The sport of biathlon combines two disciplines, shooting and cross-country skiing. That leads to the obvious question which of those two elements has the bigger effect on the overall result.

I came up with the idea to use the three statistical values which I talk about often – shooting percentage, shooting time and skiing speed – and put them into relation with the Overall World Cup rank. One way to do that is by looking at the data as a **system of linear equations **(a general system looks like this):

Of course the four sets of data are incompatible (i.e. the World Cup rank is a dimensionless quantity, the shooting time has a physical dimension, seconds). A way around is making all four values *ranks*, more precisely a ranking for each category among athletes with World Cup points. That leads to a linear system which looks something like this:

This system of linear equations is *overdetermined*, i.e. there are more equations than unknowns, and *inconsistent*, i.e. it has no solution. Luckily there are ways to finding an **approximate solution**, for example the method of least squares. The linear system *Ax = b* has the approximate (least squares) solution *x = (A’A) ^{-1}A’b*.

After finding the least squares solution, the ratio between x1, x2 and x3 gives us the approximate influence of the shooting percentage, the shooting time and the skiing speed for an athlete’s World Cup rank.

Men | Non-Team Shooting % | Shooting Time | Skiing Speed % |

Top 10 | 0.19 | 0.13 | 0.67 |

Top 20 | 0.49 | 0.01 | 0.50 |

Top 30 | 0.50 | -0.04 | 0.54 |

Top 40 | 0.23 | 0.06 | 0.71 |

Top 60 | 0.26 | 0.10 | 0.64 |

Top 104 | 0.24 | 0.07 | 0.70 |

Average: | 0.32 | 0.05 | 0.63 |

**Influence of Shooting and Skiing**on World Cup rank |

**Men**

(0 = no effect, 1 = single factor)

Among all 104 male athletes who won World Cup points last season, the skiing speed was clearly the most important factor. The skiing speed influence on the World cup rank varied between 50 % and 71 %, depending on what group you look at.

There is the very unusual effect that for the men’s top 20 and top 30 athletes the shooting percentage briefly becomes very important, while it plays a much smaller role overall (top 104) and for the top 10. My best guess would be that that’s the region where the chart of the overall skiing pace flattens out, and therefore the shooting briefly becomes a more important factor.

Women | Non-Team Shooting % | Shooting Time | Skiing Speed % |

Top 10 | 0.24 | 0.10 | 0.66 |

Top 20 | 0.29 | 0.09 | 0.62 |

Top 30 | 0.43 | 0.05 | 0.53 |

Top 40 | 0.35 | 0.06 | 0.60 |

Top 60 | 0.23 | 0.02 | 0.75 |

Top 98 | 0.17 | 0.02 | 0.81 |

Average: | 0.28 | 0.06 | 0.66 |

**Influence of Shooting and Skiing**on World Cup rank |

**Women**

(0 = no effect, 1 = single factor)

The results for the women don’t look fundamental different. The skiing speed is slightly more important (81 %) for where an athlete is eventually ranked in the Overall World Cup. Also like for the men, there is the same curious effect that the shooting percentage effect reaches its maximum for the top 30 athletes.

Some general observations:

- The
**skiing pace**is the most important factor for every group listed above. Overall its influence on the World Cup rank last season was about 65 % on average, pretty much across the board, both for men and women. **Shooting percentages**play a lesser, but still significant role, with a 25-30 % influence.- While
**shooting times**have by far the smallest effect, it’s not negligible. The shooting speed accounts for about one sixth to one seventh of the total shooting influence.

The Overall World Cup rank last season was (very roughly!) determined like this: **Shooting accuracy 30 %, Shooting speed 5 %**, **Skiing speed 65 %**. Interestingly, that seems to be true for men and women alike.

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