Psychology general abilities Druzhinin Vladimir Nikolaevich (Doctor of Psychology)
Progressive matrices by J. Raven
J. Raven was a student of Charles Spearman. In 1936, he, together with L. Penrose, proposed the Progressive Matrices test to measure the level of development of general intelligence. According to Spearman, the best way to determine intelligence is to test abstract relationships. The test items are based on Gestalt theory and Spearman's theory of intelligence. It is assumed that the subject initially perceives the task as a whole, then identifies patterns of change in the elements of the image, after which the selected elements are included in the whole image and the missing part of the image is found.
The material chosen was abstract geometric figures with an internal pattern organized according to a certain law (see Fig. 16).
Three main versions of the test were constructed: 1) a simpler color test intended for children from 5 to 11 years old, 2) a black and white version for children and adolescents from 8 to 14 years old and adults from 20 to 65 years old, 3) a version of the test , designed in 1977 by J. Raven in collaboration with D. Court and intended for individuals with high intellectual achievements. The last option includes not only a non-verbal, but also a verbal part.
The test can be carried out both with a time limit for completing tasks, and without a limit (at the request of the researcher).
The color version of the test uses three series, differing in difficulty level. Each series contains 12 matrices. The second option consists of 5 series (A, B, C, D, E) of 12 tasks each, arranged in increasing difficulty. The difficulty of the tasks increases from series A to series E. The subject completes the first 5 tasks of series A with the help of an experimenter, the rest - independently. The subject must choose the correct answer from 6-8 proposed ones. The number of answer options increases as the difficulty of the series increases.
Raven assumed that during the test the subject learns and completing the previous task prepares him to perform the subsequent, more difficult one.
In the series A the subject must complete the missing part of the image. He must demonstrate the ability to differentiate elements and identify connections between gestalt elements, as well as complete the missing part of the structure, comparing it with samples.
In the series IN the subject must find analogies between pairs of figures, differentiating their elements.
When performing a series WITH you need to solve the problem by determining the principle of changing figures vertically and horizontally.
In the series D It is required to determine the pattern of permutation of figures horizontally and vertically.
Series E to solve it, it requires analyzing the figures of the main image and composing the missing figure in parts.
For each correct solution, 1 point is assigned, the number is calculated right decisions in each series and the total number of points, which are converted into either standard scores (stans) or a standard IQ. Based on the batch results, a “variability index” is also calculated. There are distributions of the number of correct solutions by series, obtained from a standardization sample, corresponding to the total score. The tabular distribution is compared with that obtained during testing of the subject, and the differences between the expected and empirical estimates are summed without taking into account the sign. The “variability index” characterizes the reliability of the results and is aimed at identifying subjects who solved tasks by guessing or feigned a low result (who did not solve simple problems).
The normal value of the index is 0-4; with a value of 7, the subject’s answers are considered unreliable.
As noted earlier, this interpretation can hardly be considered the only possible one. Our research shows that the solution to the Raven test is probabilistic in nature, since the lack of interest in simple tasks, incorrect understanding of the task (provoked by the test material itself) leads to the fact that the test taker, while solving complex tasks, may make mistakes in simple ones.
Raven's test reliability ranges from 0.70 to 0.89; the average difficulty of test items is 0.32; correlation with success schooling(assessment of academic performance) – 0.72. The correlation with IQ according to the D. Wechsler test (WAIS) is 0.70-0.74 (adults) and 0.91 (children 9-10 years old), with arithmetic tests - up to 0.87. Our research has shown that the Raven test is not metrologically flawless.
Tasks D12 and E8 are designed so poorly that the probability of their correct solution (0.13 and 0.14) does not exceed the significant probability of solving this task by chance (p = 0.125). The tasks either contain the wrong idea, or the form of the material makes it possible for the test subject to make logical constructions that were not intended by the developer.
In the task of EY, in addition to correct option answer (No. 6), there are two (No. 1 and No. 2), the frequency of which subjects choose is statistically significantly higher than the probability of a random answer.
For task C12, the estimate of the probability of choosing the correct answer (No. 2) exceeds the limit statistical significance, but the estimate of the probability of choosing the false option (No. 4) significantly exceeds the probability of the correct choice.
That is, the very structure of an incorrectly constructed task leads the subject to a false answer. This state of affairs is a consequence of the conflict between two theories that are eclectically based on the test: perceptual (Gestalt theory) and the theory of general intelligence. The perceptual characteristics of the task prevent the subject from performing a consistent logical analysis of it. Thus, it is not the strategy “from the whole to the details” that comes into conflict with the strategy “from the details to the whole,” but the perceptual pattern contradicts the logical one.
According to Yusupov’s model, the number of tasks in a level test should not exceed 7, and in the Raven test there are 60 of them (in the abbreviated version - 30). The test is clearly informationally redundant. Basic tasks that reliably differ in difficulty level can only be considered: B8, A12, C4 (or D6), D8 (or D10, or E2), C8, E6, EY, E12.
The number of easy tasks in the Raven test is excessively large - they account for almost half of all test tasks. Moreover, there is no correspondence between empirical and standard item difficulty (n = 0.543).
To assess the complexity of a task, we proposed a modified complexity indicator:
where n is the number of solvers test, N – total standardization sample size.
In the first section of the chapter it was already mentioned that the version of calculating the test score proposed by the test authors does not justify itself and should be replaced with a more reliable one, taking into account the empirical complexity of the task.
The success of the test, as we have established, depends both on the level of development of “speed intelligence” and on the cognitive ability that determines the solution of complex tasks.
Perhaps the second factor is based on the differentiation of an individual’s cognitive experience, associated with such features of intelligence as cognitive complexity, conceptual development, and verbal competence.
Factorization of the correlation matrix of the application of the Raven test, a test for the diagnosis of cognitive “simplicity-complexity” (author A. G. Shmelev) and the “Concepts” test (for generalization of concepts) showed that two independent factors are distinguished - non-verbal and verbal, having equal weight, but the technique for generalizing concepts is included in the nonverbal factor.
Factor analysis of the intercorrelation matrix of seven techniques (cognitive simplicity-complexity test, Raven's test, Concepts technique, Eysenck test, etc.) revealed 3 equally significant factors: non-verbal – perceptual ( maximum load on the Raven test), speed intelligence (maximum load - Eysenck test and geometric generalization test) and the factor of verbal competence (positive load on the Concepts test and an indicator of cognitive simplicity). At the same time, a negative and significant loading of this factor on Raven’s test was found. Consequently, the more complex and differentiated the cognitive experience of the subject, the more successfully he solves the Raven test, which requires analytical work, and the worse he copes with tasks on conceptual generalization.
The differentiation of cognitive structures is one of the main parameters of intelligence that determines the success of solving analytical thinking problems, including in the Raven test.
From the book Holotropic Consciousness by Grof Stanislav From the book History of Psychology. Crib author Anokhin N V21 PROGRESSIVE PSYCHOLOGICAL CONCEPTS IN THE USA In the second half of the 18th century. The struggle of the American colonies against the subordination of England began. This determined the main shifts in the ideological life of the country.1. A strong educational movement is rising with its apology
From the book Teach Yourself to Think! by Buzan TonyPROGRESSIVE READING METHODS: FASTER AND FASTER In addition to the tips above general Some of the readers may find the following information useful, which, however, usually accompanies the work of a qualified
From the book Ariadne's Thread, or Journey through the Labyrinths of the Psyche author Zueva ElenaPROGRESSIVE “MEMORY CARDS” Based on the content of the previous chapters and taking into account that the brain perceives information better if it is structured in such a way as to “slip” like a coin into a machine, and also taking into account the measuring ability of the brain
From the book Club of Psychological Fighters. Fuck fear author Ivanov Alexey AlekseevichParental matrices In the transmission of “moral heritage”, the second signaling system is involved - speech - word. Hence the “pedigree”. And speech, like the unique design of a human hand, is different for everyone. Moreover, the first signal system in the original sense
From the book Shadows of the Mind [In Search of the Science of Consciousness] by Penrose RogerRitual of burning the “matrix of fear” Get alone. Make sure that no one disturbs you and that nothing interferes with you. Choose a suitable place (the kitchen is also suitable). Turn on meditative music or remain in silence - do what is comfortable for you. For burning
From the book The Self-Liberating Game author Demchog Vadim Viktorovich From the book Transpersonal Psychology. New approaches author Tulin Alexey7. Scenario of the information-quantum matrix To begin with, it is important to note that the word “matrix” means the womb, i.e. that which bears, feeds, crystallizes one or another form of reality.91 There are many ways of modeling and accumulation
From the book Areas of the Human Unconscious: Data from LSD Research [with drawings from patients!] by Grof StanislavBasic perinatal matrices Grof's map of consciousness consists of an empirical spectrum of extraordinary states of consciousness. Grof identifies the first, biographical level, which includes various traumatic events in a person’s life after birth. In the center
STIMULUS MATERIAL
A series of cards (matrices) with tasks of increasing complexity. This test is intended for children from 4-5 years old (series A cards), 6-8 years old (AB series cards) and up to 10 years old (B series cards). To solve the problem, children need to find the principle on which the structure is built. given matrix, and choose the correct answer from the options below.
Instructions
Look carefully at the picture. You need help fixing a damaged rug, and to do this you need to find a patch that does not differ in design from this rug. Below are given different variants, choose the correct one.
Carrying out the test
Children are sequentially presented with matrices in order of increasing complexity, starting with series A and ending with a series corresponding to the child’s age. Although there is practically no limit on the decision time, it is important to determine the child’s reaction time, that is, the time interval between receiving the instruction and answering. If the child finds it difficult to choose the correct option, a hint with a detailed explanation of how to solve this problem is possible. This hint does not reduce the objectivity of the test as the level of difficulty increases. If the child answers quickly but incorrectly, you can make the task easier by covering half of the pictures offered for choice.
First of all, the number of correct answers is counted (each 1-point solution is worth 1 point), then the sum of the points received and their percentage of the total number of answers. According to the percentage of correct answers, five levels are distinguished intellectual development children:
1st level(over 95%) - particularly highly developed intelligence;
2nd level(75-94%) - above average intelligence;
3rd level(25-74%) - average intelligence;
4th level(5-24%) - below average intelligence;
Level 5(below 5%) - intellectual defect. In addition to the level of intellectual development of children, the Raven test makes it possible to analyze the process of solving a problem.
The speed of the child’s response (reaction time) allows us to identify impulsive children, that is, children who respond without thinking, almost immediately after they hear the instructions. The reaction time of such children is 15-20 seconds. A large number of incorrect answers (up to 50%) are associated not with intellectual difficulties, but with impaired attention, low concentration, and also with the inability to plan their activities. These defects, while not actually defects in thinking, can significantly reduce a child’s performance at school, and therefore it is very important to identify them in time and, if possible, correct them. Reducing the field of view helps reduce the spread of attention, and therefore the number of correct answers increases when we cover three of the six options offered for analysis. When working with such children, it is necessary to specifically highlight the main stages of orientation and the sequence of operations necessary For correct solution to the problem. It is possible to attach specially drawn activity diagrams that will also help the child organize it.
Analysis of the process of solving tasks in the Raven test also helps to identify the learning ability of children, which often corresponds to their abilities. So, for children who find it difficult to complete the first, fairly easy tasks, it is necessary to explain the way to solve them. If children immediately grasp the adult’s explanations and quickly learn to solve problems of this type, they can transfer the learned technique to more difficult, final tasks. Therefore, the general low level of work of these children (as well as the errors they made in other tests, in particular in the Kogan test) is associated not with an intellectual defect, but with low level knowledge, which can be easily adjusted during training.
Sometimes children cope well with the first tasks, but do not solve more difficult ones, despite the help of an adult. In this case, we can talk about their low learning ability and the need for more attention from adults, more complete and lengthy explanations of new material.
Thus, in the Raven test it is important not only to calculate the total number of correct answers, but also to find out which tasks the child solved - only the first ones or the last ones too.
Theoretical provisions:
Raven's progressive matrices are based on 2 theories:
1. The theory of apperception of forms developed by Gestalt psychology.
In accordance with this theory, each task can be considered as a specific whole, consisting of a number of interconnected elements. It is assumed that first there is a global assessment of the task - the matrix, and then the implementation of analytical perception with the test subject highlighting the principle adopted when developing the series. On final stage the selected elements are included in a holistic image, which helps to detect the missing detail of the image.
2. Ch. Spearman’s theory of neogenesis.
... develops the considered provisions of the theory of perception of forms.
The test is developed in the tradition English school intelligence studies, according to which best way measurement factors G– the task of identifying relationships between abstract figures.
Productive ability – this is the ability to generate new solutions, the ability to perceive and establish relationships.
Since perception is, first of all, conceptual process, an essential element productive capacity is the ability to generate new, mainly non-verbal concepts that make it possible to think freely and clearly.
Short story creations:
The test was proposed by L. Penrose and J. Raven in 1936.
Progressive matrices were developed as tests to measure the ability to think clearly. This is one of best tests nonverbal intelligence.
The most famous are 2 main options progressive matrices Ravena - black and white and color.
The color version is intended for examination of children aged 5 to 11 years and for persons over 65 years of age.
Black-and-white SMPs are intended for examining children and adolescents aged 9 to 14 years and adults aged 20 to 65 years.
Black and white dough material The SPM consists of 60 matrices or compositions with a missing element.
The tasks are divided into 5 series (A, B, C, D, E) with 12 tasks of the same type, but of increasing complexity in each series. The difficulty of the tasks also increases as you move from series to series.
This progressive order ensures standard training in the method of operation.
The subject must choose the missing element of the matrix among 6–8 proposed options.
If necessary, the subject performs the first 5 tasks of series A with the help of an experimenter.
When developing the test, an attempt was made to implement the principle of “progressiveness”, which consists in the fact that completing previous tasks and their series is, as it were, preparing the subject for performing subsequent ones.
ð Learning to perform more difficult tasks occurs.
Five series provide 5 opportunities to master the way of thinking necessary to solve problems, and 5 progressive assessments of the subject’s intellectual activity.
Color option Raven's progressive matrices consist of 3 series - A, A B, B, 12 matrices in each series.
Validity and reliability data:
The reliability coefficient of the test, according to various researchers, varies from 0.70 to 0.89.
The predictive validity of the test (in relation to academic performance criteria) is 0.72.
Target: measuring the general level of development of intellectual abilities.
Application area: from 5 to 65 years old, no cultural or gender restrictions.
Organization and procedure of the examination:
The test can be carried out both individually (especially for young children), and in a group version, as well as during self-testing.
The group can be any size. The study is carried out in a quiet room. Each subject works at a separate table, which eliminates cheating. All subjects sit facing the experimenter.
Pencils and answer sheets are distributed. Test takers are asked to enter the necessary information about themselves in the appropriate lines of the answer forms. After they do this, the test booklets are distributed. Subjects are asked not to open the booklets until everyone is ready.
* All subjects, regardless of their age, are given exactly the same series of tasks, in exactly the same order, and they are all asked to work at a constant speed convenient for them, without stopping or delay, from the beginning of the scale to its end.
* It is preferable to work without time restrictions unless it is related to specific research purposes.
* Having people work slowly and at their own pace provides a more reliable measure of their ability to think clearly.
Stimulus material: a set of test booklets, individual answer sheets and a pencil.
* To maintain the steady interest of the subject and to avoid the negative impact of fatigue, each task is very clearly designed, neatly drawn and everything is done to make it pleasant to look at.
Processing the results:
Processing the results obtained is simple. Each correct solution is worth 1 point. The total amount of points received is calculated (provided that the subject completed all the series from beginning to end), as well as the number of correct solutions in each of the 5 series.
Primary scores from the tables are converted in accordance with age standards into percentiles or stanina.
An essential step in quantitative processing is the calculation of the “variability index”. The index is determined based on tables of distribution of the number of correct solutions in each of the 5 series.
Options for distribution in the table are determined in accordance with the total score in all series.
The differences between the expected and actual estimates in each series (without taking into account the sign, “modulo”) are summed up.
The author points out that if the variability (inconsistency) index exceeds 2 units, then this indicates instability in the assessment of general intellectual abilities. however, the overall test score is considered valid even when the inconsistency exceeds 2 units.
Normal values of the inconsistency index: from 0 to 4.
Critical value: 7 or more – test data is considered unreliable.
Comparison of the real and expected distribution of the number of correct decisions in the series is aimed at identifying subjects who complete the task by guessing.
The value of the variability index may be significantly higher than the critical value if the subject is set to simulate a low result on the test (demonstration of inability to solve the simplest problems).
Episode description:
Each series of tasks is compiled according to certain principles:
Serie A. When working with matrices of this series, the following basic thought processes are implemented:
a) differentiation of the main elements of the structure and disclosure of connections between them;
b) identifying the missing part of the structure and comparing it with the presented samples.
*Analysis of the structure of the whole and restoration of integrity (perception of integrity)
Series B. It comes down to finding an analogy between two pairs of figures. The subject reveals this principle through gradual differentiation of elements.
*Analogy between pairs of figures, classification through symmetry.
Series C. The tasks in this series contain complex changes in figures in accordance with the principle of their continuous development, enrichment vertically and horizontally.
*Analysis of progressive changes in structure, enrichment of figures in the horizontal and vertical directions.
Series D Compiled on the principle of rearranging figures in a matrix in horizontal and vertical directions.
*Analysis of the pattern of alternation of figures in a holistic structure.
Series E. The most difficult. The process of solving tasks in this series consists of analyzing the figures of the main image and subsequent “assembling” the missing figure in parts (analytical-synthetic mental activity).
*Synthesis of the missing figure in parts according to the algebraic principle (addition/subtraction).
Algorithm for constructing a diagnosis:
Level I. Superintellectual - if the score is equal to or greater than the 95th percentile for a given age group.
Level II. Intellectual capabilities are clearly above average - if the score is equal to or greater than the 75th percentile; II+ if the score is between the 90th and 95th percentile.
Level III. Average intelligence - if the score is in the range from 25 to 75 percentile; III+ if the score is higher than the median (50th percentile); III-, if the score is below the median.
Level IV. Intellectual capabilities are clearly below average - if the score is equal to or less than the 25th percentile; VI- if the score is equal to or less than the 10th percentile.
Level V Significant decline in intelligence - if the score is equal to or less than the 5th percentile for the corresponding age group.
- - - - - - - - - - - - - - - - - - - - - -
Advantages: the test is free from cultural influence (only norms are adapted); free from the influence of socio-economic parameters; education, etc.
The test tasks appeal to 3 main mental processes:
Voluntary attention
Perception of the whole
Flexibility of thinking
Flaw: cannot be used twice on the same person.
Time for this technique is limited!
Junior students = 20 minutes
8 – 12 years = 20 minutes
Adults = 15 minutes
Ticket 43.
Short selection test.
Brief history of the creation of the methodology and general theoretical principles: There is a group of tests of general mental abilities that were specially developed for rapid preliminary assessment of personnel. The first test in this direction was the Otis Self-Administered Test. The test in most cases had a good validity coefficient for selection for the position of clerk, computing machine operator, workers, foremen, etc. For highly skilled employees, there was little correlation between test scores and job success.
The most famous adaptation of the Otis test is the Wonderlick test. Unlike the Otis test, the Wonderlic test works well on a sample of highly qualified employees and managers. As A. Anastasi notes, the reliability of the test, despite its brevity, is quite satisfactory."
The structure of the adapted test corresponds to the structure of general abilities. The starting premise for analyzing the test structure was following definition learning ability:
Learning ability – this is quite important feature, which is necessary to master any specialty. Existing system training of specialists practically eliminates the possibility of an individual approach.
Data on the validity and reliability of the methodology:
Test reliability was determined using the Kuder-Richardson method, which takes into account the consistency of answers across all test items. The alpha coefficient calculated using the Kuder-Richardson formula was α = 0.86. Reliability was calculated based on two samples: 6th grade secondary school students (60 people) and persons with higher education, mostly communications engineers (140 people).
For research purposes, simultaneously with CAT, the subject performed numerical and verbal tests G.Yu.Eysenka. The correlation of PT with IQ according to the Eysenck numerical test was 0.68, and with IQ according to verbal 0.61 (significance level p<0,001).
Success in completing the COT weakly correlates with the degree of independence of choices from previous success or failure in the aspiration test; correlation coefficient – 0.22, significance level: p<0,1.
A comparison of the CAT with other tests of special abilities has not been carried out. For the 2 studied samples, PT distribution graphs were obtained, indicating the heterogeneity of the samples. These distributions indicate the dependence of PT on the level of education for samples as a whole: with a higher level of education, PT shifts upward due to a decrease in the weakest group, solving 6-9 tasks, and an increase in the average group, solving 18-24 tasks. The “strong” group, solving more than 30 tasks, in both samples is small and quite stable in size (5-7%).
It can be assumed that CAT affects some genetically determined aspects of mental speed and general ability, at least those aspects that do not depend on the level of education.
Target: determination of the integral indicator “general abilities”. The test provides for the diagnosis of the ability to generalize and analyze material, flexibility of thinking, inertia of thinking, distractibility of attention, literacy, orientation, spatial imagination, and mathematical abilities.
Application area : The methodology for determining the general abilities of the COT is intended for use in the selection and distribution of personnel in industry, the army, and the education system. It can be used in any situation related to the determination of learning ability.
Organization : Group and individual examination: the test can be carried out in a group version. In this case, it is desirable that the subjects work at separate tables. First, individual forms are distributed and signed. Individual test books are then distributed.
Brief description of the test: the test consists of 50 tasks with 3–5 answer options, of which only one is correct. In addition, there are open tasks (without answer options).
Examination procedure : instructions within the text given on the first page. All examples are understood. The experimenter answers the subjects' questions. It is especially noted that during the test, questions about its content are not allowed. In addition, it is emphasized that the test execution time is strictly limited - 15 minutes. On command: “Stop! Time's up" - all forms are turned over and handed over to the experimenter.
During the test, it is advisable for the experimenter to make sure that the forms of the subjects are signed.
Processing the results : The total sum of answers matching the key is recorded. Processing one form takes 15-20 seconds. PT has median norms, which divide the sample into two parts and the results are assessed according to the “hit or miss” principle. If the subject’s PT is equal to or higher than the median for a given group of specialists, then the subject may be allowed to undergo further testing on tests of special abilities (if they exist and are adapted). Or, depending on the PT, the test taker may be assigned to one or another study group if the test is used to justify the differentiation of students according to their level of learning/
Brief description of the scales :
CAT is intended to determine the integral indicator “general abilities” and provides for the diagnosis of the following “critical points” of intelligence:
1. Ability to summarize and analyze material :When completing tasks on proverbs, it is necessary to abstract from a specific phrase, move to the area of interpretation of meanings and find intersections of meanings in this area, and then return to specific phrases.
2. Flexibility of thinking .If the subject performs tasks of this type incorrectly (tasks: find a word that is opposite in meaning; determine whether 2 words are similar or different in meaning), it can be assumed that associations are chaotic in nature and are not inhibited.
3. Inertia of thinking. Switchability .The arrangement of tasks provides the ability to quickly switch from one type of activity to another. The alternation of different types of tasks in the text can make it difficult for people with inert connections from past experience to solve them. Such persons have difficulty changing their chosen way of working, are not inclined to change the course of their judgments, or switch from one type of activity to another. Their intellectual processes are inactive, the pace of work is slow. At the same time, the quality of their mental products is low.
4. Emotional components of thinking. Distractibility. Thinking must be actively directed towards an object, a task. A number of test items reduce the test score in people prone to emotional destruction of thought processes (tasks: quite difficult). Such subjects begin to smile and turn to the experimenter. In the “level of aspiration” test, these individuals are characterized by their impulsive response to successes and failures.
5. Speed and accuracy of perception. Distribution and concentration of attention. Some of the test tasks relate to the so-called clerical abilities (tasks: compare pairs of rows of numbers or words). The work of a highly qualified specialist or manager involves working with a lot of different documents that need to be processed in a short time, highlighting the main contents, comparing numbers, etc.
6. Use of language. Literacy. Some of the test tasks are designed for the ability to correctly use the language, literacy, and basic knowledge of a foreign language (within the alphabet) (tasks: determining relationships, searching for an extra word, differences in figures, etc.).
7. Choosing the optimal strategy. Orientation. According to the instructions, the test allows solving problems in any order. Some subjects who find it easier to do either numerical or verbal problems, having reflected on this fact, look through all the tasks and solve the simplest ones for themselves, and then return to another type of problem.
8. Spatial imagination. The test presents four tasks related to operations in two-dimensional space.
9. Mathematical abilities.
Thus, the test indicator (PT) is a complex multiparameter indicator of a person’s general abilities.
Math statistics is a branch of mathematics that studies approximate methods for collecting and analyzing data from experimental results to identify existing patterns, i.e. finding laws of distribution of random variables and their numerical characteristics.
In mathematical statistics, it is customary to distinguish two main areas of research::
1. Parameter estimation population.
2. Testing statistical hypotheses (some a priori assumptions).
The basic concepts of mathematical statistics are: population, sample, theoretical distribution function.
General population is the set of all conceivable statistical data from observations of a random variable.
X G = (x 1, x 2, x 3, ..., x N, ) = (x i; i=1,N)
The observed random variable X is called the sample feature or factor. The general population is a statistical analogue of a random variable; its volume N is usually large, so a part of the data is selected from it, called a sample population or simply a sample.
X B = (x 1, x 2, x 3, ..., x n, ) = (x i; i=1,n)
Х В М Х Г, n £ N
Sample
is a set of randomly selected observations (objects) from the general population for direct study. The number of objects in the sample is called the sample size and is denoted by n. Typically the sample is 5%-10% of the population.Using a sample to construct patterns that govern an observed random variable allows one to avoid its continuous (mass) observation, which is often a resource-intensive process, or even simply impossible.
For example, a population is a set of individuals. Studying an entire population is time-consuming and expensive, so data is collected from a sample of individuals who are considered representative of that population, allowing inferences to be made about that population.
However, the sample must satisfy the condition representativeness, i.e. provide a reasonable representation of the population. How to form a representative (representative) sample? Ideally, they strive to obtain a randomized sample. To do this, a list of all individuals in the population is made and they are randomly selected. But sometimes the costs of compiling a list may turn out to be unacceptable, and then they take an acceptable sample, for example, one clinic, hospital, and study all the patients in this clinic with a given disease.
Each sample element is called a variant. The number of repetitions of variants in a sample is called frequency of occurrence. The quantity is called relative frequency options, i.e. is found as the ratio of the absolute frequency of variants to the entire sample size. A sequence of options written in ascending order is called variation series.
Let's consider three forms of variation series: ranked, discrete and interval.
Ranked series
- this is a list of individual units of the population in ascending order of the characteristic being studied.Discrete variation series is a table consisting of columns or rows: a specific value of the feature x i and the absolute frequency n i (or relative frequency ω i) of the manifestation of the i-th value of the feature x.
An example of a variation series is the table
Write the distribution of relative frequencies.
Solution: Let's find the relative frequencies. To do this, divide the frequencies by the sample size:
The distribution of relative frequencies has the form:
| 0,15 | 0,5 | 0,35 |
Control: 0.15 + 0.5 + 0.35 = 1.
A discrete series can be represented graphically. In a rectangular Cartesian coordinate system, points with coordinates () or () are marked, which are connected by straight lines. Such a broken line is called frequency polygon.
Construct a discrete variation series (DVR) and draw a polygon for the distribution of 45 applicants according to the number of points they received in the admission exams:
39 41 40 42 41 40 42 44 40 43 42 41 43 39 42 41 42 39 41 37 43 41 38 43 42 41 40 41 38 44 40 39 41 40 42 40 41 42 40 43 38 39 41 41 42.
Solution: To construct a variation series, we place the different values of the characteristic x (variants) in ascending order and write down its frequency under each of these values.
Let's construct a polygon for this distribution:
Rice. 13.1. Frequency polygon
Interval variation series used for a large number of observations. To construct such a series, you need to select the number of intervals of the characteristic and set the length of the interval. If there are a large number of groups, the interval will be minimal. The number of groups in a variation series can be found using the Sturges formula: 
(k is the number of groups, n is the sample size), and the width of the interval is 
where is the maximum; - the minimum value is an option, and their difference R is called range of variation.
A sample of 100 people from the population of all medical university students is being studied.
Solution: Let's calculate the number of groups: . Thus, to compile an interval series, it is better to divide this sample into 7 or 8 groups. The set of groups into which observation results are divided and the frequency of obtaining observation results in each group is called statistical totality.
To visually represent the statistical distribution, use a histogram.
Frequency histogram is a stepped figure consisting of adjacent rectangles built on one straight line, the bases of which are identical and equal to the width of the interval, and the height is equal to either the frequency of falling into the interval or the relative frequency ω i.
Observations of the number of particles entering the Geiger counter within a minute gave the following results:
21 30 39 31 42 34 36 30 28 30 33 24 31 40 31 33 31 27 31 45 31 34 27 30 48 30 28 30 33 46 43 30 33 28 31 27 31 36 51 34 31 36 34 37 28 30 39 31 42 37.
Based on these data, construct an interval variation series with equal intervals (I interval 20-24; II interval 24-28, etc.) and draw a histogram.
Solution: n = 50
The histogram of this distribution looks like:
Rice. 13.2. Distribution histogram
Task options
№ 13.1. Every hour, the voltage in the electrical network was measured. The following values (B) were obtained:
227 219 215 230 232 223 220 222 218 219 222 221 227 226 226 209 211 215 218 220 216 220 220 221 225 224 212 217 219 220.
Construct a statistical distribution and draw a polygon.
№ 13.2. Observations of blood sugar in 50 people gave the following results:
3.94 3.84 3.86 4.06 3.67 3.97 3.76 3.61 3.96 4.04
3.82 3.94 3.98 3.57 3.87 4.07 3.99 3.69 3.76 3.71
3.81 3.71 4.16 3.76 4.00 3.46 4.08 3.88 4.01 3.93
3.92 3.89 4.02 4.17 3.72 4.09 3.78 4.02 3.73 3.52
3.91 3.62 4.18 4.26 4.03 4.14 3.72 4.33 3.82 4.03
Based on these data, construct an interval variation series with equal intervals (I - 3.45-3.55; II - 3.55-3.65, etc.) and depict it graphically, draw a histogram.
№ 13.3. Construct a polygon of frequency distributions of erythrocyte sedimentation rate (ESR) for 100 people.
Methods of mathematical statistics
1. Introduction
Mathematical statistics is a science that deals with the development of methods for obtaining, describing and processing experimental data in order to study the patterns of random mass phenomena.
In mathematical statistics, two areas can be distinguished: descriptive statistics and inductive statistics (statistical inference). Descriptive statistics deals with the accumulation, systematization and presentation of experimental data in a convenient form. Inductive statistics based on these data allows one to draw certain conclusions regarding the objects about which data are collected or estimates of their parameters.
Typical areas of mathematical statistics are:
1) sampling theory;
2) theory of assessments;
3) testing statistical hypotheses;
4) regression analysis;
5) analysis of variance.
Mathematical statistics is based on a number of initial concepts without which it is impossible to study modern methods of processing experimental data. Among the first of these is the concept of a general population and a sample.
In mass industrial production, it is often necessary to determine whether the quality of the product meets the standards without checking each product produced. Since the quantity of products produced is very large or the testing of products is associated with rendering them unusable, then it is not checked a large number of products. Based on this check, it is necessary to give a conclusion about the entire series of products. Of course, you cannot say that all transistors from a batch of 1 million pieces are good or bad by checking one of them. On the other hand, since the process of selecting samples for testing and the tests themselves can be time-consuming and lead to high costs, the scope of product testing should be such that it can give a reliable representation of the entire batch of products, while being of minimal size. For this purpose, we introduce a number of concepts.
The entire set of objects being studied or experimental data is called the general population. We will denote by N the number of objects or the amount of data that makes up the general population. The value N is called the volume of the population. If N>>1, that is, N is very large, then N = ¥ is usually considered.
A random sample, or simply a sample, is a portion of a population selected at random from it. The word "random" means that the probability of selecting any object from the population is the same. This is an important assumption, but it is often difficult to test in practice.
The sample size is the number of objects or the amount of data that makes up the sample and is denoted by n. In the future, we will assume that the sample elements can be assigned, respectively, numerical values x 1, x 2, ... x n. For example, in the process of quality control of manufactured bipolar transistors this could be measurements of their DC gain.
2. Numerical characteristics of the sample
2.1 Sample mean
For a particular sample of size n, its sample mean
is determined by the relationwhere x i is the value of the sample elements. Typically you want to describe the statistical properties of random random samples rather than just one of them. This means that it is being considered mathematical model, which assumes a sufficiently large number of samples of size n. In this case, the sample elements are considered as random variables Xi, taking values xi with a probability density f(x), which is the probability density of the general population. Then the sample mean is also a random variable
equal toAs before, we will denote random variables in capital letters, and the values of random variables are lowercase.
The average value of the population from which the sample is drawn will be called the general average and denoted by m x. It can be expected that if the sample size is significant, the sample mean will not differ significantly from the population mean. Since the sample mean is a random variable, the mathematical expectation can be found for it:
Thus, the mathematical expectation of the sample mean is equal to the general mean. In this case, the sample mean is said to be an unbiased estimate of the population mean. We will return to this term later. Since the sample mean is a random variable that fluctuates around the general mean, it is desirable to estimate this fluctuation using the variance of the sample mean. Consider a sample whose size n is significantly smaller than the population size N (n<< N). Предположим, что при формировании выборки характеристики генеральной совокупности не меняются, что эквивалентно предположению N = ¥. Тогда
Random variables X i and X j (i¹j) can be considered independent, therefore,
Let's substitute the result obtained into the formula for variance:
where s 2 is the variance of the population.
From this formula it follows that with increasing sample size, fluctuations of the sample average around the general average decrease as s 2 /n. Let us illustrate this with an example. Let there be a random signal with mathematical expectation and variance respectively equal to m x = 10, s 2 = 9.
Signal samples are taken at equally spaced times t 1, t 2, ...,
X(t) X 1
t 1 t 2 . . . t n t
Since the samples are random variables, we will denote them X(t 1), X(t 2), . . . , X(tn).
Let us determine the number of samples so that the standard deviation of the estimate of the mathematical expectation of the signal does not exceed 1% of its mathematical expectation. Since m x = 10, it is necessary that
On the other hand, therefore or From here we obtain that n ³ 900 samples.2.2 Sample variance
For sample data, it is important to know not only the sample mean, but also the spread of sample values around the sample mean. If the sample mean is an estimate of the population mean, then the sample variance must be an estimate of the population variance. Sample variance
for a sample consisting of random variables is determined as follows
Using this representation of the sample variance, we find its mathematical expectation
