Slepnev Pavel

In the 7th grade algebra course, the textbook edited by Telyakovsky offers material from statistics “The Arithmetic Mean, Range and Mode.” The student in his work offers examples for considering this topic that his classmates suggested.

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MU Department of Education MO "Tarbagatai district"

MBOU "Zavodskaya OOSH"

"Arithmetic mean, range and mode"

Completed by: Slepnev Pavel, 7th grade student

Scientific adviser:

Ulakhanova Marina Rodionovna,

mathematic teacher

year 2012

Introduction Page 3

Main part Page 4-9

Theory of the issue pp. 4-6

Mini-projects pp. 7-9

Conclusion Page 9

References Page 10

Introduction

Relevance

In that academic year We started studying two subjects: algebra and geometry. When studying algebra, some things are familiar to me from the 5th and 6th grade courses, some we study more thoroughly and in depth, we learn a lot of new things. What’s new to me when studying algebra is an acquaintance with some statistical characteristics: range and mode. We have already met with the arithmetic mean earlier. What also turned out to be interesting is that these characteristics are used not only in mathematics lessons, but also in life, in practice (in production, in agriculture, in sports, etc.).

Formulation of the problem

When we were solving problems for this point in class, the idea arose to create the problems ourselves and prepare presentations for them, that is, to sort of start creating our own problem book. Everyone comes up with a problem, makes a presentation for it, as if everyone is working on their own mini-project, and in class we solve everything together and discuss it. If mistakes are made, we correct them. And at the end carry out public defense these mini-projects.

The purpose of my work: study statistics.

Objectives: begin developing a statistics problem book in the form of computer presentations.

Subject of research: statistics.

Object of study: statistical characteristics ( average, scope, fashion).

Research methods:

1. Studying literature on this topic.
2. Data analysis.
3. Use of Internet resources.
4. Using Power Point.
5. Summarizing the collected materials on this topic.

Main part.

Theory of the issue

While studying the section “Statistical characteristics” we became acquainted with the following concepts: arithmetic mean, range, mode. These characteristics are used in statistics. This science studies numbers separate groups population of the country and its regions, production and consumption of various types of products, transportation of goods and passengers various types transport, Natural resources and so on.

“Statistics knows everything,” Ilf and Petrov asserted 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, machines, bicycles, monuments there are in the country, lighthouses and sewing machines... How much life, full of ardor, passions and thoughts, looks at us from statistical tables!..” This ironic description gives a fairly accurate idea of ​​statistics (from the Latin status - state) - the science that studies, processes and analyzing quantitative data on a wide variety of mass phenomena in life.

Economic statistics studies changes in prices, supply and demand for goods, predicts the growth and decline of production and consumption.

Medical statistics studies the effectiveness of various drugs and treatment methods, the likelihood of a certain disease depending on age, gender, heredity, living conditions, bad habits, predicts the spread of epidemics.

Demographic statistics studies the birth rate, population size, and its composition (age, national, professional).

There are also financial, tax, biological, and meteorological statistics.

IN school course algebra we look at concepts and methods descriptive statistics, which is engaged in primary processing of information and calculation of the most indicative numerical characteristics. According to the English statistician R. Fisher: “Statistics can be characterized as the science of reducing and analyzing material obtained from observations.” The entire set of numerical data obtained in the sample can (conditionally) be replaced by several numerical parameters, some of which we have already considered in lessons - arithmetic mean, range, mode. results statistical research are widely used for practical and scientific inferences, so it is important to be able to determine these statistical characteristics.

Statistical characteristics are found everywhere these days. For example, the population census. Thanks to this census, the state will know how much money is needed for the construction of housing, schools, hospitals, how many people need housing, how many children there are in the family, the number of unemployed, salary levels, etc. The results of this census will be compared with the last one, they will see whether the country has improved during this time or the situation has become worse, it will be possible to compare the data with the results in other countries. In industry great importance has fashion. For example, a product that is in great demand will always be sold, and factories will have a lot of money. And there are many such examples.

The results of statistical studies are widely used for practical and scientific conclusions.

Definition 1. The arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by the number of terms.

Example: When studying the workload, a group of 12 7th grade students was identified. They were asked to note on a certain day the time (in minutes) spent on completing homework in algebra. We received the following data:

23, 18, 25, 20, 25, 25, 32, 37, 34, 26, 34, 25. With this series of data, you can determine how many minutes, on average, students spent on algebra homework. To do this, you need to add the indicated 12 numbers and divide the resulting sum

at 12: ==27.

The resulting number 27 is called the arithmetic mean of the series of numbers under consideration.

The arithmetic mean is important characteristic number of numbers, but sometimes it is useful to consider others average.

Definition 2. The mode of a series of numbers is a number that occurs in this series more often than others.

Example: When analyzing information about the time spent by students on algebra homework, we may be interested not only in the arithmetic mean and the range of the obtained data series, but also in other indicators. For example, it is interesting to know what time consumption is typical for a selected group of students, i.e. which number occurs most often in the data series. It is easy to see that in our example this number is 25. They say that the number 25 is the mode of the series under consideration.

A series of numbers may have more than one mode, or may not have a mode at all. For example, in the series of numbers 47, 46, 50, 47, 52, 49, 45, 43, 54, 52, 47, 52, two modes are the numbers 47 and 52, since each of them occurs three times in the series, and other numbers – less than three times.

There is no mode in the number series 69, 68, 66, 70, 67, 62, 71, 74, 63, 73, 72.

The mode of a data series is usually found when one wants to identify some typical indicator. Mode is an indicator that is widely used in statistics. One of the most frequent use fashion is the study of demand. For example, when deciding what weight packs to pack butter in, what flights to open, etc., demand is first studied and fashion is identified - the most common order.

However, finding the arithmetic mean or mode does not always allow one to draw reliable conclusions based on statistical data. If we have a series of data, then in order to make valid conclusions and reliable forecasts based on them, in addition to the average values, we must also indicate how much the data used differ from each other. One of statistical indicators the difference or spread of data is the range.

Definition 3. The range of a series of numbers is the difference between the largest and smallest of these numbers.

Example: In the example above, we found that, on average, students spent 27 minutes on algebra homework. However, analysis of the data series shows that the time spent by some students differs significantly from 27 minutes, i.e. from the arithmetic mean. The highest consumption is 37 minutes, and the lowest is 18 minutes. The difference between the largest and least expense time is 19 minutes. In this case, another statistical characteristic is considered - scope. The range of a series is found when one wants to determine how large the spread of data in a series is.

Mini projects

And now I would like to present the results of our work: mini-projects for creating a statistics problem book.

I work at the Super-auto showroom as the chief manager of the sales department. Our salon provided cars to participate in the all-wheel drive game. Last year at the exhibition and sale our cars were a success! The sales results are as follows:

The sales department needs to summarize the results of the exhibition:

1. How many cars were sold per day on average?
2. What is the spread in the number of cars during the exhibition and sale period?
3. How many cars were most often sold per day?

Answer: on average, 150 cars were sold per day, the range of the number of cars sold was 150, most often 100 cars were sold per day.

I, Anastasia Volochkova, was invited to the jury for the final of the Ice and Fire competition. The competition took place in the city of St. Petersburg. Three pairs of the strongest skaters reached the finals: 1 pair. Batueva Alina and Khlebodarov Kirill, 2nd couple. Selyanskaya Yulia and Kushnarev Pavel, 3 pair. Zaigraeva Anastasia and Afanasyev Dmitry. Jury: Anastasia Volochkova, Elena Malysheva, Alexey Dalmatov. The jury gave the following scores:

Find the arithmetic mean, range and mode in the series of estimates for each pair.

Answer:

This year I visited St. Petersburg for a ballroom dancing competition. Three beautiful couples took part in the competition: Elena Sushentsova and Kirill Khlebodarov, Alina Batueva and Pavel Slepnev, Victoria Dzhaniashvili and Valery Tkachev.

The couples received the following scores for their performances:

Find average rating, scope and fashion.

Answer:

I am the director of the fashion clothing and accessories store “Fashion”. The store makes a good profit. Sales figures for last year:

For the first 2-3 months, the profit reached 2 million per month. Afterwards, the profit increased to 4 million. The most successful months were: December and May. In May, we mainly bought dresses for proms, and in December for New Year's celebrations.

Question to my chief accountant: what are the results of our work for the year?

Answer:

We organized a tuning workshop “Turbo”. During the first week of our work, we earned: on the first day - $120,000, on the second day - $350,000, on the third day - $99,000, on the fourth day - $120,000. Calculate what is our average income per day, what is the gap between the highest and lowest earnings and what amount is repeated most often?

Answer: arithmetic mean – $172,250, range – $251,000, mode – $120,000.

Conclusion

In conclusion, I want to say that I love this topic. Statistical characteristics are very convenient and can be used everywhere. In general, they compare, strive for progress and help to find out the opinion of the people. In the course of working on this topic, I became acquainted with the science of statistics, learned some concepts (arithmetic mean, range and mode) where this science can be applied, and expanded my knowledge in computer science. I think that our problems as examples for mastering these concepts will be useful to others! We will continue to get acquainted with this science and create our own problems!

So my journey into the world of mathematics, computer science and statistics has ended. But I think it's not the last. There is still a lot I want to know! As Galileo Galilei said: “Nature formulates its laws in the language of mathematics.” And I want to master this language!

Bibliography

1. Bunimovich E.A., Bulychev V.A. « Probability and statistics in mathematics course secondary school", M.: Pedagogical University“First of September”, 2005
2. Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. “Algebra, 7th grade”, M: “Prosveshcheniye”, 2009
3. Makarychev Yu.N., Mindyuk N.G. « Algebra. Elements of statistics and probability theory”, grades 7 – 9. – M.: Education, 2005.

Review

The subject of the student's research is statistics.

The object of the study is statistical characteristics (arithmetic mean, range, mode).

To become familiar with the theory of the issue, the student studied scientific sources, Internet resources.

The chosen topic is relevant for students who show interest in mathematics, computer science, and statistics. For his age, sufficient material was analyzed, data was selected, and generalized. The student has sufficient knowledge of ICT.

The work is completed in accordance with the requirements.

At the end of the study, a conclusion is drawn and a practical product is presented: presentations of problems in statistics. I am glad that a person is so passionate about mathematics.

Scientific supervisor: Ulakhanova MR,

mathematic teacher

Solving problems on the topic: “Statistical characteristics. Arithmetic mean, range, mode and median

Algebra-

7th grade

Historical information

• Arithmetic mean, range and mode are used in statistics - a science that deals with obtaining, processing and analyzing quantitative data about a variety of mass phenomena occurring in nature and society.
• The word "statistics" comes from Latin word status, which means “state, state of affairs.” Statistics studies the size of individual population groups of the country and its regions, production and consumption
• various types of products, 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.

Average– the quotient of dividing the sum of all numbers by the number of terms

• Scope– the difference between the largest and smallest number of this series
• Fashion is the number that occurs most often in a set of numbers
• Median– of an ordered series of numbers with an odd number of terms is the number written in the middle, and the median of an ordered series of numbers with an even number of terms is the arithmetic mean of the two numbers written in the middle. The median of an arbitrary series of numbers is the median of the corresponding ordered series.

• Average ,

• scope and fashion
• are used in statistics - science,
• which is engaged in receiving,

processing and analysis

quantitative data on various

• mass phenomena occurring

in nature and

• Society.

Task No. 1

• Series of numbers:
• 18 ; 13; 20; 40; 35.
• Find the arithmetic mean of this series:

• Solution:
• (18+13+20+40+35):5=25,5
• Answer: 25.5 – arithmetic mean

Problem No. 2

• Series of numbers:
• 35;16;28;5;79;54.

• Find the range of the series:

• Solution:

• The largest number is 79,
• The smallest number is 5.
• Row range: 79 – 5 = 74.
• Answer: 74

Problem No. 3

• Series of numbers:
• 23; 18; 25; 20; 25; 25; 32; 37; 34; 26; 34; 2535;16;28;5;79;54.

• Find the range of the series:

• Solution:
• The greatest time consumption is 37 minutes,
• and the smallest is 18 minutes.
• Let's find the range of the series:
• 37 – 18 = 19 (min)

Problem No. 4

• Series of numbers:
• 65; 12; 48; 36; 7; 12
• Find the mode of the series:

• Solution:

• Fashion of this series: 12.
• Answer: 12

Problem No. 5

• A series of numbers can have more than one mode,
• or maybe not.

• Row: 47, 46, 50, 47, 52, 49, 45, 43, 53, 47, 52
• two modes - 47 and 52.

• The row: 69, 68, 66, 70, 67, 71, 74, 63, 73, 72 has no fashion.

Problem No. 5

• Series of numbers:
• 28; 17; 51; 13; 39
• Find the median of this series:

• Solution:

• First put the numbers in ascending order:
• 13; 17; 28; 39; 51.
• Median – 28.
• Answer: 28

Problem No. 6

The organization kept daily records of letters received during the month.

As a result, we received the following series of data:

39, 42, 40, 0, 56, 36, 24, 21, 35, 0, 58, 31, 49, 38, 24, 35, 0, 52, 40, 42, 40,

39, 54, 0, 64, 44, 50, 37, 32, 38.

For the resulting series of data, find the arithmetic mean,

What is the practical meaning of these indications?

Problem No. 7

The cost (in rubles) of a pack of Nezhenka butter in the neighborhood stores is recorded: 26, 32, 31, 33, 24, 27, 37.

How much does the arithmetic mean of this set of numbers differ from its median?

Solution.

Let's sort this set of numbers in ascending order:

24, 26, 27, 31, 32, 33, 37.

Since the number of elements of the series is odd, the median is

value in the middle number series, that is, M = 31.

Let's calculate the arithmetic mean of this set of numbers - m.

m= 24+ 26+ 27+ 31+ 32+ 33+ 37 = 210 ═ 30

M – m = 31 – 30 = 1

Creative

Average arithmetic series numbers – This is the sum of these numbers divided by the number of terms.

The arithmetic mean is called the average value of a number series.

Example: Find the average arithmetic numbers 2, 6, 9, 15.

Solution. We have four numbers. This means that their sum must be divided by 4. This will be the arithmetic mean of these numbers:
(2 + 6 + 9 + 15) : 4 = 8.

Average geometric series numbers- this is the root nth degree from the product of these numbers.

Example: Find the average geometric numbers 2, 4, 8.

Solution. We have three numbers. This means we need to find the third root of their product. This will be the geometric mean of these numbers:

3 √ 2 4 8 = 3 √64 = 4

Scope series of numbers is the difference between the largest and smallest of these numbers.

Example: Find the range of numbers 2, 5, 8, 12, 33.

Solution: The largest number here is 33, the smallest is 2. So the range is 31:

Fashion series of numbers is the number that appears in a given series more often than others.

Example: Find the mode of the series of numbers 1, 7, 3, 8, 7, 12, 22, 7, 11, 22, 8.

Solution: The number 7 appears most often in this series of numbers (3 times). It is the mode of a given series of numbers.

Median.

In an ordered series of numbers:

Median of an odd number of numbers is the number written in the middle.

Example: In a series of numbers 2, 5, 9, 15, 21, the median is the number 9, located in the middle.

Median of an even number of numbers is the arithmetic mean of the two numbers in the middle.

Example: Find the median of the numbers 4, 5, 7, 11, 13, 19.

Solution: There is an even number of numbers (6). Therefore, we are looking for not one, but two numbers written in the middle. These are numbers 7 and 11. Find the arithmetic mean of these numbers:

(7 + 11) : 2 = 9.

The number 9 is the median of this series of numbers.

In an unordered series of numbers:

Median of an arbitrary series of numbers is called the median of the corresponding ordered series.

Example 1: Find the median of an arbitrary series of numbers 5, 1, 3, 25, 19, 17, 21.

Solution: Arrange the numbers in ascending order:

1, 3, 5, 17 , 19, 21, 25.

In the middle is the number 17. It is the median of this series of numbers.

Example 2: Let's add to our arbitrary row numbers one more number so that the series becomes even, and we find the median:

5, 1, 3, 25, 19, 17, 21, 19.

Solution: We build an ordered series again:

1, 3, 5, 17 , 19 , 19, 21, 25.

The numbers 17 and 19 were in the middle. Find their average value:

(17 + 19) : 2 = 18.

The number 18 is the median of this series of numbers.

First level

Statistics. Basic concepts and definitions (2019)

Lyudmila Prokofievna Kalugina (or simply “Mymra”) in the wonderful film “ Love affair at work“Novoseltseva taught: “Statistics is a science, it does not tolerate approximation.” In order not to fall under the hot hand of the strict boss Kalugina (and at the same time easily solve tasks from the Unified State Examination and State Examination with elements of statistics), we will try to understand some concepts of statistics that can be useful not only in thorny path conquering the exam on the Unified State Examination, but also simply in Everyday life.

So what is Statistics and why is it needed? The word “statistics” comes from the Latin word “status”, which means “state and state of affairs”. Statistics deals with the study of the quantitative side of mass social phenomena and processes in numerical form, identifying special patterns. Today statistics is used in almost all areas public life, ranging from fashion, cooking, gardening to astronomy, economics, medicine.

First of all, when getting acquainted with statistics, it is necessary to study the basic statistical characteristics used for data analysis. Well, let's start with this!

Statistical characteristics

To the main statistical characteristics data samples (what kind of “sampling” is this!? Don’t be alarmed, everything is under control, it’s unknown word just for intimidation, in fact, the word “sample” simply means the data that you are going to study) include:

1. sample size,
2. sample range,
3. average,
4. fashion,
5. median,
6. frequency,
7. relative frequency.

Stop, stop, stop! How many new words! Let's talk about everything in order.

Volume and Scope

For example, the table below shows the height of the players of the national football team:

This selection is represented by elements. Thus, the sample size is equal.

The range of the presented sample is cm.

Average

Not very clear? Let's look at our example.

Determine the average height of the players.

Well, shall we get started? We have already figured out that; .

We can immediately safely substitute everything into our formula:

Thus, the average height of a national team player is cm.

Or like this example:

For a week, 9th grade students were asked to decide how to more examples from the problem book. The number of examples solved by students per week is given below:

Find the average number of problems solved.

So, in the table we are presented with data on students. Thus, . Well, let's first find the amount ( total) of all solved problems by twenty students:

Now we can safely begin to calculate the arithmetic mean of the solved problems, knowing that:

Thus, on average, 9th grade students solved each problem.

Here's another example to reinforce.

Example.

On the market, tomatoes are sold by sellers, and prices per kg are distributed as follows (in rubles): . What is the average price of a kilogram of tomatoes on the market?

Solution.

So, what's in in this example equals? That's right: seven sellers offer seven prices, which means ! . Well, we’ve sorted out all the components, now we can start calculating the average price:

Well, did you figure it out? Then do the math yourself average in the following samples:

Answers: .

Mode and median

Let's look again at our example with the national football team:

What is the mode in this example? What is the most common number in this sample? That's right, this is a number, since two players are cm tall; the growth of the remaining players is not repeated. Everything here should be clear and understandable, and the word should be familiar, right?

Let's move on to the median, you should know it from your geometry course. But it’s not difficult for me to remind you that in geometry median(translated from Latin as “middle”) - a segment inside a triangle connecting the vertex of the triangle with the middle opposite side. Keyword MIDDLE. If you knew this definition, then it will be easy for you to remember what a median is in statistics.

Well, let's get back to our sample of football players?

Did you notice in the definition of median important point, which we haven’t met here yet? Of course, “if this series is ordered”! Shall we put things in order? In order for there to be order in the series of numbers, you can arrange the height values ​​of football players in both descending and ascending order. It is more convenient for me to arrange this series in ascending order (from smallest to largest). Here's what I got:

So, the series has been sorted, what other important point is there in determining the median? That's right, an even and an odd number of members in the sample. Have you noticed that even the definitions are different for even and odd quantities? Yes, you're right, it's hard not to notice. And if so, then we need to decide whether we have an even number of players in our sample or an odd one? That's right - there are an odd number of players! Now we can apply to our sample a less tricky definition of the median for an odd number of members in the sample. We are looking for the number that is in the middle in our ordered series:

Well, we have numbers, which means there are five numbers left at the edges, and height cm will be the median in our sample. Not so difficult, right?

Now let’s look at an example with our desperate children from grade 9, who solved examples during the week:

Are you ready to look for mode and median in this series?

To begin with, let's order this series of numbers (arrange from the smallest number to the largest). The result is a series like this:

Now we can safely determine the fashion in this sample. Which number occurs more often than others? That's right! Thus, fashion in this sample is equal.

We have found the mode, now we can start finding the median. But first, answer me: what is the sample size in question? Did you count? That's right, the sample size is equal. A is even number. Thus, we apply the definition of median for a series of numbers with an even number of elements. That is, we need to find in our ordered series average two numbers written in the middle. What two numbers are in the middle? That's right, and!

Thus, the median of this series will be average numbers and:

- median the sample under consideration.

Frequency and relative frequency

That is frequency determines how often a particular value is repeated in a sample.

Let's look at our example with football players. We have before us this ordered series:

Frequency is the number of repetitions of any parameter value. In our case, it can be considered like this. How many players are tall? That's right, one player. Thus, the frequency of meeting a player with height in our sample is equal. How many players are tall? Yes, again one player. The frequency of meeting a player with height in our sample is equal. By asking and answering these questions, you can create a table like this:

Well, everything is quite simple. Remember that the sum of the frequencies must equal the number of elements in the sample (sample size). That is, in our example:

Let's move on to the following characteristic- relative frequency.

Let us turn again to our example with football players. We have calculated the frequencies for each value; we also know the total amount of data in the series. We calculate the relative frequency for each growth value and get this table:

Now create tables of frequencies and relative frequencies yourself for an example with 9th graders solving problems.

Graphical representation of data

Very often, for clarity, data is presented in the form of charts/graphs. Let's look at the main ones:

1. bar chart,
2. pie chart,
3. bar chart,
4. polygon

Column chart

Column charts are used when they want to show the dynamics of changes in data over time or the distribution of data obtained as a result of a statistical study.

For example, we have the following data on the grades of a written test in one class:

The number of people who received such an assessment is what we have frequency. Knowing this, we can make a table like this:

Now we can build visual bar graphs based on such an indicator as frequency(on horizontal axis estimates reflected on vertical axis we set aside the number of students who received the appropriate grades):

Or we can construct a corresponding bar graph based on the relative frequency:

Let's consider an example of the type of task B3 from the Unified State Examination.

Example.

The diagram shows the distribution of oil production in countries around the world (in tons) for 2011. Among countries, the first place in oil production was occupied by Saudi Arabia, seventh place - United United Arab Emirates. Where did the USA rank?

Answer: third.

Pie chart

To visually depict the relationship between parts of the sample under study, it is convenient to use pie charts.

Using our table with the relative frequencies of the distribution of grades in the class, we can construct a pie chart by dividing the circle into sectors proportional to the relative frequencies.

A pie chart retains its clarity and expressiveness only with a small number of parts of the population. In our case, there are four such parts (in accordance with possible estimates), so the use of this type of diagram is quite effective.

Let's look at an example of the type of task 18 from the State Examination Inspectorate.

Example.

The diagram shows the distribution of family expenses during a seaside holiday. Determine what the family spent the most on?

Answer: accommodation.

Polygon

The dynamics of changes in statistical data over time are often depicted using a polygon. To construct a polygon, mark in coordinate plane points, the abscissas of which are moments in time, and the ordinates are the corresponding statistical data. By connecting these points successively with segments, a broken line is obtained, which is called a polygon.

Here, for example, we are given the average monthly air temperatures in Moscow.

Let's make the given data more visual - we'll build a polygon.

The horizontal axis shows the months, and the vertical axis shows the temperature. We build the corresponding points and connect them. Here's what happened:

Agree, it immediately became clearer!

A polygon is also used to visually depict the distribution of data obtained as a result of a statistical study.

Here is the constructed polygon based on our example with the distribution of scores:

Let's consider typical task B3 from the Unified State Exam.

Example.

In the figure, bold dots show the price of aluminum at the close of exchange trading on all working days from August to August of the year. The dates of the month are indicated horizontally, and the price of a ton of aluminum in US dollars is indicated vertically. For clarity, the bold points in the figure are connected by a line. Determine from the figure what date the aluminum price at the close of trading was the lowest for the given period.

Answer: .

bar chart

Interval data series are depicted using a histogram. A histogram is a stepped figure 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 frequency or relative frequency. Thus, in a histogram, unlike a regular bar chart, the bases of the rectangle are not chosen arbitrarily, but are strictly determined by the length of the interval.

For example, we have the following data on the growth of players called up to the national team:

So we are given frequency(number of players with corresponding height). We can complete the table by calculating the relative frequency:

Well, now we can build histograms. First, let's build based on frequency. Here's what happened:

And now, based on the relative frequency data:

Example.

To the exhibition innovative technologies Representatives of the companies arrived. The chart shows the distribution of these companies by number of employees. The horizontal line represents the number of employees in the company, the vertical line shows the number of companies with given number employees.

What percentage are companies with a total number of employees of more than one person?

Answer: .

Brief summary

Sample size- the number of elements in the sample.

Sample range- difference between maximum and minimum values sample elements.

Arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by their number (sample size).

Mode of number series- the number most often found in a given series.

Medianordered series of numbers with an odd number of terms- the number that will be in the middle.

Median of an ordered series of numbers with an even number of terms- the arithmetic mean of two numbers written in the middle.

Frequency- the number of repetitions of a certain parameter value in the sample.

Relative frequency

For clarity, it is convenient to present data in the form of appropriate charts/graphs

• ELEMENTS OF STATISTICS. BRIEFLY ABOUT THE MAIN THINGS.

Statistical sampling - a specific number of objects selected from the total number of objects for research.

Sample size is the number of elements included in the sample.

Sample range is the difference between the maximum and minimum values ​​of sample elements.

Or, sample range

Average of a series of numbers is the quotient of dividing the sum of these numbers by their number

The mode of a series of numbers is the number that appears most frequently in a given series.

The median of a series of numbers with an even number of terms is the arithmetic mean of the two numbers written in the middle, if this series is ordered.

Frequency represents the number of repetitions, how many times over a period of time some event occurred, manifested itself specific property object or observed parameter reached this value.

Relative frequency is the ratio of frequency to the total number of data in the series.

Objectives: to give concepts, algorithms for finding the arithmetic mean and median, range and mode of a number of numbers, to show the significance of this topic in practical human activity; acquiring practical skills to perform these tasks; increasing the level of mathematical training required by the new standards.

• equip students with a system of knowledge on the topic “Determination of the probability of events, the arithmetic mean and median of a set of numbers”;
• develop skills in applying this knowledge when solving a variety of problems of varying complexity;
• prepare students for passing the State Examination Test;
• develop independent work skills.

During the classes

1. Theoretical part.

1). Finding the probability of events.

In everyday life, in practical and scientific activities, certain phenomena are often observed and certain experiments are carried out.

In the process of observation or experiment one encounters some random events, i.e., such events that may or may not happen. For example, getting heads or tails when tossing a coin, hitting a target or missing a shot, winning a sports team in a meeting with an opponent, losing or a draw - all these are random events.

Patterns random events studies a special branch of mathematics called probability theory. Probability theory methods are used in many fields of knowledge.

The origin of probability theory occurred in search of an answer to the question: how often does this or that event occur in a large series of tests occurring under the same conditions with random outcomes.

In order to assess the probability of an event of interest to us, it is necessary to conduct a large number of experiments or observations, and only after that can the probability of this event be determined.

For example, throwing a die. When throwing a die, the chances of each number from 1 to 6 appearing on its top face are the same. They say there are 6 equally possible outcomes experience with dice rolling: roll 1,2,3,4,5, and 6 points.

Outcomes in this experiment are considered equally possible if the chances of these outcomes are equal.

Outcomes in which some event occurs are called favorable outcomes for that event.

Definition: the ratio of the number of favorable outcomes N (A) of event A to the number of all equally possible outcomes N of this event is called the probability of event A.

Scheme for finding the probability of an event.

To find the probability of a random event A during a certain test, you should:

• find the number N of all equally possible outcomes of a given test;
• find the number N(A) of those favorable trial outcomes in which event A occurs;
• find the ratio N(A)/N; this is the probability of event A

For example: 1 . A box contains 10 red, 7 yellow and 3 blue balls. What is the probability that a ball taken at random will be yellow?

Solution. Equally possible outcomes - (10+7+3)=20

Favorable outcomes-7

2. There are 5 black balls in the box. What is the smallest number of white balls that must be placed in this box so that the probability of drawing a black ball out of the box at random is no more than 0.15?

Solution: Let x be white balls.

2) Determining and finding the arithmetic mean and median of a series of numbers.

Definition: the arithmetic mean of several numbers is a number equal to the ratio of the sum of these numbers to their number.

The arithmetic mean of a set of numbers x 1 , x 2 , x 3 , x 4 , x 5 is usually denoted as x.

For example, the arithmetic mean of five numbers will be written like this:

X = (x 1 +x 2 +x 3 +x 4 +x 5)/5

Example: find the student’s average grade in mathematics if over the past period he received: 3,4,4,5,3,2,4,3.

Solution: (3+4+4+5+3+2+4+3)/8=3.5

Definition: a median is a number that divides a set of numbers into two parts of equal numbers, so that on one side of this number all values ​​are greater than the median, and on the other less. Instead of "median" you could say "middle".

Scheme for finding the median of a set of numbers:

To find the median of a set of numbers:

• arrange a number set (write in ascending order);
• simultaneously cross out the “largest” and “smallest” numbers of a given set of numbers until one number or two numbers remain;
• if one number remains, then it is the median (for an odd set of numbers);
• if there are two numbers left, then the median will be the arithmetic mean of the two remaining numbers (for an even set of numbers).

The median is usually denoted by the letter M.

Example: find the median of a set of numbers: 9,3,1,5,7.

Solution: write the numbers in ascending order: 1,3,5,7,9.

Cross out 1 and 9, 3 and 7. The remaining number 5 is the median. M=5

Example: find the median of a set of numbers 2,3,3,5,7,10.

Solution: cross out 2 and 10, 3 and 7. To find M you need: (3+5)/2= 4. M=4

Determining and finding scope and mode.

Definition: The range of a series of numbers is the difference between the largest and smallest of these numbers.

The range of a series is found when one wants to determine how large the spread of data in a series is.

Definition: The mode of a series of numbers is the number that appears in a given series more often than others.

A series of numbers may have more than one mode, or may not have a mode at all.

Example: In a physical education lesson, 14 schoolchildren were high jumping, and the teacher was recording their results. The result was the following series of data (in cm):

125, 110, 130, 125, 120, 130, 140, 125, 110, 130, 120, 125, 120, 125.

Find the median, range and mode of measurement.

Solution: write down all measurement options in ascending order, separating groups of identical results with spaces:

110, 110, 120, 120, 120, 125, 125, 125, 125, 125, 130, 130, 130, 140.

The measurement range is 140-110=30.

125-met greatest number times, i.e. 5 times; it is a mode of measurement.

2. Practical part.

1). Tasks for independent decision on probability theory.

1. For every 100 light bulbs, on average there are 4 defective ones. What is the probability that a light bulb taken at random will turn out to be working? Answer: 0.96.

2. On average, there are 8 defective CDs per 400 CDs. What is the probability that a CD taken at random will be good? Answer: 0.98.

3. 17 points out of 50 are colored in Blue colour, and 13 of the remaining points are colored orange. What is the probability that a randomly selected point will be colored? Answer: 0.6.

4. One letter is randomly selected from the word “mathematics”. What is the probability that the selected letter occurs only once in this word? Answer: 0.3.

5. One letter is randomly selected from the word “certification”. What is the probability that the chosen letter will be the letter "a"? Answer: 0.2

6. Of the 30 ninth-graders, 4 chose an exam in physics, 12 in social studies, 8 in a foreign language, and the rest in literature. What is the probability that the selected student will take the literature exam. Answer: 0.2.

7. Test in mathematics consists of 15 problems: 4 problems in geometry, 2 problems in probability theory, the rest in algebra. The student made a mistake in one problem. What is the probability that a student made a mistake in an algebra problem? Answer: 0.6.

8. Out of 1000 cars produced in 2007-2009, 150 have a defective brake system. What is the probability of buying a faulty car? Answer: 0.15.

9. Participating in the rhythmic gymnastics competition are: 3 gymnasts from Russia, 3 gymnasts from Ukraine and 4 gymnasts from Belarus. The order of performance will be determined by drawing lots. Find the probability that a gymnast from Russia will compete first. Answer 0.3

10. There are 18 gymnasts performing at the rhythmic gymnastics championship, among them 3 gymnasts from Russia, 2 gymnasts from China. The order of performance is determined by drawing lots. Find the probability that a gymnast from either Russia or China will compete last? Answer: 5/18.

11. From a class of 12 boys and 8 girls, 1 person on duty is chosen by lot. What is the probability that it will be a boy? Answer: 0.6.

12. 2 coins are thrown at the same time. What is the probability of them landing on 2 heads? The answer is 0.25.

2)Problems on finding the arithmetic mean and median, range and mode of a set of numbers.

Milling crews spent on processing one part different time(in min.), presented as a data series: 40; 37; 35; 36; 32; 42; 32; 38; 32. How much does the median of this set differ from the arithmetic mean? Answer: 0.

5 apple tree seedlings were planted in the garden, the height of which in centimeters is as follows: 168, 13, 156, 165, 144. How much does the arithmetic mean of this set of numbers differ from its median? Answer: 3, 8

6 pear trees growing in the garden gave a harvest, the mass of which (in kg) for each of the trees is as follows: 29, 35, 26, 28, 32, 36. How much does the arithmetic mean of this set of numbers differ from its median? Answer: 0.5

The time the cashier served each of several store customers formed the following series of data: 2 minutes. 42 sec., 3 min. 2 sec., 3 imn. 7 sec., 2 min. 54 sec., 2 min. 48 sec. Find the mean and median of this data series. Answer: 2 min. 55 sec., 2 min. 54 sec.

The time between seven calls received by the taxi service formed the following series of data: 34 seconds, 45 seconds, 1 minute. 16 sec., 38 sec., 43 sec., 52 sec. Find the mean and median of this data series. Answer: 48 sec., 44 sec.

Literature : Mordkovich, A. G., I. M. Smirnova. Tutorial for educational institutions(basic level) - M.: Mnemosyne, 2009. - 164 p.