Processing of statistical data. Presentation on the topic "elements of statistical data processing" What is statistical data processing for?

Atyusheva Anna

In the work, using the example of processing data on the progress of 7th grade students, the main statistical characteristics are considered, the collection and grouping of statistical data is carried out, statistical information is clearly presented, and the analysis of the data obtained is carried out.

The work contains an accompanying presentation.

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Municipal autonomous educational institution "Gymnasium No. 24"

XXII scientific conference MAGNI

Statistical data processing

MAOU "Gymnasium No. 24" Atyusheva Anna

Consultant: math teacher

Shchetinina Natalia Sergeevna

Magadan, 2016

Introduction ……………………………………………………………………………………………… 3

1. Basic concepts used in statistical data processing ……………………… .5
2. Research part ………………………………………………… ............................ ...... 7

2.1. Statistical processing of data on the progress of students in grade 7 "B" ………………… ..7

2.2 Visual presentation of data using histograms ……………………………………………………………………………………………… 18

2.3. Comparative characteristics of the educational activity of students according to the results of the 1st and 2nd quarters .................................................................. 21

2.4. Analysis of the questionnaire survey of students in grade 7 "B" for parental control over the progress of children ............................................................. 23

Conclusion ………………………………………………………………………………………… ... 27

Literature ……………………………………………………………………………………………… 28

Introduction

Any of us, opening a book or newspaper, turning on the TV or getting to the train station, is constantly faced with a tabular form of information presentation. These are the lesson timetable, train timetable, multiplication table and much more. All information is presented in the form of diagrams or graphs.

You need to be able to process and analyze such information. Without data processing, comparison of events, it is impossible to trace the development of a particular problem.

In the course of algebra, we studied statistical characteristics that are widely used in various studies. I was interested in the practical application of the studied characteristics, and the opportunity to process the data so that the information presented would clearly determine the course of development of a particular problem and, as a consequence, the result of its solution. As such a problem, I decided to consider the performance of my class in the quarters of the first half of the year.

Object research area- algebra

Object of study- statistical characteristics

Subject of study- academic performance of 7 "B" grade students in quarters of the first half of the year

Hypothesis: We believe that using the example of processing data on the performance of students in grade 7B, we will not only get acquainted with the main statistical characteristics, but also learn on our own:

• collect and group statistical data;
• visually present statistical information;
• analyze the data obtained.

Target: learn to process, analyze, and visualize the available information.

Tasks:

• study statistical characteristics;
• collect information on student performance in grade 7 in quarters

the first half of the year;

• process information;
• carry out a visual presentation of information using histograms;
• analyze the data obtained, draw appropriate conclusions.

Basic concepts used in statistical data processing

Statistics is a science that deals with obtaining, processing and analyzing quantitative data on various mass phenomena occurring in nature and society. The word "statistics" comes from the Latin word "status", which means "state, state of affairs."

The simplest statistical characteristics are arithmetic mean, median, range, mode.

• Arithmetic meana series of numbers is called the quotient of dividing the sum of these numbers by the number of terms. Usually, the arithmetic mean is found when they want to determine the average value for a certain series of data: the average wheat yield per hectare in the region, the average output of one working team per shift, the average score of the certificate, the average air temperature at noon in this decade, etc.
• Median an ordered series of numbers with an odd number of members is called the number written in the middle, and the median of an ordered series of numbers with an even number of members is called the arithmetic mean of two numbers written in the middle. Note that it is more convenient and faster to work with a number series if it is ordered, i.e. a row in which each subsequent number is not less (or not more) than the previous one.
• Fashion a series of numbers is called the number most often found in a given series. A number of numbers may have more than one mod or no mod at all. The mode of a data series is usually found when one wants to identify some typical indicator. Note that the arithmetic mean of a series of numbers may not coincide with any of these numbers, and the mode, if it exists, must necessarily coincide with two or more numbers in the series. In addition, unlike the arithmetic mean, the concept of "mode" refers not only to numerical data.
• In a sweep a series of numbers is the difference between the largest and the smallest of these numbers. The range of a series is found when they want to determine how large the spread of data in a series is.

Let's show the definition of each of the characteristics using the example of a series of numbers: 47,46,52,47,52,47,52,49,45,43,53,53,47,52.

Arithmetic mean 48,7.

It is found like this: we determine the sum of the numbers and divide it by their number.

(47+46+52+47+52+47+52+49+45+43+53+53+47+52):14=48,7.

Median of this series of numbers will be the number 48.

It is found like this: we order a series of numbers, choosing the one that is in the middle. If the number of numbers is even, then we find the arithmetic mean of the two in the middle of the row of numbers.

43,45,46,47,47,47, 47,49 ,52,52,52,52,53,53

(47+49):2=48

Fashion of this series of numbers will be the numbers 47 and 52 ... These numbers are repeated most often.

47 ,46, 52 , 47 , 52 , 47 , 52 ,49,45,43,53,53, 47 , 52 .

In a sweep of this series of numbers will be 10.

It is found like this: choose the largest and smallest number in the series and find the difference between these numbers.

47,46,52,47,52,47,52,49,45, 43, 53 ,53,47,52

53-43=10

Research part

Statistical processing of data on the performance of students in grade 7 "B"

Let's move on to processing information. Let's compose tables for each of the subjects, consisting of three lines, the first will contain a series of data. Each variant from this series was actually observed in the sample for a certain number of times. This number is called the multiplicity of the options. So let's put in the second line the multiplicity of the corresponding option. Let's get the distribution table of the sample.

If we add all the multiplicities, then we get the number of all measurements made during the sampling - the sample size (In our case, this number is 24, which corresponds to the number of students in the class).

In the third line, the ratio, expressed as a percentage, is called the frequency of the options.

Frequency options =

In general, if a table of relative frequencies is compiled based on the results of the study, then the sum of the relative frequencies is equal to 100%.

I quarter

Russian language.

Let's sort the data of the sample (marks): 3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4 , 4.5.

Average mark in the subject:(average).

Frequency allocation table

Literature.

Let's order the data of the sample (marks): 3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5 , 5.5.

Average mark in the subject:(average).

Algebra.

Let's sort the data of the sample (marks): 3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4 , 5.5.

Average mark in the subject:(average).

The largest number of students in the subject have "4, 3" (fashion)

About half of the students in the Russian language study at 4 (median)

Story.

Let's sort the data of the sample (marks): 3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4 , 4.5

Average mark in the subject:(average).

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Social science.

Let's order the data of the sample (marks): 3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5 , 5.5

Average mark in the subject:(average).

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Geography.

Let's sort the data of the sample (marks): 3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5.5 ,5

Average mark in the subject:(average).

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Physics.

Let's sort the data of the sample (marks): 3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4 , 4.5

Average mark in the subject:(average).

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Biology.

Let's sort the data of the sample (marks): 3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5 ,5

Average mark in the subject:(average).

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

LIFE SAFETY FUNDAMENTALS.

Let's sort the data of the sample (marks): 4,4,4,4,4,4.4.5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5 ,5

Average mark in the subject:(average).

Let's sort the data of the sample (marks): 3,4,4,4.4,4,4,4,4,4,5,5,5,5,5,5,5,5.5,5,5.5,5,5,5

Average mark in the subject:(average).

The largest number of students in the subject have "5" (fashion)

About half of the students in the Russian language study at 5 (median)

English language.

Let's sort the data of the sample (marks): 3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5.5,5,5 ,5

Average mark in the subject:(average).

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Informatics.

Let's sort the data of the sample (marks): 3,4,4,4,4.4,4,4,4,4,4,4,4,4,5,5,5,5.5.5,5,5,5,5

Average mark in the subject:(average).

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Technology.

Let's sort the data of the sample (marks): 3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,55,5,5,5,5,5

Average mark in the subject:(average).

The largest number of students in the subject have "5" (fashion)

About half of the students in the Russian language study at 4.5 (median)

Now let's collect similar information on the results of the second quarter.

Russian language.

Let's sort the data of the sample (marks): 3,3,3.3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4 ,4

Average mark in the subject:(average)

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Literature.

Let's sort the data of the sample (marks): 3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5 , 5.5

Average mark in the subject:(average)

The largest number of students in the subject have "3" (fashion)

About half of Russian language students study grade 3 (median)

Algebra.

Let's sort the data of the sample (marks): 3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5 , 5.5

Average mark in the subject:(average)

The largest number of students in the subject have "3" (fashion)

About half of Russian language students study grade 3 (median)

Story.

Let's sort the data of the sample (marks): 3,3,3,3,3,3,3,3,3,4.4,4,4,4,4,4,4,4,4,4,4,4,4,4 ,5

Average mark in the subject:(average)

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Society.

Let's sort the data of the sample (marks): 3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5 , 5.5

Average mark in the subject:(average)

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Geography.

Let's sort the data of the sample (marks): 3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5 , 5.5

Average mark in the subject:(average)

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Physics.

Let's sort the data of the sample (marks): 3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,44,5,5 ,5

Average mark in the subject:(average)

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Biology.

Let's sort the data of the sample (marks): 3,3,3,4,4,4,4,4,4,4.4,4,4,4,4,4,4,4,4,5,5,5,5,5 ,5

Average mark in the subject:(average)

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

LIFE SAFETY FUNDAMENTALS.

Let's sort the data of the sample (marks): 3,4,4,5,5,5,5,5.5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5 ,5

Average mark in the subject:(average)

The largest number of students in the subject have "5" (fashion)

About half of the students in the Russian language study at 5 (median)

History and society of the native land.

Let's sort the data of the sample (marks): 3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5 , 5.5

Average mark in the subject:(average)

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

English language.

Average mark in the subject:(average)

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Informatics.

Let's sort the data of the sample (marks): 3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5 , 5.5

Average mark in the subject:(average)

The largest number of students in the subject have "4" (fashion)

About half of the students in the Russian language study at 4 (median)

Technology.

Let's sort the data of the sample (marks): 3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5 , 5.5

Average mark in the subject:(average)

The largest number of students in the subject have "5" (fashion)

About half of the students in the Russian language study at 4 (median)

Data visualization with histograms

For a visual presentation of data obtained as a result of a statistical study, various methods of their presentation are widely used.

We will use histograms for clarity of the data. A histogram is a stepped shape made up of closed rectangles. The base of each rectangle is equal to the length of the interval, and the height is equal to the multiplicity of the variant or relative frequency. Thus, in a histogram, unlike a conventional bar chart, the bases of the rectangle are not chosen arbitrarily, but are strictly determined by the length of the interval.

Comparative characteristics of student performance in the first quarter subjects

Comparative characteristics of student performance in the subjects of the second quarter

conclusions

According to the results of the first quarter, it is clearly seen that the most difficult for students to cope with such subjects as: Russian language and algebra, subjects for which "three" is an assessment that is a priority in relation to other marks. This means that the quality in these subjects is lower than in others.

It is also clear that the high level of triplets in subjects such as literature, history, society, physics, English. It is also sad to have triples in subjects such as technology, biology, geography.

According to the results of the second quarter, the number of triples and fives significantly decreased, that is, students distributed their strengths in all subjects, and not according to separately preferred ones.

Histogram of the distribution of the average score in the subjects of the first quarter

Histogram of the distribution of the average score in the subjects of the second quarter

Conclusion

To create these diagrams, we used such a statistical characteristic as the arithmetic mean. It is clearly seen that in the second quarter, knowledge of the Russian language, history and society of the native land, computer science deteriorated. Improved in history, society, physics, biology, life safety, English. But at the same time, the diagrams show that more significant changes for the better occurred only in physics and the English language.

Comparative characteristics of the educational activity of students according to the results of the first and second quarters

Histogram of the quality of knowledge in the subjects of the first quarter

Histogram of the quality of knowledge in the subjects of the second quarter

By combining both histograms into one, it is much easier to see the picture of class performance in comparison. And individually it is easier to see for which subjects the quality is higher. For example, in the first quarter the quality is less than 60% in subjects - algebra, Russian, history, in the second - Russian, literature, algebra, physics. It is already clear that the most difficult for students are the Russian language, algebra. And the percentage of quality in all subjects is not very different 66% - the first quarter, 68% - the second. That is, the jump-like quality in subjects, which is clearly visible on the comparison diagram, suggests that students are not particularly trying to improve their level of knowledge, and do not hold their positions in one or another subject area.

Chart comparing all items by quality for the 1st and 2nd quarters

During the second quarter, the number of good and excellent students in the Russian language, society, biology, English, and technology increased significantly. The number in literature, algebra, life safety, IORK and computer science has slightly decreased. And you can see a strong drop in the quality of physics, which is associated with the students' unpreparedness for lessons.

And again we come to the conclusion that children learn “in leaps and bounds”, and there are no special preferences in the direction of education (humanitarian subjects, physics and mathematics, subjects of the natural cycle).

Analysis of the questionnaire survey of 7 "B" grade students on the subject of parental control over the progress of children

Based on the results of the above study, we decided to conduct a survey among students of grade 7 "B" for parental control over the teaching of children (questionnaires, see Appendix)

The sample size is 22 people.

Parents check homework

Conclusion

Almost a quarter of students on this issue without parental control, which of course affects their academic performance.

Number of homework checks per week

Median = 0,0,0,0,0,0,1,1,2,2,3,3,3,3,4,4,5,7,7,7,7,7 = (3 + 3 ): 2 = 3

Arithmetic mean = 3

Conclusion

On average, the assignment is checked three times a week. Given the spasmodic learning curve, this is not enough.

Median = 0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,5,5,6,7, 7.7 = (2 + 2): 2 = 2

Arithmetic mean = 3 (on average, diaries are checked by parents 3 times a week)

The amount of time students spend doing homework

• Swing R = x (max) - x (min) = 3.5 - 0.5 = 3 hours

(characterizes the magnitude of the scatter of the observed values, i.e. shows the difference between the longest and the shortest time)

• Mode M (0) = 2.5 hours ( shows the value that occurs more often than others, i.e. shows the time students spend most often)

Histogram of Students' Time spent on Homework

Conclusion

On average, homework takes 2.5 hours a day. Which is considered a normal indicator for the age of students.

Conclusion

As a result of the work done, I learned to process and analyze the available information

Knowing the statistical characteristics helped me determine the GPA in various subjects, as well as fashion and scope in those indicators of performance where it would seem impossible to determine them. Without data processing, comparison of events, it is impossible to trace the development of a particular problem. We tried not only to track the problem that had arisen - the decline in the quality of knowledge and academic performance in subjects, but also to try to find out the reason, which, in our opinion, lay in insufficient parental control over the academic performance of their children. The questionnaire survey and the results of academic performance showed that students of grade 7 "B" do not have enough skill in self-control over their learning, and parents believe the opposite.

The work done, I think, will be useful both for the class teacher in working with parents, and for my classmates to improve their results in individual subjects in the future.

Statistics is a science that studies, processes and analyzes quantitative data on a wide variety of mass phenomena in life. We have only revealed its characteristics a little for ourselves, and there is still a lot of unknown and interesting ahead.

Bibliography:

1. http://www.nado5.ru/e-book/naibolshii-obzchii-delitel

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   Statistical data processing Prepared by: 7th grade "B" student of MAOU "Gymnasium No. 24" Anna Atyusheva Consultant: mathematics teacher Natalya Sergeevna Shchetinina

   Purpose: learn to process, analyze, and visualize the available information. Objectives: to study statistical characteristics; collect information about the progress of students in grade 7 in the quarters of the first half of the year; process information; carry out a visual presentation of information using histograms; analyze the data obtained, draw appropriate conclusions.

   A hypothesis using the example of processing data on student performance, you can not only get acquainted with the main statistical characteristics, but also learn how to collect and group statistical data; visually present statistical information; analyze the received data.

   Statistics is a science that deals with obtaining, processing and analyzing quantitative data on various mass phenomena occurring in nature and society. The word "statistics" comes from the Latin word "status", which means "state, state of affairs." Simplest statistical characteristics: Arithmetic mean Median Span Mode

   Determination of each of the characteristics using the example of a series of numbers: 47,46,52,47,52,47,52,49,45,43,53,53,47,52. The arithmetic mean of this series of numbers will be the number 48.7. (47 + 46 + 52 + 47 + 52 + 47 + 52 + 49 + 45 + 43 + 53 + 53 + 47 + 52): 14 = 48.7. The median of this series of numbers will be the number 48.43,45,46,47,47,47, 47, 49, 52,52,52,52,53,53 (47 + 49): 2 = 48 The mode of this series of numbers will be be the numbers 47 and 52. 47, 46, 52, 47, 52, 47, 52, 49,45,43,53,53, 47, 52. The range of this series of numbers will be 10. 49.45, 43, 53, 53.47.52 53-43 = 10

   Problems with academic performance in grade 7 "B"

   Option 2 3 4 5 Frequency rate no options 14 9 1 Frequency% 0% 58.3% 37.5% 4.2% Russian language. Let's sort the data of the sample (marks): 3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4 , 4.5. Average score in the subject: 14 ∙ 3 + 9 ∙ 4 + 5 ∙ 124 = 8324≈3.5 (arithmetic mean). The largest number of students in the subject have "3" (fashion) Approximately half of the students in the Russian language study at 3 (median)

   For a visual presentation of data obtained as a result of a statistical study, various methods of their presentation are widely used.

   Comparative characteristics of student performance in subjects of the first quarter

   Comparative characteristics of student performance in subjects of the second quarter

   Histogram of the distribution of the average score in the subjects of the I and II quarters

   Comparison chart of all subjects by quality for the I and II quarters

   Questioning among students of grade 7 "B" on the subject of parental control over the education of children QUESTIONNAIRE 1. Do your parents check your homework? ___________________________________________________________ 2. How many times a week? ___________________________________________________________ 3. How many times a week do your parents look at your diary? ___________________________________________________________ 4. How much time on average do you spend each day on homework? ___________________________________________________________

   Parents check homework

   Number of homework checks per week Median = 0.0.0.0.0.0.1.1.2.2.3.3.3.3.4.4.5.7.7.7.7, 7 = (3 + 3): 2 = 3 Arithmetic mean = 3

   Histogram of students' time spent on homework

The purpose of the lesson:
- creating conditions for mastering the topic at the level of comprehension and primary memorization;
- to form the mathematical competence of the student's personality.

Lesson Objectives
Educational: to form an idea of ​​statistics as a science; to familiarize students with the concepts of basic statistical characteristics; to form the ability to find the scope, fashion of a number, analyze data, improve the skills of oral counting.
Developing: promote the ownership of concepts and their interpretation; development of supra-subject skills of analysis, comparison, systematization and generalization; continue the formation of a subject language, contribute to the formation of key competencies (cognitive, informational, communicative) at various stages of the lesson, contribute to the formation of a single scientific picture of the world among students by identifying interdisciplinary connections between statistics and various sciences.
Educational: foster interest in the subject under study, information culture; readiness to comply with generally accepted norms and rules, high efficiency and organization.

Technologies used: Technology of student-centered learning, information and communication technologies.
Necessary equipment, materials: multimedia projector, computer, interactive board.

During the classes

1. Organizational moment.

Checking the readiness of students for the lesson

Attendance check

2. Goal-setting.

Justification of the need to study this topic

Involving students in the lesson goal setting

And from what sources do we receive and collect information? (Estimated answers: radio, television, newspapers, magazines, telephone, people, internet, letters).

Where do people store information? (Estimated answers : in memory and on external media).
Studying at a technical school is getting information? At school, you studied general subjects, and while studying at a technical school, what else do you get? (Estimated answer: h knowledge by profession). The more we learn, the more information our memory contains.

Today I offer you another piece of information. You are trained as a mining operator. You will work on EKG-8I excavators. What is the performance of this excavator. At my request, the plant provided me with the following information. (Excavator performance - table)

Guys, is a lot of information good? Can all information be useful, high-quality? What should we be able to do in order not to get lost in the maze of information? (Estimated student answer: "Should be able to separate useful, high-quality information from low-quality"). Those. be able to handle it.

CONCLUSION: today in the lesson we will learn to process information.

3. Organization of activities for the study of new material.(students, in the process of explaining, make notes in notebooks and complete assignments)

1. Defining statistics

What are statistics? It is said that the British Prime Minister Benjamin Disraeli (1804 - 1881) answered this question as follows: "There are three types of lies: lies, blatant lies and statistics."

Statistics Is an exact science that studies methods of collecting, analyzing and processing data that describe mass actions, phenomena and processes.

(An excerpt from the novel "The Twelve Chairs" by Ilf and Petrov is read

"Statistics knows everything" it is known how much food the average citizen of the republic eats a year: it is known how many hunters, ballerinas in the country: machines, bicycles, monuments, lighthouses and sewing machines: How much life full of fervor, passions and thoughts looks at us from statistical tables! .. ".

Its name comes from the Latin word "status" - state, from this root the words stato (Italian), statistik (German), statе (English) - state arose.

Statistics study:

The main goals of studying the elements of statistics:

• the number of individual groups of the population of the country and its regions,
• production and consumption of various types of products,
• transportation of goods and passengers by various types of transport,
• natural resources and much more.

You know in which country the beginning of statistical practice was laid (in China), the first censuses of the country's population date back to the 5th century. II millennium BC

In the 19th century, it became possible to process data using formulas, mathematical laws and special characteristics. This?.... ( mat. statistics).

2. Math statistics

Math statistics- This is a branch of mathematics that studies methods of collecting, systematizing and processing the results of observations of random mass phenomena in order to identify existing patterns.

So why did Disraeli compare statistics with lies? (There was no rigorous scientific processing of information, the data was interpreted as they wanted).

Mathematical statistics has universal methods of information processing
This is what allowed the heroes of the film "Office Romance" to say the following words about statistics ( fragment of the film "Office Romance").
CONCLUSION: statistics bring information to the system.

3. Graphical presentation of information

Distribution polygon

Distribution histogram

Pie chart

4. Measurement characteristics
1. A series of data is a series of results of any measurements.

For instance: 1) measuring a person's height

2) Measurement of human (animal) weight

3) Meter readings (electricity, water, heat ...)

4) Results in the 100-meter sprint

2. The volume of the data series - the volume of the data series is the amount of all data.

For instance: a row of numbers 1 is given; 3; 6; -4; 0

its volume will be 5. Why?

3. The range of a data series is the difference between the largest and smallest numbers in a data series.

For instance: if a row of numbers is given 1; 3; 6; -4; 0; 2, then sweep this data series will be equal to 6 (since 6 - 0 = 6)

4. Mode of a data series - the mode of a data series is the number of a series that occurs most often in this series.

For example: p data poison may or may not have fashion.

So, in the data series 47, 46, 50, 52, 47, 52, 49, 45, 43, 53, each of the numbers 47 and 52 occurs twice, and the rest of the numbers - less than two times. In such cases, we agreed to consider that the series has two modes: 47 and 52.

5. Median of the row

The odd-member median is the number written in the middle.

Median with an even number of members - it is the arithmetic average of the two numbers written in the middle.

for instance: determine the median of a series of numbers

sixteen; -4; 5; -2; -3; 3; 3; -2; 3. Answer: -3

2) -1; 0; 2; one; -one; 0; 2; -one. Answer: 0

6. The arithmetic mean is the quotient of dividing the sum of the numbers in the series by their number.

for instance: given a row of numbers -1; 0; 2; one; -one; 0; 2; -one. Then the arithmetic mean will be: (-1 + 0 + 2 + 1 + (- 1) + 0 + 2 + (- 1)): 8 = 2: 8 = 0.25

4. Consolidation of the studied material.

Practical work

Exercise: to characterize the progress of Peter Ivanov's student in mathematics for the fourth quarter.

Completing of the work:

1. Collection of information:

The scores were written out from the journal: 5,4,5,3,3,5,4,4,4.

2. Processing of the received data:

Statistical processing of data is impossible without their ordering, generalization and analysis. Any results obtained must first be brought into such a form so that the maximum of useful information can be extracted from them. If the received data is too much, then they need to be grouped or summarized.

So, for grouping, it is necessary to determine the norms according to which the received data will be distributed. In this case, not only the clarity, but also the potential usefulness of the information obtained will depend on the chosen method. Correctly grouped research results are much more convenient to study and analyze.

Data processing can be applied in many areas of human activity. They can be divided into 3 main types:

1) universal methods that can be used without taking into account the scope;

2) methods for certain areas of activity, engaged in the study of real processes or phenomena;

3) methods for researching certain data.

It is clear that the more accurate the method by which the statistical processing of data is carried out, the more effective the analysis of a specific situation will be. If the first method is applicable for scientific results, the value of which will be assessed only by general scientific criteria, then the third method is used only for solving certain problems in a specific area.

In addition to a general knowledge of the methods by which the data is processed, it is also important to know how best to work with the results obtained. Statistical data processing involves the creation of tables or graphs for clarity of the information received.

At the initial stage, the information can be summarized in a table. So, for example, statistical processing of experimental data, written in tabular form, allows researchers to save researchers from additional unnecessary records of indicators, measurement values, additional factors that affect the course of the experiment. In tables, it is convenient to record not only the data of the study or experiment, but also to summarize intermediate and main results. True, for their correct construction, it is necessary to think over the required number of rows and columns in advance, write down all the necessary parameters.

The table can be done simply on a sheet of paper or directly enter the data into a computer. The second option will allow you to quickly sort the obtained data in the desired way, find the largest or, conversely, the smallest value, summarize or find by the selected group of results.

Do not forget that if competent statistical data processing requires several tables, then they must be numbered and a unique name must be thought up for each.

Graphs are a more visual way to record data. They visually show the relationship between different quantities, making it easier to understand the research results.

Knowing the basic principles of constructing tables and graphs, you can quickly and efficiently process the received data.

Methods of statistical processing of the results of an experiment are mathematical techniques, formulas, methods of quantitative calculations, with the help of which the indicators obtained during the experiment can be generalized, brought into a system, revealing the laws hidden in them.

We are talking about such regularities of a statistical nature that exist between the variables studied in the experiment.

Data Are the main elements to be classified or categorized for processing 26.

Some of the methods of mathematical and statistical analysis allow calculating the so-called elementary mathematical statistics characterizing the sample distribution of data, for example:

Sample mean,

Sample variance,

Median and a number of others.

Other methods of mathematical statistics make it possible to judge the dynamics of changes in individual statistics of the sample, for example:

Analysis of variance,

Regression analysis.

With the help of the third group of methods of sample data, one can reliably judge the statistical relationships that exist between the variables that are investigated in this experiment:

Correlation analysis;

Factor analysis;

Comparison methods.

All methods of mathematical and statistical analysis are conventionally divided into primary and secondary 27.

Methods that can be used to obtain indicators that directly reflect the results of measurements made in an experiment are called primary methods.

Secondary methods of statistical processing are called, with the help of which, on the basis of primary data, statistical patterns hidden in them are revealed.

The primary methods of statistical processing include, for example:

Determination of the sample mean;

Selective variance;

Selective fashion;

Sample median.

Secondary methods typically include:

Correlation analysis;

Regression analysis;

Methods for comparing primary statistics for two or more samples.

Let's consider methods for calculating elementary mathematical statistics, starting with a sample mean.

Arithmetic mean - it is the ratio of the sum of all data values ​​to the number of terms 28.

The average value as a statistical indicator is the average assessment of the psychological quality studied in the experiment.

This assessment characterizes the degree of its development as a whole in that group of subjects that was subjected to psychodiagnostic examination. By comparing directly the mean values ​​of two or more samples, we can judge the relative degree of development in the people making up these samples, assessed quality.

The sample mean is determined using the following formula 29:

where x cf is the sample mean or the arithmetic mean of the sample;

n - the number of subjects in the sample or private psychodiagnostic indicators, on the basis of which the average value is calculated;

x k - particular values ​​of indicators for individual subjects. There are n such indicators in total, therefore the index k of this variable takes values ​​from 1 to n;

∑ - accepted in mathematics sign of summation of the values ​​of those variables that are to the right of this sign.

Dispersion Is a measure of the dispersion of the data around the mean of 30.

The greater the variance, the greater the deviation or scatter in the data. It is determined in order to be able to distinguish from each other values ​​that have the same average, but different scatter.

The variance is determined by the following formula:

where is the sample variance, or simply variance;

An expression that means that for all x k from the first to the last in a given sample, it is necessary to calculate the differences between the particular and average values, square these differences and sum;

n is the number of subjects in the sample or primary values ​​for which the variance is calculated.

Median the value of the trait being studied is called, which divides the sample, ordered by the value of the given trait, in half.

Knowing the median is useful in determining whether the distribution of the particular values ​​of the studied trait is symmetrical and close to the so-called normal distribution. The mean and median for the normal distribution usually coincide or differ very little from each other.

If the sample distribution of characteristics is normal, then methods of secondary statistical calculations based on the normal distribution of data can be applied to it. Otherwise, this cannot be done, since serious errors can creep into the calculations.

Fashion one more elementary mathematical statistics and characteristics of the distribution of experimental data. Fashion is the quantitative value of the trait under study, which is most often found in the sample.

For symmetric feature distributions, including the normal distribution, the mode values ​​coincide with the mean and median values. For other types of distributions, asymmetric, this is not typical.

The method of secondary statistical processing, through which the connection or direct relationship between two series of experimental data is clarified, is called method of correlation analysis. It shows how one phenomenon influences another or is related to it in its dynamics. This kind of relationship exists, for example, between quantities that are in causal relationships with each other. If it turns out that two phenomena statistically reliably correlate with each other and if at the same time there is confidence that one of them can act as the cause of the other phenomenon, then this definitely implies the conclusion that there is a causal relationship between them.

There are several variations of this method:

Linear correlation analysis allows you to establish direct relationships between variables by their absolute values. These connections are graphically expressed as a straight line, hence the name "linear".

The linear correlation coefficient is determined using the following formula 31:

where r xy - linear correlation coefficient;

x, y - average sample values ​​of the compared values;

X i , at i - partial sample values ​​of the compared values;

P - the total number of values ​​in the compared series of indicators;

Dispersion, deviations of the compared values ​​from the mean values.

Rank correlation determines the dependence not between the absolute values ​​of the variables, but between the ordinal places, or the ranks occupied by them in the order of magnitude. The formula for the rank correlation coefficient is 32:

where R s is the Spearman rank correlation coefficient;

d i - the difference between the ranks of the indicators of the same subjects in ordered rows;

P - the number of subjects or digital data (ranks) in the correlated series.