### The fastest one

It is quite easy to define the fastest biathlete in a concrete race, using skiing speed statistics. But it is not that easy to define the fastest biathlete in the season. The main difficulty here is the selection of way to sum up sportsmen's results in different races.

Let us first try to select any 2 biathletes, say Ivanov and Petrov, and define the fastest of them. Let us assume that there were 10 races in the season, in which they both participated, and in 7 of them Ivanov was faster than Petrov, while in 3 of them Petrov was faster than Ivanov. It is quite logical to suppose that by season results generally Ivanov was faster than Petrov, and Ivanov's speed race is to Petrov's speed race as 7 is to 3. Let us call this correspondence Ivanov's and Petrov's paired rating. The concrete values of Ivanov's and Petrov's ratings do not matter, they can be equal to 7 and 3, 70 and 30, 350 and 150, the main thing is that they should be to each other as 7 to 3.

Taking a different pair of biathletes, say Sidorov and Pupkin, we can calculate their paired rating. Let us assume that there were 7 races in the season, in which they both participated, and in 5 of them Sidorov was faster than Pupkin. It means that their paired rating is 5 to 2. Making similar calculations for each pair of biathletes, we can define their paired ratings.

Now in order to make a single table for all biathletes, we need to sort out their individual ratings in such a way that the correspondence of individual ratings for each pair of biathletes would be equal to their paired rating. We can do it by making and solving a system of linear algebraic equations. The rating calculated in such a way is called E-Rating.

Following the links below you can view biathletes' speed rating calculation results. The calculation takes into account only non-team races and the results of those sportsmen who has at least 10 non-team races in the season. Zero figures in tables stand for very small rating values which have been approximated down to zero.