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# The coefficient of variation : general introduction into modern statistics

Without a doubt, the ability to orientate in modern variational methods is absolutely necessary for all modern undergraduates, regardless of their age and/or profession. Modern statistics supplies students with a great variety of different methods of collecting, classifying and comparing information that is useful in virtually all spheres of science from biology and organic chemistry to structural linguistics and sociology. Undoubtedly, one cannot compose even a simple science paper, eschewing the use of these mathematical stochastic techniques. Therefore, in order to understand these doubtlessly useful methods one has to be acquainted with fundamental concepts of modern statistics, such as the coefficient of variation, standard deviation, and the intensity variation. Certainly, the coefficient of variation is perhaps the most significant figure among these concepts because it is widely used not only in highly specialized scientific disciplines, but also in practically all spheres of common life that require precise and accurate work with numbers. Hence, let us study the most significant characteristics of the conception of the coefficient of variation in order to understand the basic concepts of modern statistics and modern financial analysis.

In fact, modern variational techniques refer to the method of optimization in which the main assignment is to determine the minimum or maximum of an integral involving several unknown functions. This method is primal to the study of modern functional analysis in the same way that the theory of minima and maxima are fundamental to the study of arithmetic calculus. During the last century, classical variational techniques have played a significant role in the solution of many problems in pure mathematical science as well as in various practical spheres of study, such as mechanics and engendering. Furthermore, in the past few decades, variational methods have experienced extraordinary flourishing in terms of diversity and the practical applicability of these techniques. Therefore, it is quite obvious that nowadays the term ‘coefficient of variation ’ is familiar to the lion’s share of specialists who are involved in different areas of knowledge, including modern statistics, management, economics, control theory and stochastic analysis. Nowadays, many scientific questions that are arising in the study of optimal designs of regression experiments can be successfully answered by modern variational techniques.

## The coefficient of variation : the general definition of the term

The coefficient of variation is used to compare dispersion of two or more characteristics that have different units of measurement. The coefficient of variation is a measure of relative dispersion, expressed as a percentage. It can be calculated using the formula: V= (σ/X)*100%; V – the value; σ – the standard deviation; X- the average value of the characteristics. In other words, the coefficient of variation allows us to categorize and analyze the received data in accordance with the frequency of appearance of some specific characteristics, which become of interest to our research. For example, let us examine a standard academic assignment, such as writing a rhetorical analysis essay. In order to understand how often the text includes some concrete linguistic characteristics, we can use the standard method of determination the coefficient of variation. This simple operation permits us to divide large unstructured clusters into several compact groups that can be easily analyzed. Naturally, the cutter understanding of the mathematical and physical sense of ‘coefficient of variation ’ also requires the understanding of the conception ‘variation’. The variation is a characteristic value of individual differences in units of the target population. In the modern economic analysis, this term plays one of the most significant roles. The measures of variations are divided into two different groups: absolute and relative. The standard deviation, quartile deviation, linear deviation appertain to the first group. The mean linear deviation is widely used in practice. For example, it allows us to analyze the rhythm of production, uniformity of materials supply. Nevertheless, this figure significantly complicates the calculations of probabilistic type, using classical mathematical statistics methods. Therefore, it is not often used in various statistical researches. The relative measures of variation are coefficient of variation, relative linear deviation, the relative measure of quartile variation, etc. These measures are calculated as the ratio of absolute values of variation to the arithmetic mean or the median. The standard deviation gives us the absolute evaluation of dispersion measures. Obviously, the standard deviation also characterizes the measure of the scattering data. Moreover, it (as opposed to the variance) can be compared with the original data. However, this figure in its pure mathematical sense is non-information, as it requires too many intermediate calculations that are quite confusing even for experienced mathematics (deviation squared, sum, average, etc.). Nevertheless, according to this definition, the comparison of several sets of variations regardless of the number of specialized characteristics is not possible. In this case, we have to use the relative measures of variation. Alternatively, to be more specific, the coefficient of variation. It also should be noted that unlike various absolute measures of the variation the coefficient of variation is measured as a percentage (when multiplied by 100%). According to this statistic indicator, we can compare the uniformity of a variety of phenomena, regardless of the scale. Hereby, it is the main reason why the coefficient of variation is so widely used in various fields of mathematical and financial study.

### The coefficient of variation and the modern methods of financial analysis

Nowadays, the coefficient of variation is the most universal indicator of the degree of dispersion of values, regardless of their scale and measurement units. The coefficient of variation can be used to compare the variation of various processes and phenomena. In general, it is assumed that if the value of the coefficient of variation is less than 33%, then the set is considered homogeneous. If the value of the coefficient of variation is more than 33%, then set is assumed mixed. In statistics, the coefficient of variation is used for comparison of the dispersion of two random variables with different units of measurement, with respect to the expected value in order to provide comparable results. In portfolio theory, this figure is used as a relative measure of risk associated with investing in a specific asset or portfolio of assets. The coefficient of variation is particularly useful in a situation where two assets have different expected returns and different levels of risk (standard deviation). For example, one investment may have a higher expected return, and the other a lower standard deviation. The modern analysis of the risk probability is based on the use of different groups of methods. In fact, the modeling of modern financial systems begins with the fundamental statistical conceptions of actions and associated variables. In fact, there exist two different series of actions in financial and actuarial statistical modeling, referred to as deterministic (or so-called non-stochastic actions) and stochastic. Stochastic actions can be described by statistical random variables whereas non-stochastic actions are deterministic in nature without a probability attachment. One can easily find a considerable number of examples that illustrate this statement by studying various thesis examples that are dedicated to the definition of non-stochastic actions in nature. Of course, some of these actions are difficult to predict even using the contemporary mathematical apparatus. However, due to the coefficient of variation we can categorize various clusters of specific units into various numerical series, according to the frequency of concrete characteristic that is demonstrated in each concrete case. Of course, in order to predict an acceptable level of the risk, we can use different methods of analysis. Here is a concise register of these techniques:

• The statistical method examines the statistics of losses and profits, the magnitude and frequency of this financial result. In addition, it allows us to compose a reliable prognosis of all potential changes, according to our original stochastic data. For example, we can calculate the statistics of profits and losses for a college paper writing service if we have the initial financial data for the first three-quarters of the fiscal year. We have to collect all accounting papers and find the standard deviation together with the coefficient of variation of several characteristic financial indexes. Therefore, if our computations are correct, we can construct a trustworthy financial prognosis for next few months or even a year.
• The method of expert evaluations is based on interviews with experts in the specific fields of management. Usually, all worksheets are written in official style and format (such as the CSE paper format). In addition, these questionnaires contain a set of questions about contemporary financial conditions of the company along with a few blank lines for possible proposals for the improvement and modernization of the contemporary financial structure. Of course, this technique is indirectly based on statistical methods because with an eye to obtaining the most objective and unprejudiced financial prognosis we have to use statistical methods of sorting and analyzing data.
• The method of analogues is based on the search of similar characteristics between similar financial objects and/or situations. Needless to say, that the search of the coefficient of variation is one of the most important objectives, which is used in this stochastic technique.

Therefore, let us summarize the facts. In the modern financial analysis, the coefficient of variation characterizes the risk per unit of expected return. In addition, the coefficient of variation also allows us to determine the likelihood of certain types of risks, if equated indexes of standard deviation differ in specific sizes, according to the considered view of business operations.

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