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9SOME ASPECTS OF USING STATISTICAL METHODS TO ESTIMATE INDICES OF WORLD-CLASS ATHLETES
A.S. Gatamov, associate professor, Honorary trainer of Azerbaijani Republic, Honorary worker of physical culture and sport
Academy of the Ministry of Emergency Situations of the Republic of Azerbaijan, Baku
Key words: athlete, level, methodology, mathematical methods, accuracy, distribution function, expectation, variance.
Introduction. Development of the modern sports industry and infrastructure sets complex goals and makes special demands to sports specialists, coaches and athletes regarding training of elite world-class athletes.
Complex and specific conditions, in which athletes train, are stipulated by numerous positive and negative factors. This particularly applies to elite athletes, who regularly participate in international competitions. In addition, training of elite world-class athletes is a set of complex physical, psychological, methodological activities, which require very coordinated implementation. Inefficiency and poor choice of technological sequence of measures when training athletes lead not only to greater material and labour costs, but also unpredictable moral and psychological injuries.
The theoretical positions and patterns resulting from the process of theoretical and experimental research, as well as the techniques of assessing the fitness level in most cases apply equally to both mid-level and world-class athletes.
One of the reasons for the failure regarding the fitness level of elite athletes is the lack of statistical data on their physical, tactical, technical and other types of fitness level, as well as the lack of clear guidelines on the application of the results of these indices in specific situations. The purpose of the study was to analyse the use of mathematical statistics in the evaluation of results of world-class athletes.
Organization of the research. Measures requiring prompt decisions; factors determining the athletes' level; methods and principles applied in the training procedure of highly skilled specialists; means applied in physical, technical, tactical, psychological and theoretical training; development of prospective, routine and operating training plans; forecasting and modeling of athletes' activities; and numerous other issues require a system approach to addressing the training of elite athletes. Undoubtedly, it is impossible to solve the above-mentioned set of problems with the existing classic
methods and means. Therefore, a new approach is necessary to solve this issue. One of them is the use of the latest methods of mathematical statistics. For this purpose, at the first stage we can use primary statistical data obtained in training procedures or various competitions, as well as conduct a special research.
Results and discussion. The statistical evaluation of the actual fitness of athletes in various competitions may require works including four primary stages:
a) accumulation of statistical information in real competitive conditions or simulated training process by conducting laboratory, development and accelerated testings;
b) analysis and compilation of statistical data by taking into account the specificity of training sessions and athletes' characteristics;
c) selection and justification of the indices for the evaluation of the athletes' fitness in the beginning, depending on the level of performed training sessions and the results to which the structure of training sessions may lead at this stage;
d) statistical processing of the accumulated empirical data on the athletes' fitness level before competition for the purpose of determining the appropriate patterns inherent to the applied methods as well as applied training procedure.
A specially organized controlled testing or observation in conditions close to those of official competitions must be the main source of information on the athletes' fitness before competitions. The statistical data on the athletes' physical fitness must be accurate and complete. This accuracy is achieved with the exact calculation of the training sessions' duration and the quality of performed workouts, as well as the completeness of information and the precise definition of the scope of the required data. Moreover, the data about athletes' fitness before competitions must definitely include information about training modes and conditions of training sessions. Incomplete and fragmentary, albeit reliable, information can result in wrong conclusions in many cases.
The main advantage of the statistical data on the athlete's condition in actual conditions is in the fact that they fully reflect the athlete's training modes, his accurate performance of both a set of techniques and some particular one, his general physical fitness and many other indicators. The results achieved by athletes in various competitions are dependent on a number of factors. The impact of a considerable amount of these factors is random. Therefore, the regularities that characterize the athletes' fitness level before competitions can be defined based on the statistical data, using the methods of the probability theory and mathematical statistics that help to reveal the probability of the appearance of the data we are interested in (for example, results from various competitions) within a certain period of time. Distribution function of a random variable is its most complete characteristic.
Athlete's achievements, in terms of probability theory, are - in fact - the results achieved in various competitions, determined by different methods from the probability distribution functions of oe or
another random variable that characterizes athlete's overall indices (number of wins, number of defeats, nonparticipation in competitions, etc.). So, the goal of the mathematical treatment of statistical data for evaluating athlete's performance is actually the acquisition of the random variables' distribution functions.
Knowing the distribution function of a random variable, it is possible to evaluate athlete's performance using the methods of the special section of mathematical statistics; that is the estimation theory of unknown parameters. The input data for determining the distribution function of the random variable is the sum of the finite number N of its observations (xi, x2, x3..., xn). The type of the distribution function can be determined by the empirical distribution function. The
approximate equation ^(X) * F^Q applies to the unlimited increase of the number of observations, the point of which lies in the fact that for the unlimited increase of the number of
experiments, the empirical distribution function comes close to the proper distribution
function FiJi? 0f the random variable.The empirical density function can serve as an estimation for
the probability density ^ dx and, for the continuous random variables it is represented by
a histogram, while for the discrete random variables it is represented by a polygon. In practice, due to the finiteness of the number of observations N, it is possible to acquire only the empirical distribution and density functions, which is more than enough for the evaluation of an athlete's performance. However, in theoretical studies, a necessity arises quite often for the identification of the known law of distribution which the acquired results of the observation comply with. The objective of statistical data treatment, in this case, is to test the hypothesis of consistency between the sum N of the observation of the random variable and a certain theoretical distribution function.
Thus, the statistical data treatment consists of two distinct phases: the determination of the empirical distribution or density functions as well as the testing of the hypothesis about the distribution law of the investigated values, with the second phase including the amount of computational work of the first phase.
m +
According to the probability theory the random variable \ : if its distribution density is
has a normal distribution,
Where a and @ : parameters of the normal probability distribution. The normal distribution function
1 r* r fe-flJ5! , FixJ = —— f exp I--Idz
■7V2,Tiii L 20"' J . (2)
Moving on with the examination of the function
(3)
the values of which have been tabulated, the following can be written:
and
x-a.
(5)
=¿jDO*=Fm=
F (x'-a \
Where CF f: the normal probability distribution function of centered and standardized
random variable.
Since in many cases time is the random variable in the probability theory, that cannot be negative, truncated normal distribution can be applied instead of the normal distribution law [1]. The random variable takes only positive values and has a truncated normal distribution, if its distribution density is
fix:t=
exp\
r (Jf-fl^l
IT 2 (ri J
.(6)
The truncated normal distribution function is written as
.
Normalization factor C is found from
(7)
Using the function /a , the expression
(8)
can be written as
.
(9)
Taking into account the formula (9) and having replaced the variable of the integration ^a dz=0«ijf , we present the expression (7) as
=y
(10)
Where : normalization factor;
^ : parameter of the truncated normal law of the probability distribution; x: sample arithmetic mean.
After the corresponding transformations of the conditions (8):
. (11)
Delivering the value ^ form the formula (11) to the expression (10), and taking into account the property of the condition F0(-x)=1-F0(x), we finally have the expression for the distribution function of truncated normal distribution.
The expectation and variance of the random variable, distributed according to the truncated normal law, are correspondently determined from the expressions
and
By applying transformations and by taking the formula (ii) into consideration, we get
,V[C] = + \p0 Ci-.j)] (12)
and
= [1 - i\Q [u^.U > + u^o'a CO.O ) . (13) Where ratio of the parameters of the truncated normal law of the probability
distribution,
h -El
: auxiliary function,
The coefficient of variation V of the random variable, distributed according to the truncated normal law, is determined by the formula
distribution. Therefore, the parameters of the normal distribution a and ® are, respectively, equal to mathematical expectation and standard deviation of the random variable.
Within the range "+~3i7 3 the area covered by the normal distribution curve constitutes 99.73% of the total area. Practically (with an inaccuracy of 0.27%), we can assume that the whole limited area is included within the curve .
With the increasing of ff, the disperse field increases, resulting in the curve of the normal distribution law getting flatter and shorter, which corresponds to lower accuracy. The smaller the
value & , the less the dispersion of the measured parameters that evaluates athletes' results and, consequently, the more accurate observations.
Conclusion. Currently, elite world-class athletes can not be trained without using various techniques which determine the level of athletes' readiness to compete, technology of assessment of accuracy and quality of technique they perform and other complex indicators, different measurement techniques and technical methods, including measuring devices and instruments used during training and at competitions, etc.
If to determine the goals of sports professionals, trainers and athletes in the field of training of world-class athletes they are mainly to solve numerous interrelated and sequentially executed activities.
V =
+JDÏp VI - +
MIO
(14)
References
1. Babaev, S.G. The fundamentals of reliability theory oilfield equipment / S.G. Babaev. -Baku: Azneftekhim, 1976, - 95 P. (In Russian)
2. Dunin-Barkovsky, I.V., Smirnov, N.V. Probability theory and mathematical statistics in engineering / I.V. Dunin-Barkovsky, N.V. Smirnov. - Moscow: Gostekhizdam, 1955. - 87 P. (In Russian)
Corresponding author: Aydin.qafarov@baku.az
