The distribution of a random variable contains all information about its statistical properties. How many values ​​of a random variable do you need to know in order to plot its distribution? To do this, you need to explore it general population.

General population - the set of all values ​​that a given random variable can take.

The number of units in the general population is called its volume N... This value can be finite or infinite. For example, if the growth of the inhabitants of a certain city is investigated, then the volume of the general population will be equal to the number of inhabitants of the city. If any physical experiment is performed, then the volume of the general population will be infinite, since the number of all possible values ​​of any physical parameter is equal to infinity.

The study of the general population is not always possible and expedient. It is impossible if the volume of the general population is infinite. But even with finite volumes, a complete study is not always justified, since it requires a lot of time and labor, and the absolute accuracy of the results is usually not required. Less accurate results, but with significantly less effort and resources, can be obtained when examining only a part of the general population. Such studies are called selective.

Statistical studies carried out only on a part of the general population are called sample, and the studied part of the general population is called a sample.

Figure 7.2 symbolically depicts a population and a sample as a set and its subset.

Figure 7.2 Population and sample

Working with a certain subset of a given general population, often constituting an insignificant part of it, we get results that are quite satisfactory in accuracy for practical purposes. Examining a large part of the population only increases accuracy, but does not change the essence of the results, if the sample is taken correctly from a statistical point of view.

In order for the sample to reflect the properties of the general population and the results to be reliable, it must be representative(representative).

In some general populations, any part of them is representative by virtue of their nature. However, in most cases, special measures must be taken to ensure that the samples are representative.

One one of the main achievements of modern mathematical statistics is the development of the theory and practice of the method of random sampling, which ensure the representativeness of data selection.

Sample surveys always lose out in accuracy compared to a survey of the entire general population. However, this can be reconciled if the magnitude of the error is known. Obviously, the more the sample size approaches the size of the general population, the smaller the error will be. Hence, it is clear that the problems of statistical inference become especially relevant when working with small samples ( N ? 10-50).

As a result of studying the material in Chapter 2, the student should:

know

• basic concepts of the general and sample populations;
• estimation methods, types and properties of estimates of parameters of the general population;
• basic methods of statistical testing of hypotheses regarding the parameters of one-dimensional and multidimensional general populations;

be able to

• find, based on sample data, estimates of the parameters of one-dimensional and multidimensional general populations;
• analyze the properties of parameters;
• test hypotheses regarding the parameters and type of distribution of the general population;
• compare the parameters of several general populations;

own

• skills of statistical estimation of parameters of one-dimensional and multidimensional general populations;
• the skills of testing hypotheses regarding the parameters and type of distribution of the general population when conducting socio-economic research using analytical software.

Population distribution

Probabilistic-statistical methods of data analysis assume that the laws governing the variable under study (random variable) are completely determined by the complex of conditions for its observation. Mathematically, these patterns are set by the corresponding probability distribution law. However, when conducting statistical research, the concept of the general population is more convenient.

Thus, the mathematical concepts "general population", "random variable" and "probability distribution law" corresponding to a given set of conditions can be considered synonyms in a certain sense.

The general population name the set of all conceivable observations that could be made under a given set of conditions.

Since the definition deals with mentally possible observations (or objects), the general population is an abstract concept, and it should not be confused with real populations subject to statistical research. So, having examined even all enterprises of a sub-sector, we can consider them as representatives of a hypothetically possible broader set of enterprises that could function within a set of conditions.

The general population can be either finite or infinite. The ultimate population takes place, for example, in a survey of family budgets, when a sample is taken from the population of families actually present in the country. The income and expenditure of the selected families is then monitored. Endless the general population is observed, for example, in scientific research, when we are interested in the average result of a large number of experiments.

In the simplest case, the general population is a one-dimensional random variable NS with a distribution function that determines the probability that NS will take a value less than a fixed real number.

In the general case, general populations are studied that include several features (usually more than two). The considered set of features is denoted by a vector having k component, each of which characterizes the corresponding feature. To analyze a vector X multidimensional statistical methods are used.

Thus, the object of research in multivariate analysis is a random vector X, or a random point in ft-dimensional Euclidean space, the system To random (one-dimensional) variables, ft-dimensional random variable

The distribution function of a random vector is called a deterministic non-negative quantity determined by the formula

where is a dimensional vector of fixed real numbers.

Deterministic non-negative quantity F (X)

Distinguish:

• continuous k-dimensional random variables, all components of which are continuous (one-dimensional) random variables;
• discrete k-dimensional random variables, all components of which are discrete random variables;
• mixed k-dimensional random variables, among the components of which there are both discrete and continuous random variables.

Distribution function F (X) for continuous k-dimensional random variable is continuous by definition.

The density of the probability distribution of the continuous k-dimensional random variable satisfies the condition

Density f (X) has the following properties:

The area bounded at the top by the density plot is always equal to one:

where through k the total number (multiplicity) of integrals is indicated;

The probability of hitting a point () in some area G is equal to

It follows from the definition of density that if we integrate the joint distribution density of two quantities NS 1, NS 2 one by one, for example, within infinite limits, then we obtain the probability density of another quantity:

Similarly, we have

Probability densities, distribution functions of subsystems, random variables of the system To random variables are called private or marginal distributions .

Conditional distributions random vector X called the distribution of the subsystem, its components, provided that the rest of the components are fixed. These components will be separated from the non-fixed by a forward slash.

For a continuous random variable, for example, the formulas that determine the density of the conditional distribution of a two-dimensional random variable () are valid, which is a subsystem of the system () provided that the last three components are fixed in it:

Subsystem, component and additional subsystem of vector components X are called independent(stochastically, probabilistically) if the equality

In particular, the components of the vector X are called independent, if

In the case of independence, similar formulas are valid for the products of densities or probabilities of marginal distributions and the coincidence of conditional distributions with the corresponding marginal ones (23].

http://www.hi-edu.ru/e-books/xbook096/01/index.html?part-011.htm- a very useful site!

The sampling method of research is the main statistical method. This is natural, since the volume of objects under study is usually infinite (and even if it is finite, then it is very difficult to enumerate all objects, you have to be content with only a part of them, a sample).

General and sample population

The general set is the totality of all the elements investigated in a given experiment.

A sample population (or sample) is a finite set of objects randomly selected from the general population.

The volume of a population (sample or general) is the number of objects in this population.

An example of a general and sample population

Let's say that the psychological predisposition of a person to dividing a given segment in relation to the golden ratio is being investigated. Since the origin of the very concept of the golden section is dictated by the anthropometry of the human body, it is clear that in this case the general population is any anthropogenic creature that has reached physical maturity and acquired final proportions, that is, the entire adult part of humanity. The volume of this collection is practically endless.

If this predisposition is investigated exclusively in the artistic environment, then the general population is people who are directly related to design: artists, architects, designers. There are also a lot of such people, and we can assume that the volume of the general population in this case is also infinite.

In both cases, for research we are forced to limit ourselves to reasonable sample sizes, choosing as representatives of one or another set of students of technical specialties (as people far from the artistic world) or students of the specialty design (as people directly related to the world artistic images).

Representativeness

The main problem of the sampling method is the question of how accurately the objects selected from the general population for the study represent the studied characteristics of the general population, that is, the question of the representativeness of the sample.

So, a sample is called representative (representative) if it accurately enough represents the quantitative ratios of the general population.

Of course, it is difficult to say what exactly is hidden behind the vague wording accurate enough... Questions of representativeness are generally the most controversial in any experimental study. There are many examples, which have already become classical, when insufficient representativeness of the sample led experimenters to absurd results.

As a rule, questions of representativeness are resolved with the help of peer review, when the scientific community accepts the point of view of a group of authoritative specialists about the correctness of the research carried out.

An example of representativeness

Let's go back to the segment division example. The issues of representativeness of samples lie here at the very foundation of the study: in no case should we mix groups of subjects on the basis of their belonging to the artistic environment.

Statistical distribution of the observed trait

Observed value frequency

Suppose that, as a result of testing in a sample of volume, the observed feature took the values ​​,, ..., and the value was observed once, the value-times, etc., the value was observed once. Then the frequency of the observed value is called the number, the values ​​are the number, and so on.

Relative frequency of the observed value

The relative frequency of the observed value is the ratio of frequency to sample volume:

It is clear that the sum of the frequencies of the observed feature should give the sample size

and the sum of the relative frequencies should give one:

These considerations can be used for control when compiling statistical tables. If the equalities are not observed, then an error was made in recording the results of the experiment.

Statistical distribution of the observed value

The statistical distribution of the observed feature is the correspondence between the observed values ​​of the feature and the corresponding frequencies (or relative frequencies).

As a rule, the statistical distribution is written in the form of a two-line table, in which the observed values ​​of the feature are indicated in the first line, and the corresponding frequencies (or relative frequencies) in the second:

If the observed feature is characterized by a continuous random variable taking values ​​from the interval, then its statistical distribution is described by the frequencies of hitting the partial intervals:

General population (in English - population) - the totality of all objects (units) about which the scientist intends to draw conclusions when studying a specific problem.

The general population consists of all objects that are subject to study. The composition of the general population depends on the objectives of the study. Sometimes the general population is the entire population of a certain region (for example, when the attitude of potential voters to a candidate is studied), most often several criteria are set that determine the object of research. For example, men 30-50 years old who use a certain brand of razor at least once a week, and have an income of at least $ 100 per family member.

Sampleor sample population- a lot of cases (subjects, objects, events, samples), using a certain procedure, selected from the general population to participate in the study.

Sample characteristics:

· Qualitative characteristics of the sample - who exactly we choose and what methods of constructing the sample we use for this.

· The quantitative characteristic of the sample - how many cases we select, in other words, the sample size.

The need for sampling

· The subject of research is very extensive. For example, the consumers of the products of a global company are a huge number of geographically dispersed markets.

· There is a need to collect primary information.

Sample size

Sample size- the number of cases included in the sample. For statistical reasons, it is recommended that the number of cases be at least 30 - 35.

Dependent and independent samples

When comparing two (or more) samples, their dependence is an important parameter. If it is possible to establish a homomorphic pair (that is, when one case from sample X corresponds to one and only one case from sample Y and vice versa) for each case in two samples (and this basis of the relationship is important for the characteristic measured on the samples), such samples are called dependent... Examples of dependent selections:

· pairs of twins,

· two measurements of any sign before and after the experimental exposure,

· husbands and wives

· etc.

If there is no such relationship between the samples, then these samples are considered independent, for example:

· men and women,

· psychologists and mathematicians.

Accordingly, dependent samples always have the same size, while the volume of independent samples may differ.

Samples are compared using various statistical criteria:

· Student's t-test

· Wilcoxon test

· Mann-Whitney U test

· Sign criterion

· and etc.

Representativeness

The sample can be considered representative or unrepresentative.

An example of a non-representative sample

In the United States, one of the most famous historical examples of non-representative sampling is considered to be during the 1936 presidential election. Leitrery Digest magazine, which successfully predicted the events of several preceding elections, was wrong in its predictions, sending out ten million trial ballots to its subscribers, as well as to people selected from phone books all over the country and people from car registration lists. In 25% of the returned ballots (almost 2.5 million), the votes were distributed as follows:

· 57% preferred Republican nominee Alf Landon

· 40% chose the then Democratic President Franklin Roosevelt

As is known, Roosevelt won the actual elections, gaining more than 60% of the votes. The Leitrery Digest's mistake was that in wanting to increase the representativeness of the sample — since they knew that most of their subscribers considered themselves Republicans — they expanded the sample by selecting people from phone books and registration lists. However, they did not take into account the realities of their day and, in fact, recruited even more Republicans: during the Great Depression, mainly representatives of the middle and upper class (that is, the majority of Republicans, not Democrats) could afford to own phones and cars.

Types of plan for building groups from samples

There are several main types of group building plan:

1. Research with experimental and control groups, which are placed in different conditions.

2. Study with experimental and control groups using a pairwise selection strategy

3. Research using only one group - experimental.

4. Research using a mixed (factorial) design - all groups are placed in different conditions.

Sample types

Samples are divided into two types:

· probabilistic

· improbable

Probability samples

1. Simple probabilistic sampling:

oSimple resampling. The use of such a sample is based on the assumption that each respondent is equally likely to be included in the sample. On the basis of the list of the general population, cards are compiled with the numbers of the respondents. They are placed in a deck, shuffled and a card is taken out of them at random, a number is recorded, then returned back. Then the procedure is repeated as many times as we need the sample size. Minus: repetition of selection units.

The procedure for constructing a simple random sample includes the following steps:

1. you need to get a complete list of members of the general population and number this list. Recall that such a list is called the sampling frame;

2. determine the expected sample size, that is, the expected number of respondents;

3. extract from the table of random numbers as many numbers as we need sample units. If there should be 100 people in the sample, 100 random numbers are taken from the table. These random numbers can be generated by a computer program.

4.select from the base list those observations whose numbers correspond to the written out random numbers

· Simple random sampling has obvious advantages. This method is extremely easy to understand. The research results can be extended to the target population. Most approaches to obtaining statistical inference involve collecting information using simple random sampling. However, the simple random sampling method has at least four significant limitations:

1. It is often difficult to create a sampling frame that allows simple random sampling.

2. A simple random sample can result in a large population, or a population spread over a large geographic area, which significantly increases the time and cost of data collection.

3. The results of using a simple random sample are often characterized by low precision and a higher standard error than the results of using other probabilistic methods.

4. Application of the SRS may result in an unrepresentative sample. Although samples obtained by simple random selection, on average, adequately represent the entire population, some of them are extremely inaccurately representative of the studied population. This is especially likely with a small sample size.

· Simple non-repeating sampling. The sampling procedure is the same, except that the cards with the respondent's numbers are not put back into the deck.

1. Systematic probability sampling. It is a simplified version of simple probability sampling. Based on the list of the general population, respondents are selected at a certain interval (K). The value of K is determined by chance. The most reliable result is achieved with a homogeneous general population, otherwise the step size and some internal cyclical patterns of the sample may coincide (mixing of the sample). Cons: Same as for simple probability sampling.

2. Serial (nested) sampling. Sampling units are statistical series (family, school, team, etc.). The selected elements are subjected to continuous examination. The selection of statistical units can be organized according to the type of random or systematic sampling. Negative: Possibility of greater homogeneity than in the general population.

3. Regional sampling. In the case of a heterogeneous population, before using probability sampling with any selection technique, it is recommended to divide the population into homogeneous parts, such a sample is called a regionalized sample. Zoning groups can be both natural formations (for example, city districts), and any feature underlying the study. The characteristic on the basis of which the division is carried out is called the characteristic of stratification and regionalization.

4. "Convenient" selection. The “convenience” sampling procedure consists of establishing contacts with “comfortable” sampling units — a group of students, a sports team, friends and neighbors. If it is necessary to obtain information about the reaction of people to a new concept, such a sample is quite reasonable. Convenient sampling is often used for preliminary testing of questionnaires.

Improbability sampling

Selection in such a sample is carried out not according to the principles of randomness, but according to subjective criteria - availability, typicality, equal representation, etc.

1. Quota sample - the sample is built as a model that reproduces the structure of the general population in the form of quotas (proportions) of the studied characteristics. The number of sample elements with a different combination of the studied characteristics is determined so that it corresponds to their share (proportion) in the general population. So, for example, if the general population is represented by 5000 people, of which 2000 are women and 3000 are men, then in the quota sample we will have 20 women and 30 men, or 200 women and 300 men. Quota samples are most often based on demographic criteria: gender, age, region, income, education, and others. Cons: Usually such samples are unrepresentative. several social parameters cannot be taken into account at once. Pros: readily available material.

2. Snowball method. The sample is constructed as follows. Each respondent, starting with the first one, is asked for the contacts of his friends, colleagues, acquaintances who would fit the selection conditions and could take part in the study. Thus, with the exception of the first step, the sample is formed with the participation of the research objects themselves. The method is often used when it is necessary to find and interview hard-to-reach groups of respondents (for example, respondents with a high income, respondents belonging to the same professional group, respondents with any similar hobbies / hobbies, etc.)

3. Spontaneous sampling - sampling of the so-called "first comer". Often used in television and radio interrogations. The size and composition of spontaneous samples is not known in advance, and is determined by only one parameter - the activity of the respondents. Cons: it is impossible to establish what general population the respondents represent, and as a result, it is impossible to determine representativeness.

4. Route survey - often used when the unit of study is the family. All streets are numbered on the map of the settlement in which the survey will be carried out. Large numbers are selected using a table (generator) of random numbers. Each large number is considered as consisting of 3 components: street number (first 2-3 numbers), house number, apartment number. For example, the number 14832: 14 is the street number on the map, 8 is the house number, 32 is the apartment number.

5. Regional sampling with a selection of typical objects. If, after regionalization, a typical object is selected from each group, i.e. an object that, according to most of the characteristics studied in the study, approaches the average, such a sample is called zoned with a selection of typical objects.

Group building strategies

The selection of groups for their participation in the psychological experiment is carried out using various strategies that are needed in order to ensure the maximum possible observance of internal and external validity.

· Randomization (random selection)

· Pairwise selection

· Stratometric sampling

· Approximate Modeling

· Engaging real groups

Randomization, or random selection, is used to create simple random samples. The use of such a sample is based on the assumption that each member of the population is equally likely to be included in the sample. For example, to make a random sample of 100 university students, you can put pieces of paper with the names of all university students in a hat, and then get 100 pieces of paper out of it - this will be a random selection (Goodwin J., p. 147).

Pairwise selection- a strategy for constructing sample groups, in which the groups of subjects are composed of subjects equivalent in terms of side parameters that are significant for the experiment. This strategy is effective for experiments using experimental and control groups with the best option - attracting twin pairs (mono- and dizygotic), as it allows you to create ...

Stratometric sampling - randomization with the allocation of strata (or clusters). With this method of sampling, the general population is divided into groups (strata) with certain characteristics (gender, age, political preferences, education, income level, etc.), and subjects with the corresponding characteristics are selected.

Approximate Modeling - drawing up limited samples and generalizing the conclusions about this sample to a wider population. For example, when 2-year university students participated in a study, the data from that study was extended to “people aged 17 to 21”. The admissibility of such generalizations is extremely limited.

Approximate modeling is the formation of a model that, for a clearly defined class of systems (processes), describes its behavior (or the necessary phenomena) with acceptable accuracy.

Section 2. Sample and general population

General and sample population.

Statistical population

General (includes all units of observation that can be attributed to it in accordance with the purpose of the study.) The general population can be considered not only within specific industries or territorial boundaries, but also limited by other characteristics (gender, age) and their combination.

Thus, depending on the purpose of the study and its tasks, the boundaries of the general population change; for this, the main features that limit it are used.

Sample (part of the general population, which should be representative of the general population and most fully reflect its properties). Based on the analysis of the sample population, you can get a fairly complete picture of the patterns inherent in the entire general population.

The sample must be representative, that is, the sample must contain all the elements and in the same ratio as in the general population. In other words, the sample must reflect the properties of the general population, that is, correctly represent it. Representativeness must be quantitative and qualitative.

Quantitative - based on the law of large numbers and means a sufficient number of elements in the sample, calculated according to special formulas and tables.

Qualitative - based on the law of probability and means the correspondence (uniformity) of the signs that characterize the elements of the sample in relation to the general population.

Sampling methods:

- random sampling - selection of observation units at random.

-Mechanical sampling - arithmetic approach to the selection of observation units typological sample- when forming, the general population is preliminarily divided into types with the last. selection of observation units from each typical group. In this case, the number of units can be selected in proportion to the size of the typical group and disproportionately - Serial sampling (nested selection)- is formed by selecting not individual observation units, but whole groups, series, or nests, which include observation units organized in a separate way

Multi-stage selection method - according to the number of stages, one-stage, two-stage, three-stage, etc. are distinguished. directed selection method- allows you to identify the influence of unknown factors when establishing the influence of known

Algorithms for parametric criteria.

Parametric tests are used for samples with a normal distribution. The formula for calculating these criteria contains the sample parameters: mean, variance, etc. Therefore, they are called parametric. The normality of the distribution law must be statistically proven using one of the goodness-of-fit criteria: Pearson's test, Fisher's F-test,-criterion Kolmogorov, etc.

In some cases, parametric tests are more powerful than nonparametric tests. The latter have a higher probability of occurrence of an error of the second kind - acceptance of a false null hypothesis.

Parametric methods include the following:

- Student's criterion

- Fisher criterion

- Methods of one-way analysis

- Methods of two-factor analysis

Student's criterion

Appointment.
The criterion allows you to evaluate the differences in the mean values ​​of samples with a normal distribution.

Description of the criterion.

The criterion is applicable for comparing the mean values ​​of two samples obtained before and after the influence of a certain factor.

Two-sample t-test for independent samples

For two unrelated samples (observations do not belong to the same group of objects), two calculation options are possible:

• when variances are known
• when the variances are unknown but equal to each other.

Where

square deviation. Here and - variance estimates.

Consider first the equal samples. In this case

In the case of equally numbered samples , expression

In both cases, the number of degrees of freedom is calculated using the formulas

It is clear that with numerical equality of samples

The empirical value of the Student's criterion is compared with the critical value (according to table 1 of the appendix) for a given number of degrees of freedom.

Null hypothesis.

Let's calculate an example in laboratory work.

Example.

The psychologist measured the time of a complex sensorimotor choice reaction (in ms) in the control and experimental groups. The experimental group (X) consisted of 9 highly qualified athletes. The control group (Y) consisted of 8 people who were not actively involved in sports. The psychologist confirms the hypothesis that the average speed of the complex sensorimotor reaction of choice in athletes is higher than the same value in people who are not involved in sports.

The arithmetic mean values ​​X and Y :, in the control group.

Then

^ The number of degrees of freedom k = 9 + 8 - 2 = 15

According to the application table for a given number of degrees, we find

Building an axis of significance

That. The differences found by the psychologist between the experimental and control groups are significant at more than 0.1% level, or in other words, the average speed of the complex sensorimotor reaction of choice in the group of athletes is significantly higher than in the group of people who are not actively involved in sports.

In terms of statistical hypotheses, this statement sounds like this: hypothesis H0 about the similarity is rejected and at the 0.1% significance level an alternative hypothesis H1 is accepted - about the difference between the experimental and control groups.

Two-sample t-test for dependent (linked) samples

Associated samples are understood as observations for one group of objects, and all observations are pairwise associated with each object of research and characterize its state before exposure and after exposure to a certain factor.

Hypotheses

: the sample mean does not differ from zero.

: The sample mean is different from zero.

1. The normality of the distribution law is preliminarily checked according to one of the goodness-of-fit criteria.

2. Calculated (i = 1..n) - pairwise differences variant, and the measurement results for i- th object before and after the impact of some factor. The quantity will be considered independent for different objects and normally distributed

3. Calculated (preferably in tabular form): the sum of pairwise differences and auxiliary parameters.

4. It is calculated - the empirical value of the criterion by the degrees of freedom according to the formula

Where n is the size of the sample.

5.Found empirical meaning Student's test is compared with the critical value(according to table 1 of the appendix) for a given number of degrees of freedom.
Null hypothesisat a given level of significanceaccepted if the empirical value.

The critical value for a selected probability and a given number of degrees of freedom can be found by using the Excel built-in function TDFRONT.

Example.

The psychologist assumed that as a result of training, the time for solving equivalent problems (i.e. having the same solution algorithm) will significantly decrease. To test the hypothesis, the solution time (in minutes) of the first and third tasks was compared in eight subjects.

The solution to the problem is presented in the table.