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.
What is the arithmetic mean?
- The arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by the number of terms
- share
- Number Average (Mean), Arithmetic Mean (Arithmetic Mean) - the average value characterizing any group of observations; is calculated by adding the numbers from this series and then dividing the resulting sum by the number of summed numbers. If one or more numbers included in the group differ significantly from the rest, then this can lead to a distortion of the resulting arithmetic mean. Therefore, in this case, it is preferable to use the geometric mean (geometric mean) (it is calculated in a similar way, but here the arithmetic mean of the logarithms of the values of the observations is determined, and then its antilogarithm is found) or - which is most often used - to find the median (average value from a series of values arranged in ascending order). Another method for obtaining the average value of any value from a group of observations is to determine the mode (mode) - an indicator (or set of indicators) that evaluates the most frequent manifestations of any variable; more often this method is used to determine the average value in several series of experiments.
For example: the numbers 1 and 99, add and divide by two:
(1+99)/2=50 - arithmetic mean
If we take the numbers (1,2,3,15,59) / 5 \u003d 16 - the arithmetic mean, etc., etc. - The arithmetic mean (in mathematics and statistics) is one of the most common measures of central tendency, which is the sum of all fixed values divided by their number.
This term has other meanings, see the average meaning.
The arithmetic mean (in mathematics and statistics) is one of the most common measures of central tendency, which is the sum of all fixed values divided by their number.It was proposed (along with the geometric mean and the harmonic mean) by the Pythagoreans 1.
Special cases of the arithmetic mean are the mean (of the general population) and the sample mean (of samples).
The Greek letter is used to denote the arithmetic mean of the entire population. For a random variable for which the mean value is defined, there is a probabilistic mean or mathematical expectation of the random variable. If the set X is a collection of random numbers with a probability mean, then for any sample xi from this population = E(xi) is the expectation of this sample.
In practice, the difference between and bar(x) is what is a typical variable, because you can see the sample rather than the entire population. Therefore, if the sample is presented randomly (in terms of probability theory), then bar(x) , (but not) can be treated as a random variable that has a probability distribution on the sample (probability distribution of the mean).
Both of these quantities are calculated in the same way:
bar(x) = frac(1)(n)sum_(i=1)^n x_i = frac(1)(n) (x_1+cdots+x_n).
If X is a random variable, then the expectation of X can be thought of as the arithmetic mean of the values in repeated measurements of X. This is a manifestation of the law of large numbers. Therefore, the sample mean is used to estimate the unknown mathematical expectation.In elementary algebra, it is proved that the average of n + 1 numbers is greater than the average of n numbers if and only if the new number is greater than the old average, less if and only if the new number is less than the average, and does not change if and only if the new the number is the average. The larger n, the smaller the difference between the new and old averages.
Note that there are several other means, including the power mean, Kolmogorov mean, harmonic mean, arithmetic geometric mean, and various weighted mean.
Examples edit wiki text
For three numbers, you need to add them and divide by 3:
frac(x_1 + x_2 + x_3)(3).
For four numbers, you need to add them and divide by 4:
frac(x_1 + x_2 + x_3 + x_4)(4).
Or easier 5+5=10, 10:2. Because we added 2 numbers, which means that how many numbers we add, we divide by that much.Continuous random variable edit wiki text
For a continuously distributed value f(x), the arithmetic mean over the interval a;b is defined in terms of the definite integral: Some problems in the application of the mean Lack of robustness robust statistics, which means that the arithmetic mean is strongly influenced by large deviations. It is noteworthy that for distributions with large skewness, the arithmetic mean - You add up the numbers and divide how many of them it was like this 33 + 66 + 99 = add up 33 + 66 + 99 = 198 and divide how many were read out for us 3 numbers are 33 66 and 99 and we need what we managed to divide like this: 33+ 66+99=198:3=66 is the orphmetic mean
- well, it's like 2+8=10 and the average is 5
- The arithmetic mean of a set of numbers is defined as their sum divided by their number. That is, the sum of all the numbers in a set is divisible by the number of numbers in that set.
The simplest case is to find the arithmetic mean of two numbers x1 and x2. Then their arithmetic mean X = (x1+x2)/2. For example, X = (6+2)/2 = 4 is the arithmetic mean of the numbers 6 and 2.
2
The general formula for finding the arithmetic mean of n numbers will look like this: X = (x1+x2+...+xn)/n. It can also be written as: X = (1/n)xi, where the summation is over the index i from i = 1 to i = n.For example, the arithmetic mean of three numbers X = (x1+x2+x3)/3, five numbers - (x1+x2+x3+x4+x5)/5.
3
Of interest is the situation where the set of numbers are members of an arithmetic progression. As you know, the members of an arithmetic progression are equal to a1+(n-1)d, where d is the step of the progression, and n is the number of the progression member.Let a1, a1+d, a1+2d,...a1+(n-1)d be members of an arithmetic progression. Their arithmetic mean is S = (a1+a1+d+a1+2d+...+a1+(n-1)d)/n = (na1+d+2d+...+(n-1)d)/n = a1+(d+2d+...+(n-2)d+(n-1)d)/n = a1+(d+2d+...+dn-d+dn-2d)/n = a1+(n* d*(n-1)/2)/n = a1+dn/2 = (2a1+d(n-1))/2 = (a1+an)/2. Thus, the arithmetic mean of the members of an arithmetic progression is equal to the arithmetic mean of its first and last members.
4
The property is also true that each member of an arithmetic progression is equal to the arithmetic mean of the previous and subsequent members of the progression: an = (a(n-1)+a(n+1))/2, where a(n-1), an, a( n+1) are consecutive members of the sequence. - Divide the sum of the numbers by their number
- when you add and divide everything
- If I'm not mistaken, this is when you add the sum of numbers and divide by the number of numbers themselves ...
- this is when you have several numbers, you add them up, and then divide by their number! let's say 25 24 65 76, add: 25+24+65+76:4=arithmetic mean!
- Vyachaslav Bogdanov answered incorrectly!!! !
Do with your words!
The arithmetic mean is the average value between two values .... It is found as the sum of numbers divided by their number ... . Or simply, if two numbers are around some number (or rather, there is some number between them in order), then this number will be cf. are. !6 + 8... cf ar = 7
- divisor gygygygygygygy
- The average between the maximum and minimum (all numerical indicators are added up and divided by their number
) - when you add the numbers and divide by the number of numbers
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:
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 may also be needed to determine the average age of the players of 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.
- N.Ya. Vilenkin. Mathematics: textbook. for 5 cells. general const. - Ed. 17th. - M.: Mnemosyne, 2005. )
- Igor had 45 rubles with him, Andrey had 28, and Denis had 17.
- With all their money, they bought 3 movie tickets. How much did one ticket cost?
