In addition to power-law averages in statistics, for a relative characteristic of the magnitude of a varying attribute and the internal structure of distribution series, structural averages are used, which are mainly represented by mode and median.

Fashion- This is the most common variant of the series. Fashion is used, for example, in determining the size of clothes, shoes, which are in greatest demand among buyers. The mode for a discrete series is the variant with the highest frequency. When calculating the mode for the interval variation series, you must first determine the modal interval (by the maximum frequency), and then the value of the modal value of the attribute according to the formula:

Median - this is the value of the feature that underlies the ranked series and divides this series into two parts equal in number.

To determine the median in a discrete series in the presence of frequencies, the half-sum of frequencies is first calculated, and then it is determined what value of the variant falls on it. (If the sorted row contains an odd number of features, then the median number is calculated by the formula:

M e \u003d (n (number of features in the aggregate) + 1) / 2,

in the case of an even number of features, the median will be equal to the average of the two features in the middle of the row).

When calculating the median for interval variation series first determine the median interval within which the median is located, and then the value of the median according to the formula:

Example. Find the mode and median.

Solution:
In this example, the modal interval is within the age group of 25-30 years, since this interval accounts for the highest frequency (1054).

Let's calculate the mode value:

This means that the modal age of students is 27 years.

Let's calculate the median. The median interval is in the age group of 25-30 years, since within this interval there is a variant that divides the population into two equal parts (Σf i /2 = 3462/2 = 1731). Next, we substitute the necessary numerical data into the formula and get the value of the median:

This means that one half of the students are under 27.4 years old, and the other half are over 27.4 years old.

In addition to the mode and median, indicators such as quartiles dividing the ranked series into 4 equal parts, deciles - 10 parts and percentiles - into 100 parts can be used.

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 are 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 met here yet? 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 an 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

That is 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 chart,
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 test 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 the countries, the first place in oil production was occupied by Saudi Arabia, the seventh place - by 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.

Using 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 construct a polygon, points are marked in the coordinate plane, the abscissas of which are points in 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 a 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 chart

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.

Representatives of companies came to the exhibition on innovative technologies. The diagram shows the distribution of these companies by the number of employees. The horizontal line shows the number of employees in the company, and the vertical line 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 the frequency to the total number of data in the series.

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 the successful passing of 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 have received a good education earn much more than those who have not received 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? Don't 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!

You can use our tasks (not necessary) and we certainly recommend them.

In order to get a hand with the help of our tasks, you need to help extend the life of the YouClever textbook that you are currently reading.

How? There are two options:

1. Unlock access to all hidden tasks in this article -
2. Unlock access to all hidden tasks in all 99 articles of the tutorial - Buy a textbook - 899 rubles

Yes, we have 99 such articles in the textbook and access to all tasks and all hidden texts in them can be opened immediately.

Access to all hidden tasks is provided for the entire lifetime of the site.

In conclusion...

If you don't like our tasks, find others. Just don't stop with theory.

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

Find problems and solve!

When studying the teaching load of students, a group of 12 seventh-graders was singled out. 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. When studying the workload of students, a group of 12 seventh graders 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 arithmetic mean of the series. The arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by the number of terms. The arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by the number of terms.(): 12=27

Row span. The range of a series is the difference between the largest and smallest of these numbers. The range of a series is the difference between the largest and smallest of these numbers. The largest time consumption is 37 minutes, and the smallest is 18 minutes. Find the range of the series: 37 - 18 = 19 (min)

Row fashion. The mode of a series of numbers is the number that occurs in this series more often than others. The mode of a series of numbers is the number that occurs in this series more often than others. The mode of our series is the number - 25. The mode of our series is the number - 25. A series of numbers may or may not have more than one mode. 1) 47,46,50,47,52,49,45,43,53,53,47,52 - two modes 47 and 52. 2) 69,68,66,70,67,71,74,63, 73.72 - no fashion.

The arithmetic mean, range and fashion, are used in statistics - a science that deals with obtaining, processing and analyzing quantitative data on a variety of mass phenomena occurring in nature and society. The arithmetic mean, range and fashion, are used in statistics - a science that deals with obtaining, processing and analyzing quantitative data on a variety of mass phenomena occurring in nature and society. 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. 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 , transportation of goods and passengers by various modes of transport, natural resources, etc.

1. Find the arithmetic mean and range of a series of numbers: a) 24,22,27,20,16,37; b) 30,5,23,5,28, Find the arithmetic mean, range and mode of a series of numbers: a) 32,26,18,26,15,21,26; b) -21, -33, -35, -19, -20, -22; b) -21, -33, -35, -19, -20, -22; c) 61,64,64,83,61,71,70; c) 61,64,64,83,61,71,70; d) -4, -6, 0, 4, 0, 6, 8, -12. d) -4, -6, 0, 4, 0, 6, 8, One number is missing in the series of numbers 3, 8, 15, 30, __, 24. Find it if: a) the arithmetic mean of the series is 18; a) the arithmetic mean of the series is 18; b) the range of the series is 40; b) the range of the series is 40; c) the mode of the series is 24. c) the mode of the series is 24.

4. In the certificate of secondary education, four friends - graduates of the school - had the following marks: Ilyin: 4,4,5,5,4,4,4,5,5,5,4,4,5,4,4; Ilyin: 4,4,5,5,4,4,4,5,5,5,4,4,5,4,4; Semyonov: 3,4,3,3,3,3,4,3,3,3,3,4,4,5,4; Semyonov: 3,4,3,3,3,3,4,3,3,3,3,4,4,5,4; Popov: 5,5,5,5,5,4,4,5,5,5,5,5,4,4,4; Popov: 5,5,5,5,5,4,4,5,5,5,5,5,4,4,4; Romanov: 3,3,4,4,4,4,4,3,4,4,4,5,3,4,4. Romanov: 3,3,4,4,4,4,4,3,4,4,4,5,3,4,4. What is the average GPA that each of these graduates graduated from high school with? Indicate the most typical grade for each of them in the certificate. What statistics did you use in your answer? What is the average GPA that each of these graduates graduated from high school with? Indicate the most typical grade for each of them in the certificate. What statistics did you use in your answer?

Independent work Option 1. Option A series of numbers is given: 35, 44, 37, 31, 41, 40, 31, 29. Find the arithmetic mean, range and mode of the rad. 2. In the series of numbers 4, 9, 16, 31, _, 25 4, 9, 16, 31, _, 25, one number is missing. missing one number. Find it if: Find it if: a) the arithmetic mean; a) the arithmetic mean is 19; which is 19; b) range of the series - 41. b) range of the series - 41. Option A series of numbers is given: 38, 42, 36, 45, 48, 45.45, 42. Find the arithmetic mean, range and mode of the rad. 2. In the series of numbers 5, 10, 17, 32, _, 26, one number is missing. Find it if: a) the arithmetic mean is 19; b) the range of the series is 41.

The median of an ordered series of numbers with an odd number of numbers is the number written in the middle, and the median of an ordered series of numbers with an even number of numbers is the arithmetic mean of the two numbers written in the middle. The median of an ordered series of numbers with an odd number of numbers is the number written in the middle, and the median of an ordered series of numbers with an even number of numbers is the arithmetic mean of the two numbers written in the middle. The table shows electricity consumption in January by residents of nine apartments: The table shows electricity consumption in January by residents of nine apartments: Apartment number Electricity consumption

Let's make an ordered series: 64, 72, 72, 75, 78, 82, 85, 91.93. 64, 72, 72, 75, 78, 82, 85, 91 - the median of this series. 78 is the median of this series. An ordered series is given: An ordered series is given: 64, 72, 72, 75, 78, 82, 85, 88, 91, 93. (): 2 = 80 - median. ():2 = 80 – median.

1. Find the median of a series of numbers: a) 30, 32, 37, 40, 41, 42, 45, 49, 52; a) 30, 32, 37, 40, 41, 42, 45, 49, 52; b) 102, 104, 205, 207, 327, 408, 417; b) 102, 104, 205, 207, 327, 408, 417; c) 16, 18, 20, 22, 24, 26; c) 16, 18, 20, 22, 24, 26; d) 1.2, 1.4, 2.2, 2.6, 3.2, 3.8, 4.4, 5.6. d) 1.2, 1.4, 2.2, 2.6, 3.2, 3.8, 4.4, 5.6. 2. Find the arithmetic mean and median of a series of numbers: a) 27, 29, 23, 31,21,34; a) 27, 29, 23, 31,21,34; b) 56, 58, 64, 66, 62, 74; b) 56, 58, 64, 66, 62, 74; c) 3.8, 7.2, 6.4, 6.8, 7.2; c) 3.8, 7.2, 6.4, 6.8, 7.2; d) 21.6, 37.3, 16.4, 12, 6. d) 21.6, 37.3, 16.4, 12, 6.

3. The table shows the number of visitors to the exhibition on different days of the week: Find the median of the specified data series. On which days of the week the number of visitors to the exhibition was greater than the median? Days of the week Mon Mon Tue Wed Wed Thu Thu Fri Fri Sat Sat Sun Sun Number of visitors

4. Below is the average daily processing of sugar (in thousand centners) by sugar industry plants in a certain region: (in thousand centners) by sugar industry plants in a certain region: 12.2, 13.2, 13.7, 18.0, 18.6 , 12.2, 18.5, 12.4, 12.2, 13.2, 13.7, 18.0, 18.6, 12.2, 18.5, 12.4, 14, 2, 17 ,eight. 14, 2, 17.8. For the given series, find the arithmetic mean, mode, range, and median. For the given series, find the arithmetic mean, mode, range, and median. 5. The organization kept a daily record of letters received during the month. As a result, we received the following series of data: 39, 43, 40, 0, 56, 38, 24, 21, 35, 38, 0, 58, 31, 49, 38, 25, 34, 0, 52, 40, 42, 40 , 39, 54, 0, 64, 44, 50, 38, 37, 43, 40, 0, 56, 38, 24, 21, 35, 38, 0, 58, 31, 49, 38, 25, 34, 0 , 52, 40, 42, 40, 39, 54, 0, 64, 44, 50, 38, 37, 32. For the presented series, find the arithmetic mean, mode, range and median. For the given series, find the arithmetic mean, mode, range, and median.

Homework. At figure skating competitions, the athlete's performance was assessed with the following points: At figure skating competitions, the athlete's performance was assessed with the following points: 5.2; 5.4; 5.5; 5.4; 5.1; 5.1; 5.4; 5.5; 5.3. 5.2; 5.4; 5.5; 5.4; 5.1; 5.1; 5.4; 5.5; 5.3. For the resulting series of numbers, find the arithmetic mean, range and mode. For the resulting series of numbers, find the arithmetic mean, range and mode.

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 on a variety of mass phenomena occurring in nature and society.
• The word "statistics" comes from the Latin word status, which means "state, state of affairs." Statistics studies the number of individual groups of the population 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- quotient from 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- an ordered series of numbers with an odd number of members is the number written in the middle, and the median of an ordered series of numbers with an even number of members is the arithmetic mean of 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
• find application in statistics - science,
• which deals with obtaining

processing and analysis

quantitative data on a variety of

• mass events taking place

in nature and

• Society.

Task #1

• Row 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

Task #2

• Row 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

Task #3

• Row 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 consumption of time - 37 minutes,
• and the smallest - 18 min.
• Find the range of the series:
• 37 - 18 = 19 (min)

Task #4

• Row of numbers:
• 65; 12; 48; 36; 7; 12
• Find the fashion of the series:

• Solution:

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

Task number 5

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

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

• Row: 69, 68, 66, 70, 67, 71, 74, 63, 73, 72 - no fashion.

Task number 5

• Row 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

Task number 6

The organization kept a daily record 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 given series of data, find the arithmetic mean,

What is the practical meaning of these indications?

Task number 7

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

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

Solution.

Sort this set of numbers in ascending order:

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

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

the value that occupies the middle of the 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 \u003d 31 - 30 \u003d 1

Creative

TEST

On the topic: "Mode. Median. Methods for calculating them"

Introduction

Mean values ​​and related indicators of variation play a very important role in statistics, which is due to the subject of its study. Therefore, this topic is one of the central in the course.

The average is a very common generalizing indicator in statistics. This is explained by the fact that only with the help of the average it is possible to characterize the population according to a quantitatively varying attribute. An average value in statistics is a generalizing characteristic of a set of phenomena of the same type according to some quantitatively varying attribute. The average shows the level of this attribute, related to the unit of the population.

Studying social phenomena and seeking to identify their characteristic, typical features in specific conditions of place and time, statisticians make extensive use of average values. With the help of averages, different populations can be compared with each other according to varying characteristics.

Averages used in statistics belong to the class of power averages. Of the power averages, the arithmetic mean is most often used, less often the harmonic mean; the harmonic mean is used only when calculating the average rates of dynamics, and the mean square - only when calculating the variation indicators.

The arithmetic mean is the quotient of dividing the sum of the options by their number. It is used in cases where the volume of a variable attribute for the entire population is formed as the sum of the attribute values ​​for its individual units. The arithmetic mean is the most common type of average, since it corresponds to the nature of social phenomena, where the volume of varying signs in the aggregate is most often formed precisely as the sum of the values ​​of the attribute in individual units of the population.

According to its defining property, the harmonic mean should be used when the total volume of the attribute is formed as the sum of the reciprocal values ​​of the variant. It is used when, depending on the material available, the weights do not have to be multiplied, but divided into options or, what is the same, multiplied by their inverse value. The harmonic mean in these cases is the reciprocal of the arithmetic mean of the reciprocal values ​​of the attribute.

The harmonic mean should be used in those cases when not the units of the population - the carriers of the attribute, but the products of these units and the value of the attribute are used as weights.

1. Definition of mode and median in statistics

The arithmetic and harmonic means are the generalizing characteristics of the population according to one or another varying attribute. Auxiliary descriptive characteristics of the distribution of a variable attribute are the mode and the median.

In statistics, fashion is the value of a feature (variant) that is most often found in a given population. In the variation series, this will be the variant with the highest frequency.

The median in statistics is called the variant, which is in the middle of the variation series. The median divides the series in half, on both sides of it (up and down) there is the same number of population units.

Mode and median, in contrast to the exponential averages, are specific characteristics, their value is any particular variant in the variation series.

Mode is used in cases where it is necessary to characterize the most frequently occurring value of a feature. If it is necessary, for example, to find out the most common wage rate in the enterprise, the market price at which the largest number of goods were sold, the size of shoes that are most in demand among consumers, etc., in these cases resort to fashion.

The median is interesting in that it shows the quantitative limit of the value of the variable characteristic, which was reached by half of the members of the population. Let the average salary of bank employees amount to 650,000 rubles. per month. This characteristic can be supplemented if we say that half of the workers received a salary of 700,000 rubles. and higher, i.e. let's take the median. The mode and median are typical characteristics in cases where the populations are homogeneous and large in number.

2. Finding the Mode and Median in a Discrete Variation Series

Finding the mode and median in a variational series, where the attribute values ​​are given by certain numbers, is not very difficult. Consider table 1. with the distribution of families by the number of children.

Table 1. Distribution of families by number of children

Obviously, in this example, the fashion will be a family with two children, since this value of options corresponds to the largest number of families. There may be distributions where all variants are equally frequent, in which case there is no fashion, or, in other words, all variants can be said to be equally modal. In other cases, not one, but two options may be the highest frequency. Then there will be two modes, the distribution will be bimodal. Bimodal distributions may indicate the qualitative heterogeneity of the population according to the trait under study.

To find the median in a discrete variation series, you need to divide the sum of frequencies in half and add ½ to the result. So, in the distribution of 185 families by the number of children, the median will be: 185/2 + ½ = 93, i.e. The 93rd option, which divides the ordered row in half. What is the meaning of the 93rd option? In order to find out, you need to accumulate frequencies, starting from the smallest options. The sum of the frequencies of the 1st and 2nd option is 40. It is clear that there are no 93 options here. If we add the frequency of the 3rd option to 40, then we get the sum equal to 40 + 75 = 115. Therefore, the 93rd option corresponds to the third value of the variable attribute, and the median will be a family with two children.

Mode and median in this example coincided. If we had an even sum of frequencies (for example, 184), then applying the above formula, we get the number of the median options, 184/2 + ½ = 92.5. Since there are no fractional options, the result indicates that the median is in the middle between 92 and 93 options.

3. Calculation of the mode and median in the interval variation series

The descriptive nature of the mode and median is due to the fact that they do not compensate for individual deviations. They always correspond to a certain variant. Therefore, the mode and median do not require calculations to find them if all the values ​​of the attribute are known. However, in the interval variation series, calculations are used to find the approximate value of the mode and median within a certain interval.

To calculate a certain value of the modal value of a sign enclosed in an interval, the following formula is used:

M o \u003d X Mo + i Mo * (f Mo - f Mo-1) / ((f Mo - f Mo-1) + (f Mo - f Mo + 1)),

Where X Mo is the minimum limit of the modal interval;

i Mo is the value of the modal interval;

fMo is the frequency of the modal interval;

f Mo-1 - the frequency of the interval preceding the modal;

f Mo+1 is the frequency of the interval following the modal.

We will show the calculation of the mode using the example given in Table 2.

Table 2. Distribution of workers of the enterprise according to the implementation of production standards

To find the mode, we first determine the modal interval of the given series. It can be seen from the example that the highest frequency corresponds to the interval where the variant lies in the range from 100 to 105. This is the modal interval. The value of the modal interval is 5.

Substituting the numerical values ​​from table 2. into the above formula, we get:

M o \u003d 100 + 5 * (104 -12) / ((104 - 12) + (104 - 98)) \u003d 108.8

The meaning of this formula is as follows: the value of that part of the modal interval, which must be added to its minimum boundary, is determined depending on the magnitude of the frequencies of the previous and subsequent intervals. In this case, we add 8.8 to 100, i.e. more than half of the interval, because the frequency of the previous interval is less than the frequency of the subsequent interval.

Let's calculate the median now. To find the median in the interval variation series, we first determine the interval in which it is located (the median interval). Such an interval will be one whose cumulative frequency is equal to or greater than half the sum of the frequencies. Cumulative frequencies are formed by gradual summation of frequencies, starting from the interval with the smallest feature value. Half the sum of the frequencies we have is 250 (500:2). Therefore, according to table 3. the median interval will be the interval with the value of wages from 350,000 rubles. up to 400,000 rubles.

Table 3. Calculation of the median in the interval variation series

Before this interval, the sum of the accumulated frequencies was 160. Therefore, in order to obtain the value of the median, it is necessary to add another 90 units (250 - 160).