# Improvement in 100-m Sprint Performance at an Altitude of 2250 m

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Corrections for Wind and Historical Trend

_{o}, using

_{o}= T + α (V

_{w}− βV

_{w}

^{2}),

_{w}is the wind reading; and α and β are constants [14]. For international standard sprinters α = 0.050 s/m, and β = 0.0556 s

^{2}/m for men and β = 0.0667 s

^{2}/m for women. The time advantage of a 2 m/s tailwind is therefore about 0.10 s for men and about 0.12 s for women.

#### 2.2. Calculation of Mean Race Time for an Olympic Games Competition

_{1}, A

_{2}, A

_{3}, … A

_{n}, for the athletes ranked 1 to n, where n is the number of athletes in the competition). The mean race time for each Olympic Games competition was then calculated from the ranking list of athletes. However, the uncertainty in a mean race time depends on the number of athletes used in the calculation [18]. In addition, including the highest-ranked athletes (i.e., 1st, 2nd, 3rd, …) in the calculation of the mean race time might not produce the lowest possible uncertainty in the mean race time. To identify the optimum selection of athletes to include in the calculation of the mean race time, the mean race time (and its associated uncertainty) was calculated using (n

_{mean}=) 5, 10, 15, 20, 25, 30, 35, or 40 athletes, and for each choice of n

_{mean}the middle-ranked athlete in the group (A

_{mid}) was systematically varied over a wide range. As an illustration of this process, if the calculation of the mean race time used five athletes and the middle-ranked athlete in the group of five was the 27th-ranked athlete (i.e., n

_{mean}= 5 and A

_{mid}= 27), then the calculation used athletes A

_{25}, A

_{26}, A

_{27}, A

_{28}, and A

_{29}. For each Olympic Games competition the uncertainty in the mean race time was plotted against A

_{mid}and n

_{mean}. These plots were examined to identify the optimum n

_{mean}and A

_{mid}that minimized the uncertainty in the mean race time. These optimum values were then used to calculate the mean race time for each of the Olympic Games competitions.

#### 2.3. Time Advantage of Mexico City

#### 2.4. Other Confounding Factors

## 3. Results

_{rank}is the mean time of the top 20 athletes in the world rankings; Y is the year; T

_{max}is the asymptotic maximum time; T

_{min}is the asymptotic minimum time; Y

_{inf}is the year at the inflection point of the curve; and a is a measure of the growth rate. The values of the regression coefficients were T

_{max}= 11.37 s, T

_{min}= 9.87 s, Y

_{inf}= 1955 years, and a = –0.054 per year for the men; and T

_{max}= 12.58 s, T

_{min}= 10.96 s, Y

_{inf}= 1962 years, and a = –0.098 per year for the women.

_{mid}) was ranked about 20–45, and a reliable value of the uncertainty (with only small random fluctuations) was attained when using greater than about 20 athletes in the calculation (Figure 2). Therefore, the mean race time for the Olympic Games competitions was taken as that calculated with n

_{mean}= 20 and A

_{mid}= 30. In a similar fashion, for the women the mean race time was taken as that calculated with n

_{mean}= 20 and A

_{mid}= 20. Figure 2 highlights that the uncertainty in the mean race time is substantially reduced when the top-ranked athletes in the competition are not included in the calculation.

_{max}, T

_{min}, Y

_{inf}, and a to be equal to those obtained from the fit to the world rankings data. When calculating the fitted curve, the mean race time at each Olympic Games competition was weighted by the 90% confidence interval in the mean race time. For both the men and women, the deviations of the mean race times at each Olympic Games competition from the fitted curve showed a clear perturbation for the Mexico City 1968 Olympic Games (Figure 3). The time advantage of Mexico City was 0.19 (±0.02) s for the men and 0.22 (±0.05) s for the women.

_{mean}) used to calculate the mean race time at the competitions. Using 15 or 25 athletes (rather than 20) changed the calculated time advantage of Mexico City by less than 0.01 s. Similarly, the time advantage of Mexico City was insensitive to the choice of middle-ranked athlete (A

_{mid}). For the men, using the 20th or 40th ranked athlete (rather than the 30th) changed the time advantage by less than 0.01 s. For the women, using the 10th or 30th ranked athlete (rather than the 20th) changed the time advantage by less than 0.01 s.

_{mean}and A

_{mid}, the non-zero altitude of some of the competition venues, boycotts at competitions, and different types of track surface. For the men, the most accurate estimate from this study is probably that with an altitude correction for the Munich 1972 Olympic Games. For this scenario the time advantage of Mexico City is 0.19 (±0.02) s. For the women, the most accurate estimate from this study is probably that with an altitude correction for Munich 1972 and with the Moscow 1980 and Los Angeles 1984 competitions removed from the data. For this scenario the time advantage of Mexico City is 0.21 (±0.05) s. The uncertainty in the time advantage of Mexico City was mainly due to the uncertainty in the mean race time at the Mexico City 1968 Olympic Games, rather than due to the uncertainty in the curve fitted to the Olympic Games competition data.

## 4. Discussion

#### 4.1. Other Competition Analyses and Mathematical Models

#### 4.2. Implications for IAAF Rules

## 5. Conclusions

## Conflicts of Interest

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**Figure 1.**These plots show the historical trend in world 100-m standards; (

**a**) men, (

**b**) women. Data points are the mean of the top 20 athletes in each year. The fitted curve is a 4-parameter logistic curve (Equation (2)). Coefficient of determination: men r

^{2}= 0.98; women r

^{2}= 0.97.

**Figure 2.**This plot shows there is an optimum choice of athletes to include in the calculation of the mean race time at an Olympic Games competition. The uncertainty (90% confidence interval) in the mean race time is lowest when the ranking of the middle athlete in the group of athletes used in the calculation is about A

_{mid}= 20–45. When the number of athletes used in the calculation of the mean race time is less than about n

_{mean}= 20, the uncertainty in the mean value is not reliable (i.e., subject to excessive random fluctuations). The optimum values were taken to be A

_{mid}= 30 and n

_{mean}= 20. Data for the men at the Mexico City 1968 Olympic Games.

**Figure 3.**These plots show the mean race times at Olympic Games competitions from 1964 to 2012 after de-trending with a curve of the same shape as the historical trend in 100-m performances; (

**a**) men, (

**b**) women. Vertical error bars indicate the 90% confidence interval in the mean. There is a substantial deviation at the Mexico City 1968 Olympic Games due to the high altitude of this site (2250 m).

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Linthorne, N.P.
Improvement in 100-m Sprint Performance at an Altitude of 2250 m. *Sports* **2016**, *4*, 29.
https://doi.org/10.3390/sports4020029

**AMA Style**

Linthorne NP.
Improvement in 100-m Sprint Performance at an Altitude of 2250 m. *Sports*. 2016; 4(2):29.
https://doi.org/10.3390/sports4020029

**Chicago/Turabian Style**

Linthorne, Nicholas P.
2016. "Improvement in 100-m Sprint Performance at an Altitude of 2250 m" *Sports* 4, no. 2: 29.
https://doi.org/10.3390/sports4020029