r²≥0.9997. We next validated the proposed model with the world marathon database including 35 million results in 2000–2021 from more than 28,000 long-distance foot races, which was then compared with the NYC marathon data. According to the two inflection times of the finish time density function (asymmetrical bell curve) of the world marathons, we classified all marathoners into three categories: super runners who finish before the left inflection time; endurable runners who finish after the right inflection time; and fast runners who finish between the two inflection times. Finally, we compared the general unit hydrograph model and two typical asymmetrical distribution functions with the world marathon data and found that the proposed model describes the data better.