Statistics is one of the oldest branches of applied mathematics, which widely uses the theoretical basis of many arithmetic definitions for the implementation of practical human activities. Even in the ancient states, it became necessary to strictly record the income of citizens by groups in order to conduct an effective taxation process. Statistical research is of great importance for economic development society, and beyond. Therefore, in this video tutorial we will look at the basic definitions of statistical characteristics.
Suppose we need to study the statistics of test completion by seventh grade students. First, we need to create an array of information that we can work with. The information, in this case, will be the numbers that determine the number of tests completed by each of the students. Consider two classes containing 15 students each. The general task included 10 exercises. The results are as follows:
7A: 4, 10, 6, 4, 7, 8, 2, 10, 8, 5, 7, 9, 10, 6, 3;
7B: 7, 5, 9, 7, 8, 10, 7, 1, 7, 6, 5, 9, 8, 10, 7.
We have received, in mathematical interpretation, two sets of numbers, each consisting of 15 elements. This array of information, by itself, is of little help in evaluating the effectiveness of task completion. Therefore, it needs to be statistically transformed. To do this, we introduce the basic concepts of statistics. A series of numbers obtained as a result of the study is called a sample. Each number (number of completed exercises) is a sampling option. And the number of all numbers (in this case, it is 30 - the sum of all students in both classes) is the sample size.
One of the main statistical characteristics is the arithmetic mean. This value is defined as a quotient obtained by dividing the sum of values of the sample variant by its size. In our case, it is necessary to add all the obtained values of the numbers and divide them by 15 (if we calculate the arithmetic mean for any one class), or by 30 (if we calculate the total arithmetic mean). In the presented example, the sum of all the number of completed tasks for class 7A will be 99. Dividing by 15, we get 6.6 - this is the arithmetic average of completed tasks for this group of students.
Working with a chaotic set of numbers is not very convenient, so very often an information array leads to an ordered data set. Let's create a variation series for class 7B using the gradual increase method, arranging the numbers from smallest to largest:
1, 5, 5, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 10, 10.
The number of occurrences of any one value in a data sample is called the sampling frequency. For example, the frequency of options "7" in the above variation series is easily determined, and it is equal to five. For display convenience, the ordered series is converted into a table showing the relationship between the standard series of variant values and the frequency of occurrence (the number of students who completed the same number of tasks).
In class 7A, the smallest sample option is "2", and the largest is "10". The interval between 2 and 10 is called the range of the variation series. For class 7B, the range of the series is from 1 to 10. The largest, in terms of frequency of occurrence, variant is called the sampling mode - for 7A this number is 7, occurring 5 times.
Lab #9
Statistical data analysis
Objective: learn how to process statistical data in spreadsheets using built-in functions; explore the possibilities of the Analysis Package in MS Excel 2010 and some of its tools: Random Number Generation, Histogram, Descriptive Statistics.
Theoretical part
Very common for processing survey data a large number objects or phenomena ( statistical data), methods of mathematical statistics are used.
Modern mathematical statistics is divided into two broad areas: descriptive and analytical statistics. Descriptive statistics covers methods for describing statistical data, presenting them in the form of tables, distributions, etc.
Analytical statistics is also called the theory of statistical inference. Its subject is the processing of data obtained during the experiment, and the formulation of conclusions that are of applied importance for various areas of human activity.
The set of numbers obtained as a result of the survey is called statistical aggregate.
sampling set(or sampling) is a set of randomly selected objects. General population is the set of objects from which the sample is made. Volume set (general or sample) is the number of objects in this set.
For statistical processing, the results of the study of objects are presented in the form of numbers x 1 ,x 2 ,…, x k. If the value x 1 observed n 1 time, value x 2 observed n 2 times, etc., then the observed values x i called options, and the number of their repetitions n i called frequencies. The procedure for counting frequencies is called data grouping.
Sample size n is equal to the sum all frequencies n i:
Relative frequency values x i is called the frequency ratio of this value n i to sample size n:
Statistical frequency distribution(or simply frequency distribution) is called a list of options and their corresponding frequencies, written in the form of a table:
| … | ||||
| … |
Relative frequency distribution called a list of options and their respective relative frequencies.
Basic statistical characteristics.
Modern spreadsheets have a huge set of tools for analyzing statistical data. The most commonly used statistical functions are built into the main core of the program, that is, these functions are available from the moment the program is launched. Other more specialized functions are included in additional routines. Specifically, in Excel, such a routine is called an Analysis ToolPak. The commands and functions of the analysis package are called Analysis Tools. We will limit ourselves to a few basic built-in statistical functions and the most useful analysis tools from the analysis suite in an Excel spreadsheet.
Mean.
The AVERAGE function calculates the sample (or general) mean, that is, the arithmetic mean of the feature of the sample (or general) population. The AVERAGE function argument is a set of numbers, usually specified as a range of cells, for example, =AVERAGE(A3:A201).
Lesson type: lesson learning new material.
The purpose of the lesson: Creation of conditions for the assimilation of the topic at the level of comprehension and primary memorization; to form the mathematical competence of the student's personality
Educational: form an idea of statistics as a science; to acquaint students with the concepts of basic statistical characteristics; to form the ability to find the arithmetic mean, range, mode, median of a series, analyze data.
Developing: promote knowledge of concepts and their interpretation; development of oversubject skills of analysis, comparison, systematization and generalization; to promote the formation of key competencies (cognitive, informational, communicative) at various stages of the lesson, to promote the formation of a unified scientific picture world by identifying interdisciplinary relationships between statistics and various sciences.
Educational: develop interest in the subject being studied information culture; readiness to comply with generally accepted norms and rules, high efficiency and organization.
Technologies used: MDO technology.
Necessary equipment, materials: multimedia projector, computer, interactive whiteboard.
Lesson plan
Organizing time. The class is divided into 4 groups.
Include a video from the film Office Romance.
WMV File (.wmv)
What do you think we are going to talk about today?
…….. right, about statistics
What is statistics? (Slide 2)
…….. this is the definition that the dictionary gives us (Slide 3)
Does statistics affect people's lives, society? Express your guesses as you wish.
Statistics as a science includes different sections: political, economic, applied, legal, medical, etc.
We will be interested in mathematical statistics. What is special about math statistics?
…….. of course with the help of mathematics (Slide 4)
Mathematical statistics has a number of characteristics. (Turn over the word “statistics” on the board).
Concepts are in front of you. (tablets on the board with the words: bisector, lunula, mules, arithmetic mean, median, mode, range, diameter, mean, maximum, optimum, invariant, constant, height) Guess which of them can be classified as statistical, what do you think?
(Suggested words put after the word statistical characteristics)
Now you will turn to texts that will help you confirm or refute your assumptions: whether the chosen concepts are statistical characteristics and how big the impact of statistics on society is. Each student received a table (Appendix 1), which he must fill out during the lesson. Let's remember the rules for working in a group: calmly, independently, in a businesslike way, with the distribution of responsibilities. The group must complete the table (Appendix 2)
Group work. Texts for groups. Annex 3. (10 min)
Protection (slide with definition + slide with task)
Be sure to fill out the checklists. (We ask each group who noted what for themselves according to this characteristic in the memo sheet) (Appendix 1.2)
Average
Putting order in the statistical characteristics
(leave only 4 characteristics)
Group 1 go to the blackboard and talk about the statistical characteristics - the arithmetic mean, the solution of the proposed problems, conclusions. (Slide 5.6).
Group 2 go to the blackboard and talk about the statistical characteristics - fashion, solving the proposed problems, conclusions. (slide 7.8)
Group 3 go to the blackboard and talk about the statistical characteristics - scope, solution of the proposed tasks, conclusions. (slide 9,10)
Group 4 go to the blackboard and talk about the statistical characteristic - the median, the solution of the proposed tasks, conclusions. (slide 11,12)
All groups came to the conclusion that there is a relationship between the life of society and statistics, the influence is great, even when we do not assume it.
Let's turn to the slides and see how statistical characteristics can manifest themselves in our everyday life. (Slides with jokes 13-19, 20)
Now we offer you to work as extras. (4 tasks of practical content are distributed) (7 minutes)
So, what statistical characteristic did you work with in the first task, what did you get
…….. fashion - eye and hair color (make a quick survey of each group)
…….. span - palm width (conduct a quick survey of each group)
what statistical characteristic did you work with in the third task, what did you get
…….. median - shoe size (conduct a quick survey of each group)
what statistical characteristic did you work with in the second task, what did you get
…….. arithmetic mean - growth (conduct a quick survey of each group)
Judging by the results, the average young man in our class looks like this (Slide 21)
And the girl is like that (Slide 22)
On such an optimistic note, we conclude our lesson.
(Answers to tasks Appendix 5)
Attachment 1.
Appendix 2
Appendix 3
Group 1. Statistics studies the number of individual groups of the population of the country and its regions, the production and consumption of various types of products, the transportation of goods and passengers by various modes of transport, Natural resources etc. The results of statistical studies are widely used for practical and scientific conclusions.
arithmetic mean A series of numbers is called a statistical characteristic, which allows you to find the quotient from 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 1 hectare in the area, the average daily milk yield from one cow on the farm, the average output of one worker, etc. Note that the arithmetic mean is found only for homogeneous values.
For example, when studying the study load of students, a group of 12 seventh graders was identified. They were asked to record on a given day the time (in minutes) it took to complete homework in algebra. We got the following data: 23, 18, 25, 20, 25, 25, 32, 37, 34, 26, 34, 25.
With this data series, we can determine how many minutes students spent on average doing their algebra homework. To do this, the indicated numbers must be added and the resulting amount divided by the quantity, i.e. in this case 12:
Wed arithm. ===27
Thus, we found that students spent an average of 27 minutes on algebra homework.
Find the arithmetic mean in the following problems:
Task 1. From the list of air pollutants from stationary sources in Khanty-Mansi Autonomous Okrug-Yugra, first select the emissions of the most common substances, and then determine the average amount of these emissions for three years, presented in the table in thousand tons.
| gaseous and liquid substances | |||
| sulfur dioxide | |||
| nitrogen oxides | |||
| carbon monoxide |
Task 2. Determine the average air temperature in the city of Uray on February 14, 2017, if it is known that on the sites: Yandex -9 oC, Gismeteo -11 oC, rp5 -16 oC, - 11 oC, meteonovosti -15 oC, meteonova -10 oC, synoptic -11 oC.
The role of statistics in our life is so significant that people often without hesitation and without realizing, constantly use elements of statistical methodology not only in work processes, but also in everyday life. Working and relaxing, shopping, meeting other children, making some decisions, a person uses a certain system, the information he has, the prevailing tastes and habits, facts, systematizes, compares these facts, analyzes them, draws a conclusion and makes certain decisions. takes concrete action. Thus, in each person there are elements of statistical thinking, which is the ability to analyze and synthesize information about the world around.
Group 2
Meaning of the word " statistics
The results of statistical studies are widely used for practical and scientific conclusions.
When processing data, statistics use some characteristics, one of which is mode. Fashion is used, for example, in determining the size of clothes, shoes, which are in greatest demand among buyers.
Fashion series - the value in the set of observations that occurs most frequently. Fashion = typical. In the series 3,4,3,5,5,4,5,3,5 mode = 5. As the most frequently occurring number.
Sometimes more than one mode occurs in the aggregate. For example: 6, 2, 6, 6, 8, 9, 9, 9, 10; mode = 6 and 9. In this case, we can say that the population is multimodal. Of the structural averages, only mode has this unique property.
There is no fashion in the series of numbers 69,68,72,74,89,87,84.
Mode as an average is used more often for non-numerical data. Among the listed car colors - white, black, metallic blue, white, metallic blue, white - fashion will be equal to white. With the help of an expert assessment, the most popular types of a product are determined with its help, which is taken into account when forecasting sales or planning their production.
Solve the following tasks:
Task 1. In the rivers of the Khanty-Mansiysk Autonomous Okrug many fish live in the Bolshoy Yugan River, inhabited by pike, perch, roach, crucian carp, ide, and burbot. Fish live in the Agan River: pike, perch, roach, sterlet, crucian carp, ide, burbot, nelma. Fish live in the Vakh River: pike, perch, roach. Fish live in the Tromgan River: pike, perch, roach, crucian carp, ide, burbot. The totality of fish of the Khanty-Mansiysk Autonomous Okrug-Yugra is multimodal (pike, perch and roach are found in all rivers in the district. Determine the most typical fish in the presented rivers.
Zalacha 2. The table shows the electricity consumption in January by residents of 9 apartments
Determine the mode of this series
Group 3. Meaning of the word " statistics has undergone significant changes over the past two centuries. The word "statistics" has the same root as the word "state" and originally meant the art and science of government: the first professors of statistics in 18th-century German universities would today be called social scientists. Because government decisions are to some extent based on data on population, industry, etc. statisticians, of course, became interested in such data, and gradually the word "statistics" began to mean the collection of data about the population, about the state, and then in general the collection and processing of data. There is no point in extracting data if there is no benefit to be derived from it. Therefore, one of the main tasks of statistics is the proper processing of information.
Today, statistics and data analysis permeate almost any modern field of knowledge: economics, advertising, marketing, business, medicine, education, etc. It determines the dynamics of development, decline or growth of social phenomena. This is a science that solves certain problems due to the availability and development of statistical methods, including thanks to the developing information technology.
When processing data, statistics uses some characteristics, one of which is the median.
Median called the value of the quantity located in the center of the ordered series.
The median divides the series into two equal parts in such a way that there are the same number of units on both sides of it. At the same time, for one half, the value of the attribute is not more than the median, for the other half, it is not less.
| The median is found according to the following algorithm: Arrange numbers in ascending order If the series contains an odd number of elements, then the median is the number in the middle; If the series contains an even number of elements, the median lies between the two middle elements of the series and is equal to the arithmetic mean calculated over these two elements. Example. Find the median of the series 16,13,15,10,19,22,25,12,18,14,19,14,16,10. Solution. Let's build a series in ascending order: 10,10,12,13,14,14,15,16,16,18,19,19,22,25, it contains an even number of elements n=14, therefore the median lies between the two middle elements of the sample - between 7-element and 8-element: 10,10,12,13,14,14,15,16,16,18,19,19,22,25 and is equal to the arithmetic mean of these elements: Me=(15+16 )/2=15.5 |
Let us give examples of the real use of the median in statistics. So, when analyzing the results shown by the participants in the race, the median allows you to select a group of athletes who showed a result above the average and put them in the next stage of the competition.
mathematical median property is that the sum of absolute (modulo) deviations from the median value gives the minimum possible value. This fact finds its application, for example, in solving transport problems, when it is necessary to calculate the construction site of an object near the road in such a way that the total length of flights to it from different places is minimal (stops, gas stations, warehouses, etc., etc.) .
Solve the following tasks:
Task 1. Current security costs environment in Khanty-Mansi Autonomous Okrug amounted to million rubles:
Find the median of this series.
Group 4. Statistics- a science that deals with obtaining, processing and analyzing quantitative data on various mass phenomena occurring in nature and society.
One of the main tasks of statistics is the proper processing of information. Of course, statistics has many others: obtaining and storing information, making various forecasts, evaluating their reliability, etc.
One of the statistical indicators of the difference or spread of data is the "Range". in a big way series is the difference between the largest and smallest of these numbers. Let's analyze the problem: When studying the workload of students, a group of 12 people was identified. They were asked to mark the time (in minutes) spent on a given day doing their algebra homework. We got the following data: 23, 18, 25, 20, 25, 25, 32, 37, 34, 26, 34, 25.
The largest time consumption is 37 minutes, and the smallest is 18 minutes. Find the range of the series:
37-18=19 minutes.
Solve the following tasks:
Task 1. The Ob River is an artery Western Siberia and carries its waters through a country like Russia. The length of the watercourse is 3650 km. The Ob River is the second among the rivers of Russia, second only to the Lena. Together with its tributary the Irtysh, the Ob is in first place in length in Russia (5410 km.) And in second place in Asia. (near the HPP), decreases to 8 m near the mouth of the Tom and increases again to 15 m in the upper reaches of the Gulf of Ob, where the river flows. Find the depth range of the Ob River.
Task 2. In the period from 17 to 19 December, the deviation of the average daily temperature from the norm in Khanty-Mansiysk Autonomous Okrug reached 16-26 degrees. And on December 21, the administration of the Beloyarsky district of the Khanty-Mansiysk Autonomous Okrug reported a cold snap to -62 ° C, in Khanty-Mansiysk - 40 °, in Surgut - 43 °, in Urai - 38 °, in Yugorsk - 42 °, in Kondinsk - 33 °. Find the temperature range of the data settlements.
Statistics studies the number of individual groups of the population of the country and its regions, the production and consumption of various types of products, the transportation of goods and passengers by various modes of transport, natural resources, etc. The results of statistical studies are widely used for practical and scientific conclusions.
The role of statistics in our life is so significant that people often without hesitation and without realizing, constantly use elements of statistical methodology not only in work processes, but also in everyday life. Working and relaxing, shopping, meeting other children, making some decisions, a person uses a certain system, the information he has, the prevailing tastes and habits, facts, systematizes, compares these facts, analyzes them, draws a conclusion and makes certain decisions. takes concrete action. Thus, in each person there are elements of statistical thinking, which is the ability to analyze and synthesize information about the world around. The results of statistical studies are widely used for practical and scientific conclusions.
| Appendix 4 Task 1. Interview 10 people from the class. Determine the most common among them hair and eye color. What statistic did you work with? Task 2. Interview 10 people from the class. Measure the width of their palms. Find the difference the largest and smallest values. What statistic is used in this task? Task 3. Interview 9 people from the class. Find out their shoe size. Line up the numbers in ascending order. Determine the median of the series. Task 4. Interview 10 people from the class. Find out their height. Find the average height respondents. What type of statistics did you work with? Appendix 5 Answers to tasks. |
||
| Average | ||
| Pike, perch, roach | ||
One of the main tasks of statistics is the proper processing of information. Of course, statistics have many other tasks: obtaining and storing information, making various forecasts, evaluating their reliability, etc. But none of these goals can be achieved without data processing. Therefore, it is first necessary to highlight the main characteristics of statistical data.
Excel spreadsheets have a huge set of tools for analyzing statistical data. The most commonly used statistical functions are built into the main core of the program, that is, these functions are available from the moment the program is launched. Other more specialized functions are included in an additional subroutine called an analysis package. The commands and functions of the analysis package are called Analysis Tools.
Consider the main characteristics of sample data.
Mean.
With the help of the average value, the sample (or general) average is calculated, that is, the arithmetic mean value of the sign of the sample (or general) population. Excel calculates the average as follows: =SUM(F4:F60)/COUNT(F4:F60). Also in Excel there is a function for calculating it: AVERAGE. The function argument is a set of numbers, usually specified as an interval of cells, for example: =AVERAGE(A3:A201).
Sample variance and sample standard deviation.
Sample variance of values random variable X is called the arithmetic mean of the squared deviations of the observed values of this quantity from their arithmetic mean:
Dispersion characterizes the deviation from the average in square units measurement of a trait, therefore, an indicator such as the standard deviation is used, which is measured in the same units as the trait under study.
The sample standard deviation is determined by the formula:
Excel has functions that separately calculate the sample variance Dv standard deviation in and general variance D G and standard deviation d. Therefore, before calculating the variance and standard deviation, you should clearly determine whether your data is a population or a sample. Depending on this, you need to use for the calculation D g and g, Dv and in.
Sample Variance Calculation Dv and sample standard deviation in done with the following functions: = SUM((4: 60 ? 28)^2)/ (COUNT(4: 60)) and = ROOT(29).
Excel has the VARP (or VAR) and STDEV (or STDEV) functions.
The argument of these functions is a set of numbers, usually given by a range of cells, for example, =VAR(B1:B48).
To calculate the general variance D r and the general standard deviation r have the VARP (or VARP) and STDEVP (or STDEVP) functions, respectively.
The arguments of these functions are the same as for the sample variance.
The volume of the population.
The volume of a sample or general population is the number of elements in the population. The COUNT (or COUNT) function determines the number of cells in a given range that contain numeric data. Empty cells or cells containing text are ignored by the COUNT function. The argument of the COUNT function is an interval of cells, for example: = COUNT (С2:С16).
To determine the number of non-empty cells, regardless of their contents, the COUNT3 function is used. Its argument is the range of cells.
Mode and median.
The mode (?) is the value of the feature that occurs more often than others in the data set. It is calculated by the MODE (or MODE) function. Its argument is the interval of cells with data. The mode is not calculated when examining the NE.
The median (?) is the value of the attribute, which divides the population into two parts equal in the number of elements. For a variation series with an odd number of members, the median is equal to the middle option, and for a series with an even number of members, it is half the sum of the two middle options. It is calculated by the MEDIAN (or MEDIAN) function. Its argument is the range of cells.
Range of variation. The largest and smallest values.
Range of variation R is the difference between the largest x max and the smallest xmin values of the sign of the population (general or sample): R=x max- x min.
For finding the greatest value x max there is a MAX (or MAX) function, and for the smallest x min is the MIN (or MIN) function. Their argument is the interval of cells. In order to calculate the range of data variation in the interval of cells, for example, from A1 to A100, enter the formula: =MAX (A1:A100)-MIN (A1:A100).
The coefficient of variation. Calculated as a percentage of the sample standard deviation to the arithmetic mean.
If the coefficient of variation is high (more than 35%), then the sample is considered heterogeneous. Therefore, the use of the average to characterize it is incorrect. In this case, the mode or median is used.
To assess the deviation of the distribution of experimental data from the normal distribution, such characteristics as asymmetry are used BUT and kurtosis E.
For a normal distribution BUT=0 and E=0.
Skewness shows how much the distribution of the data is asymmetrical with respect to the normal distribution: if BUT>0, then most of the data has values above the mean; if BUT<0, то большая часть данных имеет значения, меньшие среднего. Асимметрия вычисляется функцией СКОС. Ее аргументом является интервал ячеек с данными, например, =СКОС (А1:А100).
Kurtosis evaluates "coolness", i.e. the value of a greater or lesser rise in the maximum of the distribution of experimental data compared to the maximum of the normal distribution. If a E>0, then the maximum of the experimental distribution is higher than the normal one; if E<0, то максимум экспериментального распределения ниже нормального. Эксцесс вычисляется функцией ЭКСЦЕСС, аргументом которой являются числовые данные, заданные, как правило, в виде интервала ячеек, например: =ЭКСЦЕСС (А1:А100). [см. 5]
We get the following calculations (Figure 14).
Figure 14 Calculation of the main characteristics
We got the following values (Figure 15).
Figure 15 Values of the main characteristics
Since the value of the coefficient of variation significantly exceeds 35%, the sample is heterogeneous and the median is used as the average value.
Home > DocumentIntroduction. 2
The concept of statistics. 2
History of mathematical statistics. 3
The simplest statistical characteristics. 5
Statistical research. eight
1. ARITHMETIC AVERAGE 92. RANGE 103. MODE 104. MEDIAN 115. JOINT APPLICATION OF STATISTICAL CHARACTERISTICS 11
Perspectives and conclusion. eleven
Bibliography. 12
Introduction.
In October, at a break before the lesson, our mathematics teacher Marianna Rudolfovna checked independent work in 7th grade. Seeing what they were writing about, I did not understand a word, but I asked Marianna Rudolfovna what the words unfamiliar to me mean - range, mode, median, average. When I received the answer, I did not understand anything. At the end of the 2nd quarter, Marianna Rudolfovna invited someone from our class to make an essay on this very topic. I found this job very interesting, and I agreed. In the course of the work, such issues were considered- What is mathematical statistics? What is the meaning of statistics for the average person? Where is the acquired knowledge applied? Why can't a person do without mathematical statistics?
The concept of statistics.
STATISTICS is a science that deals with obtaining, processing and analyzing quantitative data on various phenomena occurring in nature and society. In the media, phrases such as accident statistics, population statistics, disease statistics, divorce statistics, etc. are often found. One of the main tasks of statistics is the proper processing of information. Of course, statistics have many other tasks: obtaining and storing information, making various forecasts, evaluating their reliability, etc. None of these goals can be achieved without data processing. Therefore, the first thing to do is statistical methods of information processing. There are many terms used in statistics for this. MATH STATISTICS - a branch of mathematics devoted to the methods and rules for processing and analyzing statistical dataHistory of mathematical statistics.
Mathematical statistics as a science begins with the works of the famous German mathematician Carl Friedrich Gauss (1777-1855), who, based on the theory of probability, investigated and substantiated the least squares method, which he created in 1795 and applied to process astronomical data (in order to clarify the orbit of a small planet Ceres). One of the most popular probability distributions, the normal one, is often named after him, and in the theory of random processes, the main object of study is Gaussian processes. AT
late XIX in. - the beginning of the twentieth century. a major contribution to mathematical statistics was made by English researchers, primarily K. Pearson (1857-1936) and R. A. Fisher (1890-1962). In particular, Pearson developed the chi-square test for testing statistical hypotheses, and Fisher developed analysis of variance, the theory of experiment design, and the maximum likelihood method for estimating parameters. AT 
In the 1930s, the Pole Jerzy Neumann (1894-1977) and the Englishman E. Pearson developed general theory testing statistical hypotheses, 
and Soviet mathematicians Academician A.N. Kolmogorov (1903-1987) and Corresponding Member of the USSR Academy of Sciences N.V. Smirnov (1900-1966) laid the foundations of nonparametric statistics.
In the forties of the twentieth century. Romanian mathematician A. Wald (1902-1950) built the theory of sequential statistical analysis. Mathematical statistics is rapidly developing at the present time.
The simplest statistical characteristics.
In everyday life, we, without knowing it, use such concepts as the median, mode, range and arithmetic mean. Even when we go to the store or do the cleaning. The arithmetic mean of a series of numbers is called the quotient of dividing the sum of these numbers by their number. The arithmetic mean is an important characteristic of a series of numbers, but it is sometimes useful to consider others. medium. Fashion name the number of the row that occurs most often in this row. We can say that this number is the most "fashionable" in this series. An indicator such as mode is used not only for numerical data. If, for example, a large group of students are asked which school subject they like best, then the fashion of this series of answers will be the subject that will be called most often. Mode is an indicator that is widely used in statistics. One of the most common uses of fashion is to study demand. For example, when deciding what weight packs to pack butter in, which flights to open, etc., demand is preliminarily studied and fashion is identified - the most common order. Note that in the series considered in real statistical studies, sometimes more than one mode is distinguished. When there is a lot of data in a series, all those values that occur much more often than others are interesting. Their statistics are also called fashion. However, finding the arithmetic mean or mode does not always make it possible to draw reliable conclusions based on statistical data. If there is a series of data, then, in addition to the average values, it is also necessary to indicate how the data used differ from each other. One of the statistical indicators of the difference or scatter of data is the range. scope is the difference between the largest and the smallest values a series of data. Another important statistical characteristic of a data series is its median. Usually, the median is looked for when the numbers in the series are some indicators and you need to find, for example, a person who showed an average result, a company with an average annual profit, an airline offering average ticket prices, etc. Median a series consisting of an odd number of numbers is called the number of this series, which will be in the middle if this series is ordered. The median of a series consisting of an even number of numbers is the arithmetic mean of the two numbers in the middle of this series. For example: 1. EPT for 4th grade is held every year in Perm schools and in 2010 the following average scores were obtained:| № schools | Maths | Russian language |
| Gymnasium No. 4 | 68.5 b. | 62.4 b. |
| № 55 | 53.1 b | 52.7 b. |
| № 111 | 46.9 b | 51.6 b. |
| № 40 | 48.4 b | 51.9 b. |
- My mother works at the Perm powder factory as an accountant. The salary of the employees of this enterprise ranges from 12,000 to 18,000. the difference is 6000. This is called the span. Several years ago, my parents and I were vacationing in the south in Anapa. I noticed that number 23 is most often found on car numbers - the number of the region. It's called fashion. I spent such time on homework during the week - 60 minutes on Monday, 103 minutes on Tuesday, 58 minutes on Wednesday, 76 minutes on Thursday, and 89 minutes on Friday. Having written these numbers from smallest to largest, the number 76 stands in the middle - this is called the median.
Statistical research.
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Statistics knows everything- Ilf and Petrov stated in their famous novel “The Twelve Chairs” and continued: “It is known how much food the average citizen of the republic eats per year ... It is known how many hunters, ballerinas ... machine tools, bicycles, monuments, lighthouses and sewing machines... How much life, full of ardor, passions and thoughts, looks at us from statistical tables! (from Italian stato - state, Latin status - state). 1. ARITHMETIC AVERAGE
I calculated the average electricity costs for our family during 2010:| Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Consumption, kW/h | 189 | 155 | 106 | 102 | 112 | 138 | 106 | 112 | 156 | 149 | 160 | 155 |
2. SPIN
The height of the girls in our class is very different: 151 cm, 160 cm, 163 cm, 162 cm, 145 cm, 130 cm, 131 cm, 161 cm The span is 163 - 130 \u003d 33 cm. The span determines the difference in height. When is scope needed and not needed? The range of a series is found when they want to determine how large the spread of data in a series is. For example, during the day, the air temperature in the city was recorded every hour. For the obtained series of data, it is useful not only to calculate the arithmetic mean, which shows what the average daily temperature is, but also to find the range of the series, which characterizes the fluctuation in air temperature during this day. For the temperature on Mercury, for example, the range is 350 + 150 = 500 C. Of course, a person cannot withstand such a temperature difference.3. FASHION
I wrote out my marks for December in mathematics: 4,5,5,4,4,4,4,5,5,4,5,5,4,5,5,5,5,5,5. It turned out that I got: "5" - 7, "4" - 5, "3" - 0, "2" - 0 Fashion is 5. But there is more than one fashion, for example, in natural history in October I had such marks - 4,4,5,4,4,3,5,5,5. There are two mods - 4 and 5 When is fashion needed? Fashion is important for manufacturers in determining the most popular clothing size, shoe size, juice bottle size, bag of chips, popular clothing style.4. MEDIAN
When analyzing the results shown by the participants in the 100-meter class race, knowledge of the median allows the physical education teacher to select a group of children who showed a result above the median for participation in the competition. When is the median needed and not needed? The median is more often used with other statistical characteristics, but it alone can be used to select results above or below the median.5. JOINT APPLICATION OF STATISTICAL CHARACTERISTICS
In our class for the last verification work in mathematics on the topic "Measurement of angles and their types" the following marks were obtained: "5" - 10, "4" - 5, "3" - 7, "2" - 1. Arithmetic mean - 4.3, range - 3, mode - 5, median - 4.Perspectives and conclusion.
Statistical characteristics allow you to study number series. Only together they can give an objective assessment of the situation. It is impossible to properly organize our life without knowing the laws of mathematics. It allows you to study, learn, correct. Statistics creates the foundation of accurate and indisputable facts, which is necessary for theoretical and practical purposes. Mathematicians invented statistics because society needed it. I think that the knowledge gained while working on this topic will be useful to me in my further studies and in life. While studying the literature, I learned that there are other characteristics such as standard deviation, variance, and others. However, my knowledge is not enough to understand them. About them in the future.Bibliography.
- Tutorial for students in grades 7-9 educational institutions"Algebra. Elements of statistics and probability theory. Yu.N.Makarychev, N.G.Mindyuk, edited by S.A.Telyakovsky; Moscow. Education. 2005 Articles from the supplement to the newspaper “The First of September. Maths". Encyclopedic Dictionary of a Young Mathematician /
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Educational and methodological complex for specialties 080504 State and municipal administration 080507 Management of organizations
Training and metodology complexGuidelines 6 Social sciences 21 00 Social sciences in general 21 02 Philosophy 21
GuidelinesThe State Rubricator of Scientific and Technical Information (SRSTI Rubricator) is a universal hierarchical classification of areas of knowledge adopted to systematize the entire flow of scientific and technical information.
Educational-methodical complex legal statistics higher professional education specialty 030501. 65 Jurisprudence field of study (bachelor)
Training and metodology complexThe study of statistical science plays an important role in the preparation of highly qualified lawyers - both practitioners and researchers. Specialist in the field social sciences, legal in particular, must master the basic issues
