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MATHEMATICAL BASES FOR FORECASTING TENDENCIES OF PERSONAL QUALITY DEVELOPMENT USING COMPOSITION DISTRIBUTION TABLES (analysis of personality development trends based on the concept of cumulative reasons)

Prediction of personality development is probabilistic.

this can be widely applied probability theory, mathematical statistics. Let us analyze from this point of view the obtained distribution tables & / diligence, hard work of employees /.

These are distribution tables that contain random values ​​X1, X2, X3, X4, X5. Each of the Xi values ​​takes on a particular value depending on which parameter the examination is taking. In any case, the examined according to the parameter Ak are distributed into Li groups.

n

-

> Li = DK,

-

i = 1

where DK is the number of examined according to the parameter Ak.

Then the probability of each random variable Xi is defined as

Li

___ = Pi

Dk

N

therefore, Pi = 1.

i = 1

For clarity, imagine in Euler circles the map

spaces & / Fig. 1 /, consisting of circles X1, X2, X3, X4, X5.

fig. 1

Since <Xi contains the whole set

I = 1

events, & - is a reliable event. Similarly: the set & 1 consists of the same random variables represented by other Euler circles, each of which is denoted by Zi

N

Therefore, <Z1 also takes all

i = 1

many Xi events, and therefore & 1 is a reliable event.

Therefore, the conditions are satisfied:

P / & / = 1

P / & 1 / = 1

As & is a space for Z1, so Zi is a space for Xi. Therefore, for any space Zi,

N

Pi = 1

i = 1

Thus, & is a Borel field. In probability theory, a Borel field is denoted by <& 1, Z>.

Similarly, & 2, & 3, & 4, ... & N can be considered.

Therefore, we have before us a system of / N / Borel fields of the same set of events &. These fields will be designated <& 1, Z>, <& 2, Z>, <& 3, Z>, ... <& N, Z>. The distribution of Xi in them is strictly individual.

The quantified life path of the personality of the subject crosses each of the received Borel fields, the values ​​<& 1, Z.>, <& 2, Z.>, ... <& N, Z.>. These values ​​correspond to a certain distribution of Xi. The intersection of the individual’s life path with the Borel field will be event A. <& 1, Z1 corresponds to A1, <& 2, Z.> - A2, <& 3, Z.> - A3, ... <& N, Z.> - AN. The life path of an individual is individual, it corresponds to a strictly defined chain of events - A1, A2, A3, A4, ... An with the corresponding distribution of Xi.

Moreover, the application of probability theory seems to change its aspect: from the analysis of a large number of personalities, a transition is made to the analysis of a large number of events in the life path of one person. By virtue of this, P / Xn / of each Borel field will be a random variable with respect to the individual. Find the mathematical expectation of X1, X2, X3, X4, X5 of each person.

The mathematical expectation of a discrete random variable X is called its average value. It is calculated by the formula:

- N

M / X / = Xi Pi

I = 1

Suppose the life path of a military man is characterized by the following events: A1 = Z3, A2 = Z1, A3 = Z4, A4 = Z3, and

etc. / see: Fig. 2 /. Then:

- x1 <& 1, Z3> + X1 <& 2, Z1> + X1 <& 3, Z4> + X1 <& 4, Z3>

M / X / = ------------------------------------------;

N

since P / x1 / + P / x2 / + P / x3 / + P / x4 / + P / x5 / = 1

in all Borel fields,

M / X1 / + M / X2 / + M / X3 / + M / X4 / + M / X5 / = 1

Based on the data obtained, it is possible to construct a histogram of the mathematical expectation of the development of a particular personality trait. the resulting histogram is comparable with the histogram of the health of all examined. on the basis of this, we can conclude about this subject according to the predicted quality of personality. It is physically impossible for 1OOO subjects to construct histograms for all predicted personality parameters and interpret them qualitatively.

To obtain a forecast for each subject should find a centered random variable M / X1 /, M / X2 /, M / X3 /, M / X4 /, M / X5 /. In the general form, a centered random variable is the difference between the random variable X and its mathematical expectation:

about -

X = X - M / X /

In this case, the application of probability theory again changes its aspect: not a multitude of events of a particular personality is subjected to analysis, but a multitude of mathematical expectations. If the mathematical expectation for a given parameter X of a given person is M / X1 /, M / X2 /, M / X3 /, M / X4 /, M / X5 /, then M / X1 /, M / X2 /, M / X3 /, M / X4 /, M / X5 / are random variables with respect to the entire sample, that is, their mathematical expectation will be, respectively, X1 / & /, X2 / & /, X3 / & /, X4 / & /, X5 / & /. Hence the formula of the centered random variable M / X / takes the following form:

about -

X1 = M / X1 / - X1 / & /

about -

X2 = X / X2 / - X2 / & /

about -

X3 = M / X3 / - X3 / & /

about -

X4 = M / X4 / - X4 / & /

about -

X5 = M / X5 / - X5 / & /

oh oh oh oh

If X1, X2 are positive values, and X4, X5 are negative, then the psychological diagnosis of this quality, personality parameter is unfavorable. But such results of psychodiagnostics need simplification. To do this, we offer the following method.

Assign X1, X2, X3, X4, X5 according to the sample & and in each Borel field & 1, & 2, & 3, & 4, ... & N, as well as M / X1 /, M / X2 /, M / X3 /, M / X4 /, M / X5 / of the studied personality, preference factors: K / X1 / = A1; K / X2 / = A2; K / X3 / = A3; K / X4 / = A4; K / X5 / = A5. Moreover, A1, A2 <O; A4, A5> O. These coefficients are conditional. Their diagnostic meaning is as follows: K / X. / indicates what probability X. and to what extent favorable for psychological selection.

Now you can find the index of preference / I <&> / personality on the predicted parameter. If the subject has this index more than the average value of the indices of all the subjects in the sample, then it can be argued about the relative favorableness of the forecast, less - relative unfavorable. Moreover, the degree of auspiciousness and adverseness is relative to the quality of the contingent being examined.

Similarly, preference indices for all predicted personality traits are found, and then the general preference index .

To determine it, it is necessary to evaluate the correlation, from the point of view of significance, of the predicted personality parameters, that is, weight coefficients. From these coefficients depends on what preference is given. There are not many people with equally highly developed personality traits, so you have to make a selection, meeting the most diverse combinations of predicted personality traits.

The introduced weights allow us to find the overall preference index.

Preference factors are complex in composition.

They include correction factors / Kk / and significance factors / Kz /.

Kp = Kk X Kz

Correction factors are introduced so that the mathematical expectations of the preference index of each predicted personality quality are approximately the same in statistical terms. Otherwise, psychological selection will occur with a greater “weight” of the most frequently encountered Xs, and Xs, which are smaller in absolute value (as a rule, X1 and X5 /, will be underestimated. Correction factors help achieve approximately the same deviations.

oh oh

X. from M / X. /.

When determining correction factors, one can proceed from two hypotheses. First: the average deviation of centered random variables

about

/ x./

of each predicted personality trait is directly proportional to the probability of X, that is, P / X. /. Therefore, the correction coefficients are found by dividing a larger value of Р / Х. / - one of the predicted personality qualities as a whole in the sample - by the remaining values ​​Р / Х. / of the same personal quality. We present these calculations with respect to the search for correction factors for &, that is, the diligence of employees. The distribution table of this quality is known to us / see: page
/.

The greatest probability of occurrence of X. for x4, - it is equal to

Oh, Z86. If we turn to the table of distribution of executive success of subjects of different social origin, we will see that our hypothesis (the average statistical deviation of centered random variables is

about

X.

- directly proportional to P / X. /, in this example, the probability

appearance of X4), fully justified. In this case, the correction factors will take the values:

Oh, Z86

Kk / X1 / = --------- = 5.5143

Oh, O7O

Oh, Z86

Kk / X2 / = ------- = 2.2976

Oh, 168

Oh 386

Kk / X3 / = ------- = 1.5197

Oh 254

Oh 386

Kk / X4 / = ------- = 1

Oh 386

Oh 386

Kk / X5 / = ------- = Z, 1639

Oh, 122

According to the second hypothesis, correction coefficients are found empirically. In this case, the coefficients will vary depending on the nature of the studied population, or rather the variances of X. The correction factors are found by the same calculations that have just been done. Only in the numerator is the experimentally found mathematical expectation of the probabilities of occurrence of each X. If this study were conducted on the same subjects, on the basis of which the found distribution tables were obtained, then Mm / X. / would coincide with P / X. /.

Significance coefficients are selected empirically, they vary depending on the level of cohesion, discipline of the team in which the individuals under study develop / or will have to develop / and from other factors. Very large differences (more than 1O times) of the significance coefficients for X1, X2, as well as X4 and X5 lead to a sharp separation of all the examined according to the level of development of the predicted personality quality into two groups - the best and the "worst". However, when determining the validity of the forecast, the ranking according to the severity of this quality is less observed. Naturally, this reduces the accuracy of the forecast if it is determined by searching for the correlation coefficient between the forecast result and the level of development of the assessed quality, and not by comparing the contrasting groups. Therefore, it is very important to choose the optimal values ​​of the coefficients of significance.

The optimal values ​​of significance coefficients, as well as weight coefficients, vary depending on the nature of the predicted personality quality, the criteria used to assess this quality, the nature of the sample population, the “nonlinearity” of the applied rating scale, the general level of training and education of the group of subjects, the quality of labor resources, and contingent workers, etc. The calculation of the optimal values ​​of the coefficients of significance, and with them the coefficients of preference, must be carried out again under the new conditions of application of this biographical technique. This requires complicated calculations, skilled mathematical analysis, which is not always possible to do. However, without significantly losing the accuracy of psycho-prognostic validity, the proposed methodology can use standard significance factors. So, at K3 / X1 / = -3; K3 / X2 / = -2; K3 / X3 / = -1, K3 / X4 / = 2; K3 / X5 / = 3 good results were obtained for various samples.

The presence of standard significance coefficients, and therefore preference coefficients, allows you to have a universal, although not adaptable to specific, sometimes very specific, conditions for the development of personalities, a methodology of biographical psycho-diagnostics. But since it is possible to give the values ​​of preference coefficients a universal character, to make them constant, is it possible to give five-frequency distribution tables, albeit conditional, but still some one, strictly defined value? Yes you can. However, this is not considered in this paper.

Due to the peculiarities of the contingent of workers in various samples of the examined tables, the distribution of personality traits can significantly differ from each other, overlapping differences / centered random variables / for one or another parameter of the life path. Let us illustrate this with an example. In one of the universities received a table of the distribution of discipline of students.

She was next.

D / X1 / = O, O72

D / X2 / = O, 153

D / X3 / = O, 281

D / X4 / = O, 38O

D / X5 / = O, 114

But in another university, a different table of the distribution of discipline in the whole sample was obtained:

D / X1 / = O, O9z

D / X2 / = O, 2O5

D / X3 / = O, 332

D / X4 / = O, 31O

D / X5 / = O, O6O

Naturally, in the second case, the distribution tables for D will be shifted towards X1 and X2. And it turns out that, for example, the only children in the family of the first university have a more favorable table of the distribution of discipline than the older children of the second. (By the way, this once again confirms that the system of training and education, the nature of activity, the degree of cohesion of the team, as it were, subjugates the influence of the past on personality development). A similar discrepancy is observed in other parameters of the life path. Therefore, when transferring distribution tables from one collective to another (from one sample to another), their commensurability in absolute values ​​is lost (although commensurability in relative values ​​is preserved, that is, in the first and second cases, older children in the family differ slightly from others higher level of discipline). In order to make the distribution tables commensurate in absolute values, it is necessary to take into account the degree of their displacement relative to each other. This can be done using the transfer coefficients /K.per/. it is found by dividing the corresponding probability according to X of a particular personality quality by the corresponding values ​​of P / X. / taken from another data bank (in either case, the distribution tables as a whole for the sample are taken). We find the transfer coefficients as applied to our example.

Oh, O72

K. lane / X1 / = ------------ = O, 77419

Oh, O9Z



Oh 153

K. lane / X2 / = ------------ = O, 74334

Oh, 2O5



Oh 382

K.per / X3 / = ------------ = 1, Z1746

Oh 252



Oh 31O

K. lane / X4 / = ------------ = Oh, 79487

Oh, Z9O



Oh 114

K. lane / X5 / = ------------ = 1.9

Oh, O6O

Now, in order to transfer the distribution tables from the data bank obtained on the basis of a study of one sample, each distribution table should be multiplied by the transfer coefficient when entering the general data bank.

Similar transfer coefficients should be taken into account when compiling distribution tables obtained when examining a part of workers from the entire sample.

Thus, the mathematical apparatus of probability theory makes it possible to more accurately, scientifically substantiate the methodology for predicting the development of personal qualities of the subjects on the basis of compiling distribution tables. It should be noted that this is only a general approach to the mathematical analysis of the methodology of biographical psycho-diagnostics, which contains great opportunities for further improvement. It is safe to say that almost all sections of probability theory can be applied in biographical studies of a statistical nature. These studies were carried out in accordance with the effect of cumulative causes in psychology, when the degree and nature of the influence of a mass of biographical facts on the development, functioning of certain personality traits, on its development trends are measured.

So, biographical data, biographical method occupies a special place in acmeological expert systems. This is due to the special role of biography in determining the development of personality. And if domestic psychology often denied the special role and significance of biography in the study of personality psychology, then practice, foreign psychology attached and attach particular importance to the analysis of biography. Acmeology, as it were, incorporates all the best foreign achievements and the experience of the practical study of people: practice is higher than theoretical knowledge, since it has universality, and is criterial in the knowledge of truth.
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MATHEMATICAL BASES FOR FORECASTING TENDENCIES OF PERSONAL QUALITY DEVELOPMENT USING COMPOSITION DISTRIBUTION TABLES (analysis of personality development trends based on the concept of cumulative reasons)

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