Lyudmila Prokofievna Kalugina (or simply “Mymra”) in the wonderful film “Office Romance” taught Novoseltsev: “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 the State Academic Examination with elements of statistics), we will try to understand some of the concepts of statistics that can be useful not only in the thorny path of conquering the exam in the Unified State Examination, but also just in everyday life. life.

So what is statistics and why is it needed? The word "statistics" comes from the Latin word "status" (status), which means "the state and state of affairs / things." Statistics deals with the study of the quantitative side of mass social phenomena and processes in numerical form, revealing special patterns. Today, statistics is used in almost all spheres of public life, ranging from fashion, cooking, gardening and ending with astronomy, economics, and medicine.

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

Statistical characteristics

The main statistical characteristics of a data sample (what else is a “sample”!? Don’t be scared, everything is under control, this is an incomprehensible word only for intimidation, in fact, the word “sample” means just the data that you are going to examine) include:

1. sample size,
2. sample size,
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 Span

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

This sample 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, let's get started? We have already figured out that; .

We can immediately boldly substitute everything into our formula:

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

Well, or like this example:

For a week, 9th grade students were asked to solve as many examples from the problem book as possible. The number of examples solved by students in a week is given below:

Find the average number of solved problems.

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

Now we can safely proceed to the calculation of the arithmetic mean of the solved problems, knowing that, a:

Thus, on average, 9th grade students solved the tasks.

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 is equal in this example? That's right: seven sellers offer seven prices, which means ! . Well, we figured out all the components, now we can start calculating the average price:

Well, did you understand? Then count yourself average in the following samples:

Answers: .

Mode and median

Let's go back to our soccer team example:

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 other players is not repeated. Everything should be clear and understandable here, and the word is familiar, right?

Let's move on to the median, you should know it from the geometry course. But it is not difficult for me to recall that in geometry median(translated from Latin - “middle”) - a segment inside a triangle connecting the vertex of the triangle with the middle of the 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, back to our sample of football players?

Did you notice an important point in the definition of the median that we have not yet met here? Of course, "if this row is ordered"! Shall we put things in order? In order to have an order in the series of numbers, it is possible to arrange the height values ​​of the players both in descending order and in ascending order. It is more convenient for me to build this series in ascending order (from smallest to largest). That's what I did:

So, the series has been ordered, what else is there an important point in determining the median? Correct, even and odd number of members in the sample. Noticed that even the definitions are different for even and odd numbers? Yes, you're right, it's hard not to notice. And if so, then we need to decide whether the number of players in our sample is even or odd? That's right - players, so the number is odd! 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 a number that turned out to be in the middle in our ordered series:

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

And now let's look at an example with our desperate guys from grade 9, who solved examples during the week:

Ready to look for mode and median in this series?

First, let's arrange this series of numbers (arrange from the smallest number to the largest). The result is this row:

Now we can safely determine the fashion in this sample. Which number is the most common? That's right! In this way, fashion in this sample is equal.

We found the fashion, now we can start finding the median. But first, tell me: what is the sample size in question? Did you count? That's right, the sample size is the same. A is even number. Thus, we apply the definition of the 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 in the middle. What two numbers are in the middle? That's right, and!

So the median of this series will be average numbers and:

- median considered sample.

Frequency and relative frequency

I.e frequency determines how often one or another value is repeated in the sample.

Let's look at our example with football players. Before us is such an ordered row:

Frequency is the number of repetitions of some 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 these questions and answering them, you can make 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 next characteristic - the relative frequency.

Let's go back to our soccer player example. We 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 the following table:

And now make tables of frequencies and relative frequencies yourself for an example with 9-graders solving problems.

Graphical display of data

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

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

bar chart

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

For example, we have the following data about the grades of a written control work in one class:

The number of those 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(the horizontal axis shows the grades; the vertical axis shows the number of students who received the corresponding grades):

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

Consider an example of the type of task B3 from the exam.

Example.

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

Answer: third.

Pie chart

For a visual representation of the relationship between parts of the sample under study, it is convenient to use pie charts.

From our plate with the relative frequencies of the distribution of grades in the class, we can build a pie chart by breaking the circle into sectors proportional to the relative frequencies.

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

Consider an example of the type of task 18 from the GIA.

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 is often depicted using a polygon. To build a polygon, mark in coordinate plane points, the abscissas of which are moments of time, and the ordinates are the corresponding statistical data. By connecting these points in series 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 - let's build a polygon.

Months are shown on the horizontal axis, temperatures are shown on the vertical axis. We build the corresponding points and connect them. Here's what happened:

Agree, it immediately became clearer!

A polygon is also used to visualize 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:

Consider typical task B3 from the exam.

Example.

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

Answer: .

bar graph

Interval data series are depicted using a histogram. The 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.

Here, 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, we will build on the basis of the frequency. Here's what happened:

Now, based on the relative frequency data:

Example.

to the exhibition on innovative technologies company representatives arrived. The diagram shows the distribution of these companies by the number of employees. The horizontal axis shows the number of employees in the company, and the vertical one shows the number of companies with a given number of employees.

What percentage are companies with a total number of employees more people?

Answer: .

Brief summary

Sample size- the number of elements in the sample.

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

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

Number series fashion- the number most often found in this series.

Medianan ordered series of numbers with an odd number of members is the number in the middle.

Median of an ordered series of numbers with an even number of members- 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.

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

The sample size is the number of items in the sample.

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

Or, sample range

Average 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 occurs most frequently in a given series.

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

The frequency is the number of repetitions, how many times during a certain period an event occurred, a certain property of an object manifested itself, or an observed parameter reached a given value.

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

Well, the topic is over. If you are reading these lines, then you are very cool.

Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!

Now the most important thing.

You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough ...

For what?

For successful passing the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.

I will not convince you of anything, I will just say one thing ...

People who received a good education, earn much more than those who did not receive it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Do not know...

But think for yourself...

What does it take to be sure to be better than others on the exam and be ultimately ... happier?

FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.

On the exam, you will not be asked theory.

You will need solve problems on time.

And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.

It's like in sports - you need to repeat many times to win for sure.

Find a collection anywhere you want necessarily with solutions, detailed analysis and decide, decide, decide!

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“Understood” and “I know how to solve” are completely different skills. You need both.

Find problems and solve!

Distribution range- this is a sequence of numbers indicating the qualitative or quantitative value of the trait and the frequency of its occurrence.

The types of distribution series are classified according to different principles.

According to the degree of ordering, the rows are divided into:

disordered

ordered

Unordered series- this is a series in which the values ​​of the attribute are recorded in the order in which the variants are received during the study.

Example: When studying the height of a group of students, its values ​​were recorded in cm (175,170,168,173,179).

ordered row is a series obtained from an unordered one in which the feature values ​​are overwritten in ascending or descending order. An ordered series is called a ranked series, and the ranking procedure

(ordering) is called sorting.

Example: (Height 168,170,173,175,179)

According to the type of feature, the distribution series are divided into:

attributive

variational.

Attribute series- this is a series compiled on the basis of a qualitative trait.

Variation series- This is a series compiled on the basis of a quantitative attribute.

Variational series are divided into discrete, continuous and interval.

Variational discrete, continuous and interval series are named according to the corresponding feature, which underlies the compilation of the series. For example, a row by shoe size is discrete by body weight - continuous.

Methods for representing series in practical and scientific medicine are divided into three groups:

Table view;

Analytical representation (in the form of a formula);

Graphical representation.

1. The simplest table consists of two columns or two rows, one of which contains the values ​​of the attribute x i in an ordered form, and in the other - the relative or absolute frequency of its occurrence n i , f i .

Example: Tabular View of Grades in a Group x i and the number of students who received them n i .

x i
n i

2. The graphical representation of the series is based on tabular data. Graphs are built in a rectangular coordinate system, where the values ​​of the feature are always plotted horizontally X i , and vertically absolute or relative frequency n i .

The main ways to present graphs:

Bar chart.

bar graph

Frequency polygon.

Variation (frequency) curve.

Bar chart- this is a graph of the representation of a series in the form of vertical straight lines-segments, the position of which on the horizontal is determined by the value of the feature, and the length of the segment is proportional to its absolute or relative frequency.

Example: a bar chart for group grades.

n i

5 4 3 2 XI

Typically, bar charts are built for discretely given features with a small number of options.

bar graph- this is a graph in the form of a stepped figure of rectangles adjacent to each other, the bases of which are the intervals of feature values, and the heights of the rectangles are proportional to the frequency or frequency (the number of objects that fall into the interval). The areas of the rectangles correspond to the number of groups in the given interval.

Histograms are plots of interval series. They are built mainly for large volumes of populations.

Example: Histogram of the normal distribution of red blood cells in human blood. Horizontally - cell diameter X i (mk), vertical - frequency n i the number of cells in the interval.

n i

2 4 6 8 10 12 x i

Poligon (polygon) frequencies- a graph of a series, represented by a broken line of a point - the vertices of which correspond to the midpoints of the intervals, and the height of the point above the horizontal is proportional to the frequency or frequency.

Polygons are built for continuous and discrete variational series in those cases when the average values ​​of the attribute are allocated in the intervals. Polygons are preferable to histograms for continuous distribution series

Example: a polygon of frequencies based on a histogram of the distribution of erythrocytes in human blood.

n i

2 4 6 8 10 12 x i

Variation (frequency) curve- series graph obtained under the condition that the volume of the population tends to infinity ( N→∞) , and the length of the interval itself tends to zero (Δ X→0) .

For practical statistical calculations, four groups of frequency distributions are identified as standards:

1. Rectangular distribution.

   Bell-shaped unimodal (single-top) distribution.

   Bimodal (two-top) distribution.

   Exponential Distribution:

growing,

decreasing.

n i

x i

x i

x i

x i

Rectangular distribution is subject to random equiprobable events.

A wide class of phenomena is subject to a bell-shaped symmetrical distribution (indicators of mental and physical development, height, weight, etc.). In practice, the most common symmetric unimodal distribution, so its classical form is called the normal distribution.

The bimodal distribution corresponds, for example, to the performance of students with and without a long break in their studies.

An exponentially decreasing distribution corresponds to the distribution of income in a capitalist society, (frequency decreases as income increases).

Text HTML version of the publication

Summary of the lesson of algebra in grade 7

Theme of the lesson: "MEDIAN OF THE ORDERED SERIES".

teacher of the Lake School branch of MKOU Burkovskaya secondary school Eremenko Tatyana Alekseevna
Goals:
the concept of the median as a statistical characteristic of an ordered series; to form the ability to find the median for ordered series with an even and odd number of members; to form the ability to interpret the values ​​of the median depending on the practical situation, to consolidate the concept of the arithmetic mean set of numbers. Develop independent work skills. Build an interest in mathematics.
During the classes

oral work.
Rows are given: 1) 4; one; 8; five; one; 2) ; nine; 3; 0.5; ; 3) 6; 0.2; ; 4; 6; 7.3; 6. Find: a) the largest and smallest value each row; b) the range of each row; c) the fashion of each row.
II. Explanation of new material.
Textbook work. 1. Consider the problem from paragraph 10 of the textbook. What does ordered row mean? I emphasize that before finding the median, you must always sort the data series. 2. On the board, we get acquainted with the rules for finding the median for series with an even and odd number of members:
median

orderly

row
numbers
from

odd

number

members
called the number written in the middle, and
median

ordered row
numbers
with an even number of members
is called the arithmetic mean of two numbers written in the middle.
median

arbitrary

row
called the median 1 3 1 7 5 4

corresponding ordered series.
I note that the indicators are the arithmetic mean, mode and median for

differently

characterize

data,

received

result

observations.

III. Formation of skills and abilities.
1st group. Exercises on the application of formulas for finding the median of an ordered and unordered series. one.
№ 186.
Solution: a) Number of members of the series P= 9; median Me= 41; b) P= 7, the row is ordered, Me= 207; in) P= 6, the row is ordered, Me== 21; G) P= 8, the row is ordered, Me== 2.9. Answer: a) 41; b) 207; at 21; d) 2.9. Students comment on how the median is found. 2. Find the arithmetic mean and median of a series of numbers: a) 27, 29, 23, 31, 21, 34; in) ; 1. b) 56, 58, 64, 66, 62, 74. Solution: To find the median, it is necessary to sort each row: a) 21, 23, 27, 29, 31, 34. P = 6; X = = 27,5; Me = = 28; 20 22 2 + 2, 6 3, 2 2 + 1125 ; ; ; 3636 21 23 27 29 31 34 165 66 +++++ = 27 29 2 +

b) 56, 58, 62, 64, 66, 74. P = 6; X = 63,3; Me== 63; in) ; one. P = 5; X = : 5 = 3: 5 = 0,6; Me = . 3.
№ 188
(orally). Answer: yes; b) no; c) no; d) yes. 4. Knowing that the ordered series contains T numbers, where T is an odd number, indicate the number of the term that is the median if T is equal to: a) 5; b) 17; c) 47; d) 201. Answer: a) 3; b) 9; c) 24; d) 101. 2nd group. Practical tasks for finding the median of the corresponding series and interpreting the result. one.
№ 189.
Solution: Number of row members P= 12. To find the median, the series must be ordered: 136, 149, 156, 158, 168, 174, 178, 179, 185, 185, 185, 194. Median of the series Me= = 176. Monthly output was more than the median for the following members of the artel: 56 58 62 64 66 74 380 66 +++++ =≈ 62 64 2 + 1125; ; ; 3636 1125 12456 18 1:5:5 6336 6 6 ++++ ⎛⎞ ++++ = = ⎜⎟ ⎝⎠ 2 3 67 174 178 22 xx + + =

1) Kvitko; 4) Bobkov; 2) Baranov; 5) Rylov; 3) Antonov; 6) Astafiev. Answer: 176. 2.
№ 192.
Solution: Let's arrange the data series: 30, 31, 32, 32, 32, 32, 32, 32, 33, 35, 35, 36, 36, 36, 38, 38, 38, 40, 40, 42; number of row members P= 20. Swipe A = x max- x min = 42 - 30 = 12. Mode Mo= 32 (this value occurs 6 times - more often than others). Median Me= = 35. In this case, the range shows the greatest spread of time for processing the part; the mode shows the most typical value of the processing time; median is the processing time that half of the turners did not exceed. Answer: 12; 32; 35.
IV. Summary of the lesson.
What is the median of a series of numbers? – Can the median of a series of numbers not coincide with any of the numbers in the series? – What number is the median of an ordered series containing 2 P numbers? 2 P– 1 numbers? How to find the median of an unordered series?
Homework:
№ 187, № 190, № 191, № 254. 10 11 35 35 22 xx + + =

As a result of systematization and processing of primary materials of statistical observation, ordered series of digital indicators are obtained that characterize either the change in the size of a phenomenon over time (a series of dynamics, which will be discussed in the topic "Series of dynamics"), or the distribution of population units according to various varying characteristics in statics (distribution series).

Distribution range- this is a series of digital indicators, representing the distribution of units of the population according to one attribute, the varieties of which are arranged in a certain sequence.

The elements of the distribution series are: options and frequencies.

Options ( ) the individual values ​​of the grouping attribute that it takes in the variation series are called. Variants can be expressed by positive and negative numbers, absolute and relative. Numbers that show how often certain options occur in a distribution series are called frequencies (). The number of units in each group can be expressed not only by the number of units (frequencies), but also in shares (percentage) of the total number of population units (frequently). The sum of the frequencies is 1 if they are expressed as a fraction of one, and 100% if they are expressed as a percentage.

Depending on the statistical nature of the variants, two types of distribution series are distinguished: attributive and variational.

Rows based on qualitative feature, called attributive(for example, the distribution of the population by sex, the distribution of enterprises by form of ownership, etc.).

The distribution series according to a quantitative characteristic are called variational(distribution of population by income, distribution of banks by assets).

Since the variation of a trait can be discrete (discontinuous) and continuous, then there are discrete and continuous (interval) variation series. In discrete variational series, the values ​​of the options are expressed as integers and differ from each other by a well-defined value (one or more units). Examples of discrete variational series are: the distribution of families by the number of children, the distribution of apartments by the number of rooms, etc.

With a continuous variation of a sign, its value can take on both integer and fractional values, that is, any values ​​in a certain interval (age, length of service, profit, etc.). For distribution series with equal intervals, the frequencies give an idea of ​​the degree to which the interval is filled with population units. For distribution series with unequal intervals, in order to compare the occupancy of the intervals, the distribution density is calculated, that is, the number of population units (frequency, frequency) per unit of interval width on average. Distribution density can be absolute (the ratio of frequency to the width of the interval) and relative (the ratio of frequency to the width of the interval).

The distribution series can be built on the accumulated frequencies (frequencies), which show how many units have a variant value that is not greater than a given one. Such distribution series are called cumulative.

Various graphs are used to display distribution series.

Thus, the distribution of the region's population by place of residence can be depicted using a pie chart (Fig. 5.1).

Rice. 5.1. Distribution of the population of the region by place

Linear and planar diagrams constructed in a rectangular coordinate system are used to depict variational series.

Discrete variational series, the variants of which are expressed as integers, are depicted as distribution area. The distribution polygon is a closed polygon, the abscissas of the vertices of which are the values ​​of the varying attribute, and the ordinates are the frequencies or frequencies corresponding to them (Fig. 5.2).

Fig.5.2. Distribution of singles and families of the city by the number of

living.

The graphic representation of continuous variational series is carried out using the so-called histogram. To build a histogram, on the abscissa axis, in accordance with the accepted scale, lay the boundaries of the intervals on which the rectangles are built. The heights of these rectangles are proportional to the distribution densities of the corresponding intervals. On fig. 4.3 shows a histogram of the distribution of the region's population in terms of average per capita total income per month in 2000.

Fig.5.3. Distribution of the population of the region by the size of the average per capita

total income per month in 2000. (according to the budget

family surveys).

With unequal intervals, the histogram is built only by the distribution density.

A cumulative curve (cumulative) is also used for the graphic representation of variational series. To build it, the value of a discrete feature (or the interval boundary) is plotted on the abscissa axis, and the incremental totals of frequencies or frequencies corresponding to these feature values ​​(or the upper limits of the interval) are plotted on the ordinate axis. The cumulative distribution of the population of the region by the size of the average per capita total income per month is shown in Figure 5.4.

Fig.5.4. Cumulative distribution of the region's population by size

average per capita total income per month in 2000.

(according to budget surveys of families).

With the help of cumulative curves, it is possible to graphically depict the process of concentration. For a graphical representation of the phenomenon of concentration, cumulative totals of indicators are used. To do this, it is necessary to have in the group table, in addition to the sums of the accumulated frequencies, also the sums of the accumulated values ​​of the most important features (grouping in the first place), expressed as a percentage of the total. The cumulative totals of the frequencies are plotted on the abscissa axis, and the corresponding cumulative totals of the indicators are plotted on the ordinate axis. By connecting the points found in this way with line segments, broken lines are obtained, which are called concentration curves.

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