Modern statistical methods of analysis. Statistical research

Clients, consumers - this is not just collecting information, but a full-fledged study. And the goal of any research is a scientifically based interpretation of the studied facts. The primary material must be processed, namely organized and analyzed. After interviewing respondents, the research data is analyzed. This is a key stage. It is a set of techniques and methods aimed at checking how correct the assumptions and hypotheses were, as well as answering the questions asked. This stage is perhaps the most difficult in terms of intellectual effort and professional qualifications, but it allows you to get the maximum useful information from the collected data. Data analysis methods are varied. The choice of a specific method depends, first of all, on what questions we want to answer. Two classes of analysis procedures can be distinguished:

• one-dimensional (descriptive) and
• multidimensional.

The purpose of univariate analysis is to describe one characteristic of a sample at a specific point in time. Let's take a closer look.

One-dimensional types of data analysis

Quantitative Research

Descriptive analysis

Descriptive (or descriptive) statistics are the basic and most general method data analysis. Imagine that you are conducting a survey to create a portrait of the consumer of a product. Respondents indicate their gender, age, marital and professional status, consumer preferences, etc., and descriptive statistics make it possible to obtain information on the basis of which the entire portrait will be built. In addition to the numerical characteristics, a variety of graphs are created to help visualize the survey results. All this variety of secondary data is united by the concept of “descriptive analysis”. The numerical data obtained during the study are most often presented in the final reports in the form of frequency tables. The tables can show different types of frequencies. Let's look at an example: Potential demand for the product

1. Absolute frequency shows how many times a particular response is repeated in a sample. For example, 23 people would buy the proposed product worth 5,000 rubles, 41 people – worth 4,500 rubles. and 56 people – 4399 rubles.
2. The relative frequency shows what proportion this value makes up of the total sample size (23 people - 19.2%, 41 - 34.2%, 56 - 46.6%).
3. Cumulative or accumulated frequency shows the proportion of sample elements that do not exceed a certain value. For example, a change in the percentage of respondents who are ready to purchase a particular product if its price decreases (19.2% of respondents are ready to buy a product for 5,000 rubles, 53.4% ​​- from 4,500 to 5,000 rubles, and 100% - from 4,399 to 5,000 rubles). 5000 rub.).

Along with frequencies, descriptive analysis involves the calculation of various descriptive statistics. True to their name, they provide basic information about the data collected. Let us clarify that the use of specific statistics depends on the scales in which the initial information is presented. Nominal scale used to record objects that do not have a ranked order (gender, place of residence, preferred brand, etc.). For this kind of data array, it is impossible to calculate any significant statistical indicators, except fashion— the most frequently occurring value of the variable. In terms of analysis, the situation is somewhat better with ordinal scale . Here it becomes possible, along with fashion, to calculate medians– a value that splits the sample into two equal parts. For example, if there are several price intervals for a product (500-700 rubles, 700-900, 900-1100 rubles), the median allows you to establish the exact price, more expensive or cheaper than which consumers are willing to purchase or, conversely, refuse to purchase. The richest in all possible statistics are quantitative scales , which represent rows numerical values, having equal intervals between each other and measurable. Examples of such scales include income level, age, time spent on shopping, etc. In this case, the following information becomes available: measures: mean, range, standard deviation, standard error of the mean. Of course, the language of numbers is rather “dry” and quite incomprehensible to many. For this reason, descriptive analysis is complemented by data visualization by constructing various charts and graphs, such as histograms, line, pie or scatter charts.

Contingency and correlation tables

Contingency tables is a means of representing the distribution of two variables, designed to study the relationship between them. Contingency tables can be considered a special type of descriptive analysis. It is also possible to present information in the form of absolute and relative frequencies, graphical visualization in the form of histograms or scatter diagrams. Contingency tables are most effective in determining whether there is a relationship between nominal variables (for example, between gender and the consumption of a product). IN general view The contingency table looks like this. Relationship between gender and use of insurance services

Data analysis and statistics are things of the same order. If statistics is the fundamental principle and source of information, then data analysis is a tool for its research, and often data analysis without statistics is impossible.

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Statistics is the study of any phenomena in numerical form. Statistics is used by data analysis in quantitative research. The opposite of them is qualitative, describing the situation without using numbers, in textual terms.

Quantitative analysis of statistical data is carried out on an interval scale and on a rational scale:

• an interval scale indicates how much one or another indicator is more or less than another and makes it possible to select indicators with similar properties of the ratio,
• a rational scale shows how many times one or another indicator is greater or less than another, but it contains only positive values, which will not always reflect the real state of affairs.

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Methods for analyzing statistical data

In the analysis of statistical data, one can distinguish the analytical and descriptive stages. The descriptive stage is the last, it includes the presentation of the collected data in a convenient graphical form - in graphs, charts, dashboards. The analytical stage is an analysis consisting of using one of the following methods:

• statistical observation - systematic collection of data on characteristics of interest;
• data summaries in which information can be processed after observation; it describes individual facts as part of the totality or creates groupings, divides information into groups based on any characteristics;
• determining absolute and relative statistical values; absolute value gives data quantitative characteristics individually, regardless of other data; relative quantities describe some objects or characteristics relative to others;
• sampling method - using not all data in the analysis, but only part of it, selected according to certain rules (sampling can be random, stratified, cluster and quota);
• correlation and regression analysis - identifies relationships between data and the reasons why data depend on each other, determines the strength of this dependence;
• time series method - tracks the strength, intensity and frequency of changes in objects and phenomena; allows you to evaluate data over time and makes it possible to predict phenomena.

Statistical Research Software

Statistical research can be carried out by marketing analysts:

For qualitative analysis statistical data, you must either have knowledge of mathematical statistics, or use a reporting and analytical program, or not do this. European companies have long realized the benefits of such analysis, so they either hire good analysts with a mathematical education or establish professional software for marketing analysts. Daily analysis in these companies helps them correctly organize the purchase of goods, their storage and logistics, adjust the number of personnel and their work schedules.

Solutions for automating data analysis allow marketing analysts to work with them. Today there are solutions available even to small companies, such as Tableau. Their advantages compared to analysis carried out solely by humans:

• low cost of implementation (from 2000 rubles per month – as of February 2018),
• modern graphical representation of the analysis,
• the ability to instantly move from one, more complete report, to another, more detailed one.

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Statistical methods of data analysis are usually divided into two large groups: one-dimensional methods statistical analysis and multivariate methods.

One-dimensional analysis methods- these are methods that are used in cases where there is a single measure to evaluate each element of the sample, or if there are several of these measures, each variable is analyzed separately from all the others. The focus of these methods is the analysis of average values ​​and indicators of variation of variables.

The classification of one-dimensional methods is carried out by the nature of the source data (metric or non-metric), as well as by the number and type of samples. Thus, the samples are divided into dependent (paired)- these are samples formed from one general population and independent samples are samples drawn from different general populations. In practice, samples drawn from different strata (in the case of using stratified or quota sampling), for example, men and women or groups of respondents with different income levels, are considered independent.

One-dimensional methods of data analysis include:

· Methods for testing hypotheses (z-test, t-test, F-test, χ2-test, etc.).

For more detailed testing of hypotheses, see: Gmurman V. E. Probability theory and mathematical statistics.

· Methods for analyzing statistical distribution series.

· One-way analysis of variance.

· Other methods.

Multivariate analysis methods- these are methods that are used in cases where two or more measures are used to estimate each sample element and these variables are analyzed simultaneously. The focus of this group of methods is already on the analysis of relationships, connections and similarities between variables.

The following multidimensional methods are distinguished:

1) Methods for identifying dependencies between variables are methods in which one or more variables are dependent and others are independent. This group includes:

· correlation and regression analysis;

· analysis of variance and covariance;

· discriminant analysis;

· joint analysis.

2) Methods for identifying interdependence between variables are methods that allow data to be grouped based on similarities. In these methods there is no division of variables into dependent and independent. This group includes:

· cluster analysis;

· factor analysis;

· multidimensional scaling.

The choice of data analysis methods is based on:

· goals, objectives, working hypotheses of marketing research;

· type of marketing research (exploratory or summary; descriptive or cause-and-effect);

· type of data collected - metric and non-metric variables;

· scales used in the study;

· sample size and method;

· method of data collection;

· areas of application and limitations of statistical methods of data analysis.

In fact, all previous stages of marketing research predetermine the choice of data analysis strategy. The experience and qualifications of the researcher himself play a significant role. In conclusion, we note that complex multivariate methods of statistical data analysis are not always used. Very often, the researcher is limited to only preliminary (basic) data analysis and its graphical interpretation.

Of course, it is necessary to remember that the analysis of marketing research data is not its last stage; it is followed by the development practical recommendations and generating the research report.

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• 3. Dynamics series
• Literature

1. Absolute and relative values

As a result of the summary and grouping of statistical material, the researcher finds himself in the hands of a wide variety of information about the phenomena and processes being studied. However, it would be a big mistake to dwell on the results obtained, because even grouped by given characteristics and reflected in tabular or graphical form, these data are still only a kind of illustration, an intermediate result that must be analyzed - in this case, statistical . Statisticalanalysis - This performance studied object V quality dismembered systems, those. complex elements And connections, forming V his interaction organic whole.

As a result of such an analysis, a model of the object being studied should be built, and, since we are talking about statistics, statistically significant elements and connections should be used when building the model.

In fact, statistical analysis is aimed at identifying such significant elements and connections.

Absoluteindicators(values) - total values, calculated or taken from summary statistical reports without any transformations. Absolute indicators are always nominal and are reflected in the units of measurement that were specified when drawing up the statistical observation program (the number of criminal cases initiated, the number of crimes committed, the number of divorces, etc.).

Absolute indicators are the basis for any further statistical operations, but they themselves are of little use for analysis. By absolute indicators, for example, it is difficult to judge the crime level in different cities or regions and it is practically impossible to answer the question where crime is higher and where it is lower, since cities or regions can differ significantly in population, territory and other important parameters.

Relativequantities in statistics, they are generalizing indicators that reveal the numerical form of the relationship between two compared statistical quantities. When calculating relative values, two absolute values ​​are most often compared, but it is possible to compare both average and relative values, obtaining new relative indicators. The simplest example of calculating a relative value is the answer to the question: how many times is one number greater than another?

When starting to consider relative values, it is necessary to consider the following. In principle, you can compare anything you want, even the linear dimensions of a sheet of A4 paper with the quantity of products produced by the Lomonosov Porcelain Factory. However, such a comparison will not give us anything. The most important condition for a fruitful calculation of relative values ​​can be formulated as follows:

1. The units of measurement of the quantities being compared must be the same or completely comparable. The number of crimes, criminal cases and convictions are correlated indicators, i.e. interrelated, but not comparable in units of measurement. In one criminal case, several crimes can be tried and a group of people convicted; Several convicts can commit one crime and, conversely, one convicted person can commit many acts. The numbers of crimes, cases and convictions are comparable to the population, the number of personnel in the criminal justice system, the standard of living of the people and other data for the same year. Moreover, over the course of one year, the indicators under consideration are quite comparable with each other.

2. The compared data must necessarily correspond to each other in terms of the time or territory of their receipt, or both parameters together.

Absolute size, With which are compared other Vedisguises, called basis or base comparisons, A compareAndsculpted index - size comparisons. For example, when calculating the ratio of crime dynamics in Russia in 2000-2010. 2000 data will be the baseline. They can be taken as one (then the relative value will be expressed in the form of a coefficient), or as 100 (as a percentage). Depending on the dimension of the quantities being compared, the most convenient, indicative and visual form expressions of relative magnitude.

If the value being compared is much greater than the base, the resulting ratio is better expressed in coefficients. For example, crime over a certain period (in years) increased by 2.6 times. The expression in times in this case will be more indicative than in percentages. Relative values ​​are expressed as percentages when the comparison value does not differ much from the base.

Relative quantities used in statistics, including legal statistics, are different types. The following types of relative quantities are used in legal statistics:

1. relations characterizing the structure of the population, or distribution relations;

2. the relationship of the part to the whole, or the relationship of intensity;

3. relationships characterizing dynamics;

4. relations of degree and comparison.

Relativemagnitudedistribution - This relative size, expressed V percent individual parts totality studied phenomena(crimes, criminals, civil cases, lawsuits, causes, prevention measures, etc.) To their general as a result, accepted behind 100% . This is the most common (and simplest) type of relative data used in statistics. This is, for example, the structure of crime (by type of crime), the structure of criminal records (by type of crime, by age of those convicted), etc.

statistical analysis absolute value

Attitudeintensity(ratio of part to whole) - a generalizing relative value that reflects the prevalence of a certain characteristic in the observed totality.

The most common intensity indicator used in legal statistics is crime intensity . Crime intensity is usually reflected by the crime rate , those. number of crimes per 100 or 10 thousand inhabitants.

KP= (P*100000)/N

where P is the absolute number of recorded crimes, N is the absolute population size.

A prerequisite that determines the very possibility of calculating such indicators, as mentioned above, is that all absolute indicators used are taken in one territory and for one period of time.

Relationship,characterizingdynamics, represent generalizing relative quantities, showing change in time those or other indicators legal statistics. The time interval is usually taken to be a year.

The basis (base) equal to 1, or 100%, is taken to be information about the characteristic being studied for a certain year, which was in some way characteristic of the phenomenon being studied. The base year data serves as a fixed base to which the indicators of subsequent years are percentaged.

Statistical analysis tasks often require annual (or other period) comparisons when base accepted data everyone previous of the year(month or other period). Such a base is called mobile. This is usually used in time series (time series) analysis.

RelationshipdegreesAndcomparisons make it possible to compare various indicators in order to identify which value is how much larger than the other, to what extent one phenomenon differs from or is similar to another, what is common and distinctive in the observed statistical processes, etc.

An index is a specially created relative indicator of comparison (in time, space, when compared with a forecast, etc.), showing how many times the level of the phenomenon being studied under one condition differs from the level of the same phenomenon under other conditions. The most common indices are in economic statistics, although they also play a certain role in the analysis of legal phenomena.

You cannot do without indices in cases where it is necessary to compare incommensurable indicators, the simple summation of which is impossible. Therefore, indices are usually defined as numbers-indicatorsFormeasurementsaveragespeakerstotalityheterogeneouselements.

In statistics, indices are usually denoted by the letter I (i). Uppercase letter or capital - depends on whether we are talking about an individual (private) index or a general one.

Individualindices(i) reflect the ratio of the indicator of the current period to the corresponding indicator of the compared period.

Summaryindices are used in analyzing the relationship between complex socio-economic phenomena and consist of two parts: the actual indexed value and the co-measurer (“weight”).

2. Average values ​​and their application in legal statistics

The result of processing absolute and relative indicators is the construction of distribution series. Row distribution - ThisorderedByhigh qualityorquantitativesignsdistributionunitstotality. The analysis of these series is the basis of any statistical analysis, no matter how complex it may later turn out to be.

The distribution series can be constructed based on qualitative or quantitative characteristics. In the first case it is called attributive, in the second - variational. In this case, differences in quantitative characteristics are called variation, and this sign itself - option. It is with variation series that legal statistics most often have to deal.

The variation series always consists of two columns (graph). One indicates the value of a quantitative characteristic in ascending order, which, in fact, are called options, which are designated x. The other column (graph) indicates the number of units that are characteristic of one or another option. They are called frequencies and are designated Latin letter f.

Table 2.1

Option x
Frequency f

The frequency of occurrence of a particular characteristic is very important when calculating other significant statistical indicators, namely, averages and variation indicators.

Variation series, in turn, can be discrete or interval. Discrete series, as the name suggests, are built on the basis of discretely varying characteristics, and interval series - on the basis of continuous variations. For example, the distribution of offenders by age can be either discrete (18, 19, 20 years, etc.) or continuous (up to 18 years, 18-25 years, 25-30 years, etc.). Moreover, the interval series themselves can be constructed either on a discrete or continuous basis. In the first case, the boundaries of adjacent intervals are not repeated; in our example, the intervals will look like this: up to 18 years, 18-25, 26-30, 31-35, etc. Such a series is called continuousdiscreterow. IntervalrowWithcontinuousvariation assumes that the upper limit of the previous interval coincides with the lower limit of the next one.

The very first indicator describing variation series is average quantities. They play an important role in legal statistics, since only with their help can populations be characterized by a quantitative variable attribute by which they can be compared. Using average values, we can compare sets of legally significant phenomena that interest us according to certain quantitative characteristics and draw the necessary conclusions from these comparisons.

Averagequantities reflect the most general trend (pattern), inherent in the entire mass of phenomena being studied. It manifests itself in typical quantitative characteristics, i.e. in the average value of all available (varying) indicators.

Statistics have developed many types of averages: arithmetic average, geometric average, cubic average, harmonic average, etc. However, they are practically not used in legal statistics, so we will consider only two types of averages - the arithmetic mean and the geometric mean.

The most common and well-known average is averagearithmetic. To calculate it, the sum of indicators is calculated and divided by the total number of indicators. For example, a family of 4 consists of parents aged 38 and 40 years and two children aged 7 and 10 years. We sum up the age: 38+40+7+10 and divide the resulting sum of 95 by 4. The resulting average family age is 23.75 years. Or let’s calculate the average monthly workload of investigators if a department of 8 people solves 25 cases in a month. Divide 25 by 8 and we get 3,125 cases per month per investigator.

In legal statistics, the arithmetic average is used when calculating the workload of employees (investigators, prosecutors, judges, etc.), calculating the absolute increase in crime, calculating the sample, etc.

However, in the example given, the average monthly workload per investigator is calculated incorrectly. The fact is that the simple arithmetic average does not take into account frequency the trait being studied. In our example, the average monthly workload of the investigator is as correct and informative as the “average temperature in the hospital” from the famous joke, which, as we know, is room temperature. In order to take into account the frequency of manifestations of the characteristic being studied when calculating the arithmetic mean, it is used as follows: averagearithmeticweighted or average for discrete variation series. (Discrete variation series - the sequence of changes in a characteristic according to discrete (discontinuous) indicators).

Arithmetic average weighted ( weighted average) has no fundamental differences from the simple arithmetic average. In it, the summation of the same value is replaced by multiplying this value by its frequency, i.e. in this case, each value (variant) is weighted by frequency of occurrence.

So, when calculating the average workload of investigators, we must multiply the number of cases by the number of investigators who investigated exactly that number of cases. It is usually convenient to present such calculations in the form of tables:

Table 2.2

Number of cases (option X)	Number of investigators (frequency f)	Product option to frequencies ( Xf)

2. Let’s calculate the actual weighted average using the formula:

Where x- the number of criminal cases, and f- number of investigators.

Thus, the weighted average is not 3.125, but 4.375. If you think about it, this is how it should be: the workload on each individual investigator increases due to the fact that one investigator in our hypothetical department turned out to be a slacker - or, conversely, was investigating a particularly important and complex case. But the issue of interpreting the results of statistical research will be discussed in the next topic. In some cases, namely, in cases of grouped frequencies of a discrete distribution, the calculation of the average, at first glance, is not obvious. Suppose we need to calculate the arithmetic mean for the distribution of persons convicted of hooliganism by age. The distribution looks like this:

Table 2.3

(option X)	Number of convicts (frequency f)	Middle of the interval	Product option to frequencies ( Xf)
		(21-18) /2+18=19,5

Next, the average is calculated according to general rule and for this discrete series is 23.6 years. In the case of the so-called open series, that is, in situations where the extreme intervals are determined by “less than x" or more x", the size of the extreme intervals is set similarly to other intervals.

3. Dynamics series

Social phenomena studied by statistics are in constant development and change. Social and legal indicators can be presented not only in a static form, reflecting a certain phenomenon, but also as a process occurring in time and space, as well as in the form of interaction of the studied characteristics. In other words, time series show the development of a trait, i.e. its change in time, space or depending on environmental conditions.

This series is a sequence of average values ​​during specified periods of time (for each calendar year).

For a more in-depth study social phenomena and their analysis of a simple comparison of the levels of the dynamics series is not enough; it is necessary to calculate the derived indicators of the dynamics series: absolute growth, growth rate, growth rate, average growth and gain rates, the absolute content of one percent of growth.

Calculation of indicators of dynamics series is carried out on the basis of comparison of their levels. In this case, there are two possible ways to compare the levels of a time series:

basic indicators, when all subsequent levels are compared with some initial level taken as the base;

chain indicators, when each subsequent level of a series of dynamics is compared with the previous one.

Absolute growth shows how many units the level of the current period is greater or less than the level of the base or previous period for a specific period of time.

The absolute increase (P) is calculated as the difference between the compared levels.

Base absolute increase:

P b = y i - y bases . (f.1).

Chain absolute increase:

P ts = y i - y i -1 (f.2).

The growth rate (Tr) shows how many times (by what percentage) the level of the current period is greater or less than the level of the base or previous period:

Baseline growth rate:

(f.3)

Chain growth rate:

(f.4)

The growth rate (Tpr) shows by what percentage the level of the current period is greater or less than the level of the base or previous period taken as the basis of comparison, and is calculated as the ratio of the absolute growth to the absolute level taken as the base.

The growth rate can also be calculated by subtracting 100% from the growth rate.

Base growth rate:

or (f.5)

Chain growth rate:

or (f.6)

The average growth rate is calculated using the formula of the geometric mean of the growth rates of the dynamics series:

(f.7)

where is the average growth rate;

- growth rates for individual periods;

n- number of growth rates.

Similar problems with a root exponent greater than three are usually solved using logarithms. From algebra we know that the logarithm of the root is equal to the logarithm of the radicand divided by the exponent of the root, and that the logarithm of the product of several factors is equal to the sum of the logarithms of these factors.

Thus, the average growth rate is calculated by extracting the root n degrees from the works of individual n- chain growth rates. The average growth rate is the difference between the average growth rate and one (), or 100% when the growth rate is expressed as a percentage:

or

In the absence of intermediate levels in the dynamic series, the average growth and increment rates are determined by the following formula:

(f.8)

where is the final level of the dynamic series;

- initial level of dynamic series;

n - number of levels (dates).

It is obvious that the indicators of average growth rates and increments, calculated using formulas (forms 7 and form 8), have the same numerical values.

The absolute content of 1% growth shows what absolute value 1% of growth contains and is calculated as the ratio of absolute growth to the growth rate.

Absolute content of 1% increase:

basic: (form 9)

chain: (f.10)

Computation and Analysis absolute value each percent increase contributes to a deeper understanding of the nature of the development of the phenomenon under study. The data from our example show that, despite fluctuations in growth rates and gains for individual years, the basic indicators of the absolute content of 1% of growth remain unchanged, while chain indicators characterizing changes in the absolute value of one percent of growth in each subsequent year compared to the previous , are continuously increasing.

When constructing, processing and analyzing time series, there is often a need to determine the average levels of the phenomena being studied over certain periods of time. The chronological average of an interval series is calculated at equal intervals using the simple arithmetic average formula, and at unequal intervals - using the weighted arithmetic average:

Where - average level interval series;

- initial levels of the series;

n- number of levels.

For a moment series of dynamics, provided that the time intervals between dates are equal, the average level is calculated using the average chronological formula:

(f.11)

where is the average chronological value;

y 1 ,., y n- absolute level of the series;

n - number of absolute levels of the dynamics series.

The average chronological level of the moment series of dynamics is equal to the sum of the indicators of this series, divided by the number of indicators minus one; in this case, the initial and final levels should be taken in half, since the number of dates (moments) is usually one greater than the number of periods.

Depending on the content and form of presentation of the source data (interval or moment series of dynamics, equal or not time intervals) for calculating various social indicators, for example, the average annual number of crimes and offenses (by type), the average size of working capital balances, the average number of offenders and etc., use the appropriate analytical expressions.

4. Statistical methods for studying relationships

In previous questions, we considered, so to speak, the analysis of “one-dimensional” distributions - variation series. This is a very important, but far from the only type of statistical analysis. Analysis of variation series is the basis for more “advanced” types of statistical analysis, primarily for studyinginterrelations. As a result of such a study, cause-and-effect relationships between phenomena are revealed, which makes it possible to determine which changes in characteristics affect the variations of the phenomena and processes being studied. In this case, the characteristics that cause changes in others are called factorial (factors), and the characteristics that change under their influence are called effective.

In statistical science, there are two types of connections between various characteristics and their information - functional connection (hard-deterministic) and statistical (stochastic).

For functionalconnections There is complete correspondence between the change in the factor characteristic and the change in the resultant value. This relationship is equally manifested in all units of any population. The simplest example: an increase in temperature is reflected in the volume of mercury in the thermometer. At the same time the temperature environment acts as a factor, and the volume of mercury acts as a resultant characteristic.

Functional relationships are characteristic of phenomena studied by such sciences as chemistry, physics, mechanics, in which it is possible to conduct “pure” experiments in which the influence of extraneous factors is eliminated. The fact is that a functional connection between two is possible only if the second value (resultative characteristic) depends only And exclusively from the first. This is observed extremely rarely in social phenomena.

Social and legal processes, which are the result of the simultaneous influence of a large number of factors, are described through statistical connections, that is, connections stochastically (accidentally) deterministic, when different values ​​of one variable correspond to different values ​​of another variable.

The most important (and common) case of stochastic dependence is correlationaddiction. With such a dependence, the cause does not determine the effect unambiguously, but only with a certain degree of probability. A separate type of statistical analysis is devoted to identifying such connections - correlation analysis.

Main task correlation analysis - based on strictly mathematical techniques, establish a quantitative expression of the relationship that exists between the characteristics under study. There are several approaches to how exactly correlation is calculated and, accordingly, several types of correlation coefficients: contingency coefficient A.A. Chuprov (to measure the relationship between qualitative characteristics), K. Pearson's association coefficient, as well as Spearman's and Kendall's rank correlation coefficients. In general, such coefficients show the probability with which the studied relationships appear. Accordingly, the higher the coefficient, the more pronounced the relationship between the characteristics.

Both direct and inverse correlations can exist between the factors being studied. Straightcorrelationaddiction observed in cases where changes in the values ​​of a factor correspond to the same changes in the value of the resultant attribute, that is, when the value of the factor attribute increases, the value of the resultant attribute also increases, and vice versa. For example, there is a direct correlation between criminogenic factors and crime ( with a "+" sign). If an increase in the values ​​of one characteristic causes reverse changes in the values ​​of another, then such a relationship is called reverse. For example, the higher the social control in society, the lower the crime (relationship with the “-” sign).

Both straight and feedbacks can be straight or curved.

Straight-line ( Linear) relationships appear when, with an increase in the values ​​of the factor attribute, there is an increase (direct) or decrease (inverse) in the value of the consequence attribute. Mathematically, this relationship is expressed by the regression equation: at = A + bX, Where at - sign-consequence; A And b - corresponding coupling coefficients; X - sign-factor.

Curvilinear connections are of a different nature. An increase in the value of a factor characteristic has an uneven impact on the value of the resulting characteristic. At first this connection can be direct, and then reverse. Famous example- the relationship between crimes and the age of the offenders. At first, the criminal activity of individuals increases in direct proportion to the increase in the age of the offenders (up to approximately 30 years), and then, with increasing age, criminal activity decreases. Moreover, the top of the distribution curve of offenders by age is shifted from the average to the left (towards a younger age) and is asymmetrical.

Correlation linear connections can be oneOfactorial, when the connection between one factor-sign and one consequence-sign is studied (pairwise correlation). They may also be multifactorial, when the influence of many interacting signs-factors on a sign-consequence is studied (multiple correlation).

But, no matter which correlation coefficient is used, no matter what correlation is studied, it is impossible to establish a connection between characteristics based only on statistical indicators. The initial analysis of indicators is always an analysis qualitative, during which the socio-legal nature of the phenomenon is studied and clarified. In this case, those are used scientific methods and approaches that are characteristic of the branch of science that studies this phenomenon (sociology, law, psychology, etc.). Then the analysis of groupings and averages allows us to put forward hypotheses, build models, and determine the type of connection and dependence. Only after this is it determined quantitative characteristic dependencies - actually, the correlation coefficient.

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In order to choose the right statistical method for analyzing data in psychological research, you must first understand the basic methods of statistical processing: what they are, in what cases they are used, for what purpose and what kind of result can be obtained.

The choice of statistical data analysis method depends on the purpose and objectives of the study. The main methods of statistical data analysis, widely used for processing the results of empirical research in theses or dissertations in psychology, are as follows:

• Calculation of descriptive statistics. Descriptive statistics, as a rule, are calculated in all graduate theses in psychology without exception. Most often, the mean values ​​(M) and standard deviations (SD) are calculated for all scales of all research methods, and these data are entered into the table of primary results of the empirical study attached to the work. They are considered, most often, in the first paragraph of the empirical chapter, comparing them with normative data on methods and determining whether the sample under study has any features that should be taken into account or that pose a limitation in the interpretation of the research results.
• Correlation analysis - identifying relationships between research scales. This method allows you to detect linear (direct and reverse) relationships between variables or their absence. Correlation analysis is the main method of statistical data analysis in works whose purpose is to study the influence of something on something, the dependence of A on B.
• Statistical analysis of differences is a group of methods for comparing two or more samples. This includes methods for comparing samples using the Student, Mann-Whitney, Wilcoxon, etc. tests. All of these methods make it possible to determine how statistically significant (reliable) the differences are between two or more groups of subjects. They are the main methods of mathematical processing of data in studies whose purpose is to study the characteristics of a group or to study differences between groups, including gender differences.
• Multivariate methods of statistical data analysis are used in studies with a large number of studied characteristics (scales and research methods). IN psychological research these are most often factor analysis and cluster analysis. These methods allow you to classify, generalize, reduce the number of variables being studied, divide them into groups or classes, and reach another level of generalization. Processing the results of empirical research using multivariate methods is considered the “highest class” of mathematical data processing. Theses, which use multivariate methods, as a rule, obviously claim to have an excellent estimate.