An empirical way to **estimate the race time** is to use the time
and the distance
of a recent run in the so-called **Riegel formula**. Pete Riegel proposed the
following formula in 1977:

t = t_{in} * (d/d_{in})^{1.06}

with t_{in} and d_{in} respectively the time and the distance of the
known performance and t is the estimated time to travel the distance d.

This model uses the top 10 times in the world at various distances to compute performances across distances. In his article Cameron states that to obtain the following formulas he used non-linear regression methods:

t = (t_{in}/d_{in}) * d *(13.49681 - 0.000030363 * d_{in} + 835.7114/d_{in}^{0.7905}) / (13.49681 - 0.000030363 * d + 835.7114/d^{0.7905})

with t_{in} and d_{in} respectively the time [s] and the distance [m] of the
known performance and t [s] is the estimated time to travel the distance d [m].

**VO _{2} Max** (also maximal oxygen consumption or maximal aerobic capacity)
is the maximum rate of oxygen consumption
as measured during incremental exercise.

The estimate based on the VO

VO_{2} Max = (-4.60 + 0.182258 * v + 0.000104 * v^{2}) / (0.8 + 0.1894393 * e^{-0.012778*tin} + 0.2989558 * e^{-0.1932605*tin) }

with v = d_{in}/t_{in}, where d_{in} the distance in meters
and t_{in} the time in seconds of the performance entered.
VO_{2}max is expressed in millilitres of oxygen per kilogram of body mass
per minute (e.g., ml/(kg·min)).

The same formula is then used in reverse to estimate the times by using the same
VO_{2} Max found at the beginning. There is no analytic solution to solve
this problem, but the solution can be found numerically (we use the Newton's Method).

This method is the only that uses age and gender of the runner.
With the so called **age grading prediction** we assume that
you will able to run the same age-graded performance at every distance.
The age-graded score is the percentage ratio between the
world record for your
age and gender and the time of your performance.

So, for example, a 40-year-old
man that runs 10 km in 50 minutes has a age-graded score equal to 55.4%. This value
is used to predit times at differen distances.

Distance | Unit | Time [hh:mm:ss] | Gender | Age |