Aimo Niemi, Turku 1997

**Introduction.**
It is well known that talent, age and exercise
are the three most important factors, which affect the result
of an athlete. The aim of this study is to find out the effect
of the age alone. To do this the WR results of 43 events were
analyzed using the records of September 1. 1997. Altogether the
material comprised over 500 single records of the athletes of
ages 25-95.

**Method of the analysis.** The method of the analysis was as
follows. From the WR results of an event a mathematical formula
was looked for, which would best explain the aging effect of
the records. When over one hundred two parameter formulas were
tested, the best fit was always obtained with the formula

(1) time = r_{25}/[1+(25/b)³ -(age/b)³ ].

In the formula r_{25} is a result of an imaginary
athlete of age
25. Other records of the event were then compared to this time
resulting a normalized scale, which allows the comparison of
different events. Table 1 gives examples
of the procedure and pictures 1
and 2 show graphically the same thing. The values
of the parameters r_{25} and b and their
mathematical error
estimates are given in the table 2.
Using these values a
normalized record set was created and then the hole sample was
analyzed. Again over one hundred equations were tested and the
best fits were studied. This time the top of the list was very
even showing only marginal disparity between different
formulas. When some of the most exotic were passed, the best
fit to the sample was found with the formula

(2) y² = (z-age)*(z+age)/s² , z = 98.22 ± 0.23, s = 94.02 ± 0.40.

Picture 3 shows the result graphically. In it the shape of the line is defined by the parameter z whereas parameter s fixes the scale of the y axis. If needed, we get y=1 exactly with age=25 by using values z= 98.22, s=94.99 or with sufficient accuracy to most practical purposes

(3) y² = (98+age)*(98-age) / 95².

**Discussion.** The result obtained confirms well the two previous
analyses, which the author has made with two completely
different data sets. Altogether nearly 1600 data points have
been used and so far, within the error limits, the result
obtained has always been the same. This shows that formula (2)
describes the aging effect very well and allows thus a simple
method to compare the results of the athletes of different
ages. Even the comparison of different events is possible, if
formula (3) is used together with the parameters r_{25}
from the table 2.

As an example suppose an athlete A, age=50, has got a time 1860
sec in men's 10 km running and athlete B, age=80, time 105 sec
in men's 100 m breaststroke. Using formula (3) we calculate the
aging index for A and B and get y=0.8872 for A and y=0.5958 for
B. Multiplying the results of A and B with these values we get
the age-adjusted results 1650.2 and 62.59. These are the
results A and B would be capable, if they both were 25 years
old. From table 2 we find the corresponding
reference times
1621.9 and 63.26 sec (r_{25}), which means that B is
better than A. His time is 1.1% better than reference time whereas
A's time is 1.7% slower.

TABLE 1. WR97 results of men's 100 m breaststroke and 10000 m running. Actual times are in seconds and normalized speeds are those obtained, when compared to the result of an imaginary athlete age 25.

age | 100 M BRST | 10000 M RUNNING | ||
---|---|---|---|---|

time | speed | time | speed | |

25 | 63.26 | 1 | 1621.9 | 1 |

25-30 | 64.11 | 0.9867 | - | - |

30-35 | 65.65 | 0.9636 | - | - |

35-40 | 65.67 | 0.9633 | 1637.48 | 0.9905 |

40-45 | 69.79 | 0.9064 | 1710.88 | 0.9480 |

45-50 | 72.96 | 0.8671 | 1802.56 | 0.8998 |

50-55 | 72.38 | 0.8740 | 1861.90 | 0.8711 |

55-60 | 77.87 | 0.8124 | 1949.86 | 0.8318 |

60-65 | 80.56 | 0.7852 | 2054.88 | 0.7893 |

65-70 | 86.06 | 0.7351 | 2082.80 | 0.7787 |

70-75 | 91.05 | 0.6948 | 2303.69 | 0.7040 |

75-80 | 97.55 | 0.6485 | 2523.40 | 0.6427 |

80-85 | 106.92 | 0.5917 | 2669.86 | 0.6075 |

85-90 | 132.18 | 0.4786 | 3263.00 | 0.4971 |

90-95 | 188.18 | 0.3362 | 4300.78 | 0.3771 |

TABLE 2.
Least square solutions of the parameters r_{25}
and b in the formula
WR97 = r_{25} /[1+(25/b)³ -(age/b)³ ]

event | r_{25}[sec] | b [year] | |
---|---|---|---|

men's running | 100 M | 10.14 ± 0.20 | 116.01 ± 1.34 |

200 M | 20.38 ± 0.32 | 113.28 ± 0.89 | |

400 M | 44.45 ± 0.50 | 107.59 ± 0.41 | |

800 M | 104.72 ± 1.78 | 108.50 ± 0.39 | |

1500 M | 209.24 ± 3.25 | 106.90 ± 0.29 | |

3000 M | 463.72 ± 4.89 | 107.90 ± 0.39 | |

5000 M | 771.71 ±12.65 | 106.98 ± 0.31 | |

10000 M | 1621.91 ±27.38 | 108.12 ± 0.64 | |

MARATHON | 7363.64 ±72.51 | 103.15 ± 0.58 | |

men's swimming | 50 M FREE | 22.98 ± 0.30 | 114.19 ± 0.84 |

50 M FLY | 24.44 ± 0.37 | 104.64 ± 0.69 | |

50 M BACK | 27.20 ± 0.48 | 110.72 ± 0.86 | |

50 M BRST | 29.67 ± 0.23 | 115.39 ± 0.76 | |

100 M FREE | 51.12 ± 0.70 | 111.12 ± 0.70 | |

100 M FLY | 56.57 ± 0.67 | 101.81 ± 0.42 | |

100 M BACK | 59.72 ± 0.80 | 108.53 ± 0.54 | |

100 M BRST | 63.26 ± 1.37 | 106.04 ± 0.69 | |

200 M FREE | 114.70 ± 1.34 | 110.56 ± 0.56 | |

200 M FLY | 131.77 ± 3.10 | 103.27 ± 1.48 | |

200 M BACK | 131.47 ± 1.87 | 108.15 ± 0.56 | |

200 M BRST | 143.48 ± 2.70 | 108.19 ± 0.74 | |

200 M I.M. | 126.22 ± 2.86 | 104.04 ± 0.58 | |

400 M FREE | 247.59 ± 3.21 | 110.07 ± 0.90 | |

400 M I.M. | 282.16 ± 3.70 | 106.40 ± 0.70 | |

800 M FREE | 516.82 ± 6.90 | 110.26 ± 0.94 | |

1500 M FREE | 1006.13 ±14.87 | 111.52 ± 0.77 | |

women's swimming | 50 M FREE | 27.00 ± 0.31 | 114.17 ± 0.73 |

50 M FLY | 28.09 ± 0.35 | 99.14 ± 0.33 | |

50 M BACK | 31.24 ± 0.27 | 109.74 ± 0.85 | |

50 M BRST | 34.10 ± 0.34 | 108.42 ± 0.62 | |

100 M FREE | 58.97 ± 0.62 | 108.95 ± 0.68 | |

100 M FLY | 65.56 ± 1.95 | 98.21 ± 1.23 | |

100 M BACK | 68.07 ± 0.85 | 106.99 ± 1.02 | |

100 M BRST | 76.20 ± 0.69 | 106.67 ± 0.73 | |

200 M FREE | 128.09 ± 1.58 | 103.79 ± 1.17 | |

200 M FLY | 144.61 ± 3.01 | 95.54 ± 1.13 | |

200 M BACK | 146.92 ± 2.33 | 106.64 ± 1.23 | |

200 M BRST | 167.88 ± 1.37 | 106.73 ± 0.65 | |

200 M I.M. | 149.14 ± 2.06 | 104.55 ± 0.96 | |

400 M FREE | 273.51 ± 2.79 | 105.50 ± 0.75 | |

400 M I.M. | 318.46 ± 5.12 | 104.18 ± 1.08 | |

800 M FREE | 561.90 ± 6.33 | 104.91 ± 0.53 | |

1500 M FREE | 1067.08 ±16.04 | 103.91 ± 0.65 |