What does it mean to find the arithmetic mean of numbers. How to find and calculate the arithmetic mean for two

Three children went to the forest for berries. The eldest daughter found 18 berries, the middle daughter found 15, and the younger brother found 3 berries (see Fig. 1). They brought the berries to my mother, who decided to share the berries equally. How many berries did each child get?

Rice. 1. Illustration for the problem

Decision

(yag.) - children collected everything

2) Divide the total number of berries by the number of children:

(yag.) went to every child

Answer: Each child will receive 12 berries.

In problem 1, the number received in the answer is the arithmetic mean.

arithmetic mean several numbers is called the quotient of dividing the sum of these numbers by their number.

Example 1

We have two numbers: 10 and 12. Find their arithmetic mean.

Decision

1) Let's determine the sum of these numbers: .

2) The number of these numbers is 2, therefore, the arithmetic mean of these numbers is: .

Answer: the arithmetic mean of the numbers 10 and 12 is the number 11.

Example 2

We have five numbers: 1, 2, 3, 4 and 5. Find their arithmetic mean.

Decision

1) The sum of these numbers is: .

2) By definition, the arithmetic mean is the quotient of dividing the sum of numbers by their number. We have five numbers, so the arithmetic mean is:

Answer: The arithmetic mean of the data in the numbers condition is 3.

In addition to being constantly offered to find it in the classroom, finding the arithmetic mean is very useful in everyday life. For example, suppose we want to go on holiday to Greece. To choose the right clothes, we look at the temperature in this country at the moment. However, we do not know the general picture of the weather. Therefore, it is necessary to find out the air temperature in Greece, for example, for a week, and find the arithmetic mean of these temperatures.

Example 3

Temperature in Greece for the week: Monday - ; Tuesday - ; Wednesday -; Thursday - ; Friday - ; Saturday - ; Sunday - . Calculate the average temperature for the week.

Decision

1) Calculate the sum of temperatures: .

2) Divide the amount received by the number of days: .

Answer: weekly average temperature approx.

The ability to find the arithmetic mean can also be needed to determine the average age of the players on a football team, that is, in order to establish whether the team is experienced or not. It is necessary to sum up the age of all players and divide by their number.

Task 2

The merchant was selling apples. At first he sold them at a price of 85 rubles per 1 kg. So he sold 12 kg. Then he reduced the price to 65 rubles and sold the remaining 4 kg of apples. What was the average price for apples?

Decision

1) Let's calculate how much money the merchant earned in total. He sold 12 kilograms at a price of 85 rubles per 1 kg: (rub.).

He sold 4 kilograms at a price of 65 rubles per 1 kg: (rub.).

Therefore, the total amount of money earned is: (rubles).

2) The total weight of apples sold is: .

3) Divide the amount of money received by the total weight of apples sold and get the average price for 1 kg of apples: (rubles).

Answer: the average price of 1 kg of sold apples is 80 rubles.

The arithmetic mean helps evaluate the data as a whole, without taking each value individually.

However, it is not always possible to use the concept of arithmetic mean.

Example 4

The shooter fired two shots at the target (see Fig. 2): the first time he hit a meter above the target, and the second - a meter below. The arithmetic mean will show that he hit the center exactly, although he missed both times.

Rice. 2. Illustration for example

In this lesson, we got acquainted with the concept of arithmetic mean. We learned the definition of this concept, learned how to calculate the arithmetic mean for several numbers. We also learned the practical application of this concept.

1. N.Ya. Vilenkin. Mathematics: textbook. for 5 cells. general const. - Ed. 17th. - M.: Mnemosyne, 2005.
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2. Igor had 45 rubles with him, Andrey had 28, and Denis had 17.
3. With all their money, they bought 3 movie tickets. How much did one ticket cost?

What is the arithmetic mean? How to find the arithmetic mean? Where and why is this value used?

To fully understand the essence of the problem, you need to study algebra for several years at school, and then at the institute. But in everyday life, in order to know how to find the arithmetic mean of numbers, it is not necessary to know everything about it thoroughly. In simple terms, this is the sum of numbers divided by the number of these summed numbers.

Since it is not always possible to calculate the arithmetic mean without a remainder, the value can even turn out to be fractional, even when calculating the average number of people. This is due to the fact that the arithmetic mean is an abstract concept.

This abstract value affects many areas of modern life. It is used in mathematics, business, statistics, often even in sports.

For example, many are interested in all members of a team or the average amount of food eaten per month in terms of one day. And data about how much was spent on average on any expensive event is found in all media sources. Most often, of course, such data are used in statistics: to know exactly which phenomenon has declined and which has increased; which product is most in demand and in what period; for ease of elimination of unwanted indicators.

In sports, we may come across the concept of an average when, for example, we are told the average age of athletes or goals scored in football. And how do they calculate the earned average score during the competition or at our beloved KVN? Yes, for this nothing else needs to be done, how to find the arithmetic mean of all the marks given by the judges!

By the way, often in school life, some teachers resort to a similar method, displaying quarterly and annual grades for their students. It is also often used in higher education institutions, often in schools, to calculate the average score of student performance in order to determine the effectiveness of a teacher or to distribute students according to their capabilities. There are still many areas of life in which this formula is used, but the goal is basically the same - to know and control.

In business, the arithmetic mean can be used to calculate and control income and losses, wages, and other expenses. For example, when submitting certificates to some organizations about income, just the average monthly for the last six months is required. Surprising is the fact that some employees whose responsibilities include collecting such information, having received a certificate not with average monthly earnings, but simply with income for six months, do not know how to find the arithmetic mean, that is, calculate the average monthly salary.

The arithmetic mean is a sign (price, wages, population, etc.), the volume of which does not change during the calculation. In simple words, when the average number of apples eaten by Petya and Masha is calculated, the number will be equal to half of the total number of apples. Even if Masha ate ten, and Petya got only one, then when we divide their total number in half, then we will get the arithmetic mean.

Today, many joke about Putin's statement that the average salary living in Russia is 27,000 rubles. The jokes of the wits mostly sound like this: “Or am I not a Russian? Or am I no longer living? And the whole question is just that these wits also, apparently, do not know how to find the arithmetic mean of the salaries of the inhabitants of Russia.

You just need to add up the incomes of oligarchs, business leaders, businessmen on the one hand and the salaries of cleaners, janitors, salesmen and conductors on the other. And then divide the amount received by the number of people whose incomes included this amount. So you get an amazing figure, which is expressed in 27,000 rubles.

The most common type of average is the arithmetic average.

simple arithmetic mean

The simple arithmetic mean is the average term, in determining which the total volume of a given attribute in the data is equally distributed among all units included in this population. Thus, the average annual output per worker is such a value of the volume of output that would fall on each employee if the entire volume of output was equally distributed among all employees of the organization. The arithmetic mean simple value is calculated by the formula:

simple arithmetic mean— Equal to the ratio of the sum of individual values ​​of a feature to the number of features in the aggregate

Find the average salary
Solution: (3 + 3.2 + 3.3 +3.5 + 3.8 + 3.1) / 6 = 3.32 thousand rubles.

Arithmetic weighted average

If the volume of the data set is large and represents a distribution series, then a weighted arithmetic mean is calculated. This is how the weighted average price per unit of production is determined: the total cost of production (the sum of the products of its quantity and the price of a unit of production) is divided by the total quantity of production.

We represent this in the form of the following formula:

Weighted arithmetic mean- is equal to the ratio (the sum of the products of the attribute value to the frequency of repetition of this attribute) to (the sum of the frequencies of all attributes). It is used when the variants of the studied population occur an unequal number of times.

The average wage can be obtained by dividing the total wage by the total number of workers:

Answer: 3.35 thousand rubles.

Arithmetic mean for an interval series

When calculating the arithmetic mean for an interval variation series, the average for each interval is first determined as the half-sum of the upper and lower limits, and then the average of the entire series. In the case of open intervals, the value of the lower or upper interval is determined by the value of the intervals adjacent to them.

Averages calculated from interval series are approximate.

Example 3. Determine the average age of students in the evening department.

Averages calculated from interval series are approximate. The degree of their approximation depends on the extent to which the actual distribution of population units within the interval approaches uniform.

When calculating averages, not only absolute, but also relative values ​​(frequency) can be used as weights:

The arithmetic mean has a number of properties that more fully reveal its essence and simplify the calculation:

1. The product of the average and the sum of the frequencies is always equal to the sum of the products of the variant and the frequencies, i.e.

2. The arithmetic mean of the sum of the varying values ​​is equal to the sum of the arithmetic means of these values:

3. The algebraic sum of the deviations of the individual values ​​of the attribute from the average is zero:

4. The sum of the squared deviations of the options from the mean is less than the sum of the squared deviations from any other arbitrary value, i.e.

The arithmetic mean is the sum of numbers divided by the number of these same numbers. Finding the arithmetic mean is very easy.

As follows from the definition, we must take the numbers, add them up and divide by their number.

Let's give an example: the numbers 1, 3, 5, 7 are given and we need to find the arithmetic mean of these numbers.

• first add these numbers (1+3+5+7) and get 16
• we need to divide the result obtained by 4 (number): 16/4 and we get the result 4.

So, the arithmetic mean of the numbers 1, 3, 5 and 7 is 4.

Arithmetic mean - the average value among the given indicators.

It is found by dividing the sum of all indicators by their number.

For example, I have 5 apples weighing 200, 250, 180, 220 and 230 grams.

The average weight of 1 apple is found as follows:

• we are looking for the total weight of all apples (the sum of all indicators) - it is 1080 grams,
• divide the total weight by the number of apples 1080:5 = 216 grams. This is the arithmetic mean.

This is the most commonly used indicator in statistics.

The arithmetic mean is the numbers added together and divided by their number, the answer is the arithmetic mean.

For example: Katya put 50 rubles in the piggy bank, Maxim 100 rubles, and Sasha put 150 rubles in the piggy bank. 50 + 100 + 150 = 300 rubles in the piggy bank, now we divide this amount by three (three people put money in). So 300: 3 = 100 rubles. These 100 rubles will be the arithmetic mean, each of them put in a piggy bank.

There is such a simple example: one person eats meat, another person eats cabbage, and the arithmetic mean they both eat cabbage rolls.

In the same way, the average salary is calculated ...

The arithmetic mean is the sum of all values ​​and divided by their number.

For example numbers 2, 3 , 5, 6 . You need to add them 2+ 3+ 5 + 6 = 16

Divide 16 by 4 and get the answer 4.

4 is the arithmetic mean of these numbers.

The arithmetic mean of several numbers is the sum of these numbers divided by their number.

x cf arithmetic mean

S sum of numbers

n number of numbers.

For example, we need to find the arithmetic mean of the numbers 3, 4, 5 and 6.

To do this, we need to add them up and divide the resulting amount by 4:

(3 + 4 + 5 + 6) : 4 = 18: 4 = 4,5.

I remember how I passed the final test in mathematics

So there it was necessary to find the arithmetic mean.

It's good that kind people suggested what to do, otherwise it's a disaster.

For example, we have 4 numbers.

We add the numbers and divide by their number (in this case 4)

For example, the numbers 2,6,1,1. Add 2+6+1+1 and divide by 4 = 2.5

As you can see, nothing complicated. So the arithmetic mean is the average of all numbers.

We know this from school. Whoever had a good math teacher could remember this simple action the first time.

When finding the arithmetic mean, it is necessary to add all the available numbers and divide by their number.

For example, I bought 1 kg of apples, 2 kg of bananas, 3 kg of oranges and 1 kg of kiwi in the store. How many kilograms on average I bought fruit.

7/4= 1.8 kilograms. This will be the arithmetic mean.

The arithmetic mean is the average of several numbers.

For example, between the numbers 2 and 4, the average number is 3.

The formula for finding the arithmetic mean is:

You need to add all the numbers and divide by the number of these numbers:

For example, we have 3 numbers: 2, 5 and 8.

Finding the arithmetic mean:

X=(2+5+8)/3=15/3=5

The scope of the arithmetic mean is quite wide.

For example, knowing the coordinates of two points of a segment, you can find the coordinates of the middle of this segment.

For example, the coordinates of the segment: (X1,Y1,Z1)-(X2,Y2,Z2).

We denote the middle of this segment by the coordinates X3,Y3,Z3.

Separately, we find the midpoint for each coordinate:

The arithmetic mean is the average of the given...

Those. we simply have the number of sticks of different lengths and want to know their average value ..

It is logical that for this we bring them together, getting a long stick, and then divide it into the required number of parts ..

Here comes the arithmetic mean.

This is how the formula is derived: Sa=(S(1)+..S(n))/n..

Arithmetic is considered the most elementary branch of mathematics and studies simple operations with numbers. Therefore, the arithmetic mean is also very easy to find. Let's start with a definition. The arithmetic mean is a value that shows which number is closest to the truth in several consecutive actions of the same type. For example, when running a hundred meters, a person shows a different time each time, but the average value will be within, for example, 12 seconds. Finding the arithmetic mean thus boils down to the sequential summation of all the numbers of a certain series (run results) and dividing this sum by the number of these runs (attempts, numbers). In formula form, it looks like this:

Sarif = (X1+X2+..+Xn)/n

As a mathematician, I am interested in questions on this subject.

I'll start with the history of the issue. Average values ​​have been thought about since ancient times. Arithmetic mean, geometic mean, harmonic mean. These concepts were proposed in ancient Greece by the Pythagoreans.

And now the question that interests us. What is meant by arithmetic mean of several numbers:

So, to find the arithmetic mean of numbers, you need to add all the numbers and divide the resulting amount by the number of terms.

There is a formula:



Example. Find the arithmetic mean of numbers: 100, 175, 325.

Let's use the formula for finding the arithmetic mean of three numbers (that is, instead of n there will be 3; you need to add all 3 numbers and divide the resulting amount by their number, i.e. by 3). We have: x=(100+175+325)/3=600/3=200.

What is the arithmetic mean

The arithmetic mean of several values ​​is the ratio of the sum of these values ​​to their number.

The arithmetic mean of a certain series of numbers is called the sum of all these numbers, divided by the number of terms. Thus, the arithmetic mean is the average value of the number series.

What is the arithmetic mean of several numbers? And they are equal to the sum of these numbers, which is divided by the number of terms in this sum.

How to find the arithmetic mean

There is nothing difficult in calculating or finding the arithmetic mean of several numbers, it is enough to add all the numbers presented, and divide the resulting amount by the number of terms. The result obtained will be the arithmetic mean of these numbers.




Let's consider this process in more detail. What do we need to do to calculate the arithmetic mean and get the final result of this number.

First, to calculate it, you need to determine a set of numbers or their number. This set can include large and small numbers, and their number can be anything.

Secondly, all these numbers need to be added up and get their sum. Naturally, if the numbers are simple and their number is small, then the calculations can be done by writing by hand. And if the set of numbers is impressive, then it is better to use a calculator or spreadsheet.

And, fourthly, the amount obtained from addition must be divided by the number of numbers. As a result, we get the result, which will be the arithmetic mean of this series.





What is the arithmetic mean for?

The arithmetic mean can be useful not only for solving examples and problems in mathematics lessons, but for other purposes necessary in a person’s daily life. Such goals can be the calculation of the arithmetic mean to calculate the average expense of finance per month, or to calculate the time you spend on the road, also in order to find out traffic, productivity, speed, productivity and much more.

So, for example, let's try to calculate how much time you spend commuting to school. Going to school or returning home, you spend different time on the road each time, because when you are in a hurry, you go faster, and therefore the road takes less time. But, returning home, you can go slowly, talking with classmates, admiring nature, and therefore it will take more time for the road.

Therefore, you will not be able to accurately determine the time spent on the road, but thanks to the arithmetic mean, you can approximately find out the time you spend on the road.

Let's say that on the first day after the weekend, you spent fifteen minutes on the way from home to school, on the second day your journey took twenty minutes, on Wednesday you covered the distance in twenty-five minutes, in the same time you made your way on Thursday, and on Friday you were in no hurry and returned for half an hour.

Let's find the arithmetic mean, adding the time, for all five days. So,

15 + 20 + 25 + 25 + 30 = 115

Now divide this amount by the number of days

Through this method, you have learned that the journey from home to school takes approximately twenty-three minutes of your time.

Homework

1. Using simple calculations, find the arithmetic average of the attendance of students in your class per week.

2. Find the arithmetic mean:





3. Solve the problem: