__Introduction__

Handicapping sporting events has been applied to a wide range of human and animal competitive endeavors. Wikipedia defines handicapping sporting events as “the practice of assigning advantage . . . to equalize the chances of winning.”

Equalization of performance is the essential feature of any handicapping system. Most people have passing familiarity with “golf handicaps” which according to the USGA enable “players of differing abilities to compete on an equitable basis.” Similarly, according to HorseRacing.com, handicapping involves “the practice of adding weight to horses in an effort to equalize their performance.”

Within the sport of human long distance running, handicapping the performances of runners according to age is sometimes referred to as “Age Grading.” The goal of Age Grading is to equalize the performance and thus provide a “level playing field” for runners of differing ages. For example, a 30 year old and an 80 year old can compare their Marathon performances to see who performed *better for their age*. Or commonly, a 60 year old runner might compare his or her current speed with their speed from 20 years ago after adjusting for the effects of age.

Currently, the best known methods for age handicapping long distance running leverage single age world records in track and field and in road racing. Several individuals who have loaned their expertise to this endeavor are Howard Grubb, R. C. Fair, Elmer Sterken, and Alan Jones. Most of these systems for age-grading differ only slightly based on model assumptions and the date they were developed (i.e. some models may have had access to more recent world records.) Two popular calculators are: WMA Age-grading calculator and Aging in Sports and Chess.

Nevertheless, these methods of age-grading are not without controversy. In “Age-graded performances” Howard Grubb has worried that “super-veteran (aged over 60 or so) athletes run more slowly at the moment than expected.” In a 2003 article titled “From the cradle to the grave: How fast can we run?” Elmer Sterken reached a similar conclusion, as did I in a more recent large study of U.S. based 5K races.

__Age Handicapping Based on Population__

Since equalization is the essential feature of any handicapping system and for Age Grading in particular, we need to consider the sense in which the world record for, say, a 80 year old can be equated to the world record for a 30 year old. Or similarly how can the 30-34 age group winner in a local race be equated to the 75-79 age group winner?

Among adults, almost all of the single age world record holders for marathons, half-marathons, 10K road races, and 5K road races come from a country in the developed world, or from Kenya and Ethiopia. However, combining all of these countries shows that the male population of 80 year olds is only 30% of the population of 30 year olds. Obviously there are two possible reasons for the smaller population of 80 year olds: either fewer people were born 80 years ago than 30 years ago, or more of the older group has died. In either case, __the smaller number of potential competitors makes the older group somewhat less competitive__.

Frequently (as in “How Fast Do Old Men Slow Down?” by R.C. Fair and “Age grading running races” by Alan Jones), the very best single age world records for each distance are fitted with a model in an attempt to estimate the upper, “biological limit” or frontier of human performance. Factors derived from these models are then used by the above referenced Age Grading calculators.

However, another, and potentially more generalizable, way to understand these single age world records is to view them as the speeds attained by (single year) age group winners in an extremely large “race” consisting of everyone who has lived in the last 100 years or so. Thus, for example, with the road 10K, the world record for 63 year old males was set in 1994 by Ed Whitlock. Ed, then, is the winner among all 63 year olds who have ever been alive at some point in the past century.

Whether we consider age group winners in local races with 5-year age group intervals or single age world records, it is possible to formalize the impact of the size of the underlying age group population. For example, to compare the winners of two age groups, symbolized by “j” and “k”, let

P_{j}, P_{k} = the total populations, summed over the relevant geography, that fall into the j^{th} and k^{th} age groups, respectively.

W_{j}(s), W_{k}(s) = W_{j}, W_{k =} the cumulative probability distribution functions (cdf) for the speeds, s, of the __winners__ of the j^{th} and k^{th} age groups, respectively.

As is shown in the appendix, the winners of the two age groups will be at the *same percentile among their peers* and hence have equivalent age-adjusted performances when

W_{j} = W_{k}^(P_{j}/P_{k})

where “^” is the power operator, i.e.

For example, Racing Among the Ages presented information on 1283 5K races from all across the U.S. Included among these races were 356 which are classified into the “large race” category, having between 500 and 999 total finishers. Letting “j” be males aged 75-79 and “k” be males 30-34, we can use these 356 large races to illustrate how age-group population size can be employed to provide an __equalized__ comparison of the age group winners in these two age groups.

At the last census (2010), the U.S. Census Bureau estimated the U.S. population of males aged 75-79 was P_{j} = 3,182,388 and the population of males ages 30-34 was P_{k} = 9,996,500. Therefore P_{j}/P_{k} = 0.32.

By definition, the median speed for age group winners among males 30-34 occurs at the 50^{th} percentile, i.e. when W_{k} = 0.50. Substituting into the above formula shows

W_{j} = 0.50^0.32 = 0.80

Consequently, 80^{th} percentile among the M75-79 age group *winners* is equivalent to the 50^{th} percentile among the M30-34 age group *winners*.

The inclusive median speed among 356 races occurs at the midpoint between the 178^{th} and 179^{th} fastest age group winners. For M30-34 this value is 10.30 mph (18:05), and for M75-79 this median value is 4.94 mph (37:45). However, the median of the M75-79 does __not__ represent an equivalent performance among the peer group. The 80^{th} percentile for age group winners among M75-79 is 6.26 mph (29:49). Thus, a time of 18:05 for M30-34 is equivalent to a time of 29:49 for M75-79.

In summary, the winners of the “j”th and “k”th age groups in a particular race will be at the same percentile among their peers and hence have equivalent performance if

W_{j} = W_{k}^(P_{j}/P_{k})

Note that, in order to compare the performances of different age groups, it is not necessary to know anything about the distribution of speeds for individuals *within* either underlying age group population. Nor is it necessary to know the precise sizes of the underlying age group populations, P_{k} and P_{j}. All that is needed is the population ratio and the distribution of speeds among age group *winners*.

For both age group winners in local races and single age state running records, the distribution of the *winning* speeds for each age group can be determined by examining several local races or the single age records across several different states. However, by definition, there is just one world record for each age. Nevertheless, it is possible to look at the residuals from a fitted model to estimate the distribution of speeds among single age world record holders. In doing this, we note that the standard deviations from the fitted model increase with age and must be estimated appropriately.

__Future Article on Age Handicapping Competitive Running__

In a future article, we will apply this simple but elegant formula to age handicap 11 different racing venues with distances ranging between the 5K and the Marathon and competitiveness ranging from age group winners in small local races to single age world record holders. Moreover, the age handicapping system thus obtained is both simpler and substantially more accurate than current methods.

__Appendix: Computational Outline__

By definition any event or venue that is open to all comers has a sampling intensity or “Reach” (R) that is similar for each age group in the applicable geography. However, this does not mean that the expected number of actual participants in the race will be proportional to the population (P_{i}) for each age group. The expected number of participants in a given age group will be proportional to the product of the underlying population and the fraction (F_{i}) of that population that is **F**it and motivated enough to compete in a given race or venue. Thus the expected number of participants in an age group is R(F_{i})P_{i}.

In the earlier example, we saw that the U.S. population of males between 75 and 79 is 32% of the population aged between 30 and 34. However, in 1283 U.S. based races, there were only 3% as many individuals in the M75-79 group as were in the M30-34 group. Thus, among the older group approximately 10% as many are sufficiently Fit and motivated to participate in races. Undoubtedly, physical limitations prevent many older adults from participating.

The sampling intensity or “Reach” factor, R, would not come into play for world records (except possibly for the impact of various international political considerations), i.e. it has a value of 1. However, based on marketing, each local race can have its own value for R since some individuals who are fit and willing to participate in a race may not hear about it in time to register; or since some individuals may not participate in a particular event because they have chosen another more desirable event that occurs at the same time.

For any particular event, the number of individuals in the applicable age group population who are unfit or unwilling to compete is (1-F_{i})P_{i}. Had these individuals been fit and willing to participate, we would expect R(1-F_{i})P_{i }of them to have participated in the event. Nevertheless, in evaluating an age group winner’s performance among his peers, it reasonable to consider him faster than both all of his peers who participated in the race, plus the expected number of potential participants who did not participate because they are unable or too slow to complete the race successfully. Thus the age-group winner is the fastest among R(F_{i})P_{i}+ R(1-F_{i})P_{i }= RP_{i} peers.

Then for a given distance (e.g. marathon, half marathon, 10K, 5K) and gender, let

s = speed of an individual at the event.

R = the fraction of individuals in the entire population who participate in the event among all those who are fit and otherwise capable.

P_{i} = the total population, summed over the relevant geography, that falls into the “i”th age group.

F_{i} = the fraction of the “i”th age group that is fit and motivated enough to compete in a given race or venue

E_{i}(s) = E_{i} = the cumulative probability distribution function (cdf) for speed in the “i”th age group; i.e., it is the percent of the __entire__ population falling into the “i”th age group that are slower than or equal to a speed of “s”. Note that the fraction of individuals in the “i”th age group who are either unable or unwilling to compete in the race is simply E_{i}(0).

W_{i}(s) = W_{i} = the cumulative probability distribution function (cdf) for the speed of the __winners__ in the “i”th age group.

Since W_{i} is the cdf of the maximum for a sample of size RP_{i} with cdf E_{i},

W_{i} = E_{i}^(RP_{i})_{ }

_{ }thus

E_{i }= W_{i}^(1/RP_{i})

Suppose two individuals belong to different age groups, the “k”th age group and the “j”th age group. Among their peers, their performances will be equivalent if they each achieve the same percentile; i.e. if

E_{k} = E_{j }

Consequently, the winners of these age groups will be at the same percentile among their peers when

W_{j}^(1/RP_{j}) = W_{k}^(1/RP_{k})

Simplifying this expression yields

W_{j}^(1/P_{j}) = W_{k}^(1/P_{k})

W_{j} = W_{k}^(P_{j}/P_{k})

It is important to note that this result does not depend on the functional form of the population cdf, E_{k}(s) and E_{j}(s), for either age group. Nor does it depend on knowledge of the exact population, P_{k} and P_{j}, of either age group. All that is needed is the population ratio.