What does the arithmetic mean of numbers mean? How to find and calculate the arithmetic mean for two
Three children went into the forest to pick berries. The eldest daughter found 18 berries, the middle one - 15, and younger brother- 3 berries (see Fig. 1). They brought the berries to mom, who decided to divide the berries equally. How many berries did each child receive?
Rice. 1. Illustration for the problem
Solution
(Yag.) - children collected everything
2) Divide total berries per number of children:
(Yag.) went to every child
Answer: Each child will receive 12 berries.
In problem 1, the number obtained in the answer is the arithmetic mean.
Arithmetic mean several numbers is 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.
Solution
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 equal to: .
Answer: average arithmetic numbers 10 and 12 are the number 11.
Example 2
We have five numbers: 1, 2, 3, 4 and 5. Find their arithmetic mean.
Solution
1) The sum of these numbers is equal to: .
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 the fact that it is constantly suggested to be found in lessons, finding the arithmetic mean is very useful in Everyday life. For example, let's say we want to go on holiday to Greece. To choose suitable clothing, we look at the temperature in this country at the moment. However, we will not know the overall weather picture. Therefore, it is necessary to find out the air temperature in Greece, for example, for a week, and find the arithmetic average 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.
Solution
1) Let's calculate the sum of temperatures: .
2) Divide the resulting amount by the number of days: .
Answer: Average temperature for the week is approx.
The ability to find the arithmetic mean may also be needed to determine the average age of the players on a football team, that is, in order to determine whether the team is experienced or not. It is necessary to sum up the ages of all players and divide by their number.
Problem 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?
Solution
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: (rubles).
Therefore, the total amount of money earned is equal to: (rub.).
2) The total weight of apples sold is equal to: .
3) Divide the received amount of money by total weight sold apples and get the average price for 1 kg of apples: (rub.).
Answer: the average price of 1 kg of apples sold is 80 rubles.
The arithmetic mean helps evaluate the data as a whole, without taking each value separately.
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 time he hit a meter below. The arithmetic average will show that he hit the center exactly, although he missed both times.
Rice. 2. Illustration for example
In this lesson we learned about 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 practical use this concept.
- N.Ya. Vilenkin. Mathematics: textbook. for 5th grade. general education uchr. - 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?
What is the arithmetic mean
The arithmetic mean of several quantities is the ratio of the sum of these quantities to their number.
The arithmetic mean of a certain series of numbers is the sum of all these numbers divided by the number of terms. Thus, the arithmetic mean is the average value of a 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 complicated in calculating or finding the arithmetic mean of several numbers; it is enough to add all the numbers presented and divide the resulting sum by the number of terms. The result obtained will be the arithmetic mean of these numbers.
Let's look at this process in more detail. What do we need to do to calculate the arithmetic mean and obtain 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 and their sum is obtained. Naturally, if the numbers are simple and there are a small number of them, then the calculations can be made by writing them by hand. But 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 will get a result, which will be the arithmetic mean of this series.
Why do you need the arithmetic mean?
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 everyday life. Such goals can be calculating the arithmetic average to calculate the average financial expenditure per month, or to calculate the time you spend on the road, also in order to find out attendance, productivity, speed of movement, yield and much more.
So, for example, let's try to calculate how much time you spend traveling to school. Every time you go to school or return home, you spend on travel different time, because when you are in a hurry, you walk faster, and therefore the journey takes less time. But when returning home, you can walk slowly, communicating with classmates, admiring nature, and therefore the journey will take more time.
Therefore, you will not be able to accurately determine the time spent on the road, but thanks to the arithmetic average, you can approximately find out the time you spend on the road.
Let's assume 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, and your journey took the same amount of time on Thursday, and on Friday you were in no hurry and returned for a whole half an hour.
Let's find the arithmetic mean, adding time, for all five days. So,
15 + 20 + 25 + 25 + 30 = 115
Now divide this amount by the number of days
Thanks to this method, you 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 for the week.
2. Find the arithmetic mean:
3. Solve the problem:
The topic of arithmetic mean and geometric mean is included in the mathematics program for grades 6-7. Since the paragraph is quite easy to understand, it is quickly completed, and by the end school year schoolchildren forget him. But knowledge of basic statistics is needed for passing the Unified State Exam, and also for international exams SAT. And for everyday life, developed analytical thinking never hurts.
How to calculate the arithmetic mean and geometric mean of numbers
Let's say there is a series of numbers: 11, 4, and 3. The arithmetic mean is the sum of all numbers divided by the number of given numbers. That is, in the case of the numbers 11, 4, 3, the answer will be 6. How do you get 6?
Solution: (11 + 4 + 3) / 3 = 6
The denominator must contain a number equal to the number of numbers whose average needs to be found. The sum is divisible by 3, since there are three terms.
Now we need to figure out the geometric mean. Let's say there is a series of numbers: 4, 2 and 8.
The geometric mean of a number is the product of all given numbers under the root with the power equal to the amount given numbers. That is, in the case of numbers 4, 2 and 8, the answer will be 4. This is how it turned out:
Solution: ∛(4 × 2 × 8) = 4
In both options, we got whole answers, since special numbers were taken for the example. This does not always happen. In most cases, the answer has to be rounded or left at the root. For example, for the numbers 11, 7 and 20, the arithmetic mean is ≈ 12.67, and the geometric mean is ∛1540. And for the numbers 6 and 5, the answers will be 5.5 and √30, respectively.
Could it happen that the arithmetic mean becomes equal to the geometric mean?
Of course it can. But only in two cases. If there is a series of numbers consisting only of either ones or zeros. It is also noteworthy that the answer does not depend on their number.
Proof with units: (1 + 1 + 1) / 3 = 3 / 3 = 1 (arithmetic mean).
∛(1 × 1 × 1) = ∛1 = 1(geometric mean).
Proof with zeros: (0 + 0) / 2=0 (arithmetic mean).
√(0 × 0) = 0 (geometric mean).
There is no other option and cannot be.
The concept of arithmetic mean means the result of a simple sequence of calculations average size for a series of numbers determined in advance. It should be noted that this value in given time widely used by specialists in a number of industries. For example, formulas are known when carrying out calculations by economists or workers in the statistical industry, where it is required to have a value of this type. In addition, this indicator is actively used in a number of other industries that are related to the above.
One of the features of the calculations given value is the simplicity of the procedure. Carry out calculations Anyone can do it. To do this you don't need to have special education. Often there is no need to use computer technology.
To answer the question of how to find the arithmetic mean, consider a number of situations.
The most simple option calculating a given value is calculating it for two numbers. The calculation procedure in this case is very simple:
- Initially, you need to carry out the operation of adding the selected numbers. This can often be done, as they say, manually, without using electronic equipment.
- After addition is performed and its result is obtained, division must be performed. This operation involves dividing the sum of two added numbers by two - the number of added numbers. It is this action that will allow you to obtain the required value.
Formula
Thus, the formula for calculating the required value in the case of two will look like this:
(A+B)/2
This formula uses the following notation:
A and B are pre-selected numbers for which you need to find a value.
Finding the value for three
Calculating this value in a situation where three numbers are selected will not differ much from the previous option:
- To do this, select the numbers needed in the calculation and add them to get the total.
- After this amount three will be found, you need to perform the division procedure again. In this case, the resulting amount must be divided by three, which corresponds to the number of selected numbers.
Formula
Thus, the formula necessary for calculating the arithmetic three will look like this:
(A+B+C)/3
In this formula The following notation is accepted:
A, B and C are the numbers for which you will need to find the arithmetic mean.
Calculating the arithmetic mean of four
As can already be seen by analogy with the previous options, the calculation of this value for a quantity equal to four will be in the following order:
- Four digits are selected for which the arithmetic mean must be calculated. Next, summation is performed and the final result of this procedure is found.
- Now, to get the final result, you should take the resulting sum of four and divide it by four. The received data will be the required value.
Formula
From the sequence of actions described above for finding the arithmetic mean for four, you can obtain the following formula:
(A+B+C+E)/4
In this formula the variables have the following meaning:
A, B, C and E are those for which it is necessary to find the value of the arithmetic mean.
Applying this formula, it will always be possible to calculate the required value for given quantity numbers.
Calculating the arithmetic mean of five
Performing this operation will require a certain algorithm of actions.
- First of all, you need to select five numbers for which the arithmetic mean will be calculated. After this selection, these numbers, as in the previous options, just need to be added and get the final amount.
- The resulting amount will need to be divided by their number by five, which will allow you to get the required value.
Formula
Thus, similarly to the previously considered options, we obtain the following formula for calculating the arithmetic mean:
(A+B+C+E+P)/5
In this formula, the variables are designated as follows:
A, B, C, E and P are numbers for which it is necessary to obtain the arithmetic mean.
Universal calculation formula
Conducting a review various options formulas to calculate the arithmetic mean, you can pay attention to what they have general pattern.
Therefore, it will be more practical to use a general formula to find the arithmetic mean. After all, there are situations when the number and magnitude of calculations can be very large. Therefore it would be wiser to use universal formula and not to develop an individual technology each time to calculate this value.
The main thing when determining the formula is principle of calculating the arithmetic mean O.
This principle, as can be seen from the examples given, looks like this:
- The number of numbers that are specified to obtain the required value is counted. This operation can be carried out either manually with a small number of numbers or using computer technology.
- The selected numbers are summed. This operation in most situations is performed using computer technology, since numbers can consist of two, three or more digits.
- The amount obtained by adding the selected numbers must be divided by their number. This value is determined at the initial stage of calculating the arithmetic mean.
Thus, the general formula for calculating the arithmetic mean of a series of selected numbers will look like this:
(A+B+…+N)/N
This formula contains the following variables:
A and B are numbers that are selected in advance to calculate their arithmetic mean.
N is the number of numbers that were taken to calculate the required value.
By substituting the selected numbers into this formula each time, we can always obtain the required value of the arithmetic mean.
As seen, finding the arithmetic mean is a simple procedure. However, you must be careful about the calculations performed and check the results obtained. This approach is explained by the fact that even in the simplest situations there is a possibility of receiving an error, which can then affect further calculations. In this regard, it is recommended to use computer technology that is capable of performing calculations of any complexity.
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
- divide
- Number Mean (Mean), Arithmetic Mean (Arithmetic Mean) - an average value characterizing a 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 in a group differ significantly from the rest, this may distort the resulting arithmetic mean. Therefore, in this case, it is preferable to use the geometric mean (it is calculated in a similar way, but here the arithmetic mean of the logarithms of the observation values is determined, and then its antilogarithm is found) or - which is used most often - to find the mean value (median). from a series of quantities arranged in ascending order). Another method of 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: numbers 1 and 99, add and divide by two:
(1+99)/2=50 - arithmetic mean
If you take the numbers (1,2,3,15,59)/5=16 - the arithmetic mean, etc., etc. - The arithmetic mean (in mathematics and statistics) is one of the most common measures of central tendency, representing the sum of all recorded values divided by their number.
This term has other meanings, see average meaning.
The arithmetic mean (in mathematics and statistics) is one of the most common measures of central tendency, representing the sum of all recorded values divided by their number.Proposed (along with the geometric mean and harmonic mean) by the Pythagoreans 1.
Special cases of the arithmetic mean are the mean (general population) and the sample mean (sample).
A Greek letter is used to denote the arithmetic mean of the entire population. For a random variable for which the mean value is determined, there is a probabilistic mean or expected value random variable. If the set X is a collection of random numbers with a probabilistic mean, then for any sample xi from this population = E(xi) is the mathematical expectation of this sample.
In practice, the difference between and bar(x) is that it is a typical variable, because you can see a sample rather than the entire population. Therefore, if the sample is represented randomly (in terms of probability theory), then bar(x) , (but not) can be treated as a random variable having 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 expected value of X can be considered as the arithmetic mean of repeated measurements of X. This is a manifestation of the law large numbers. Therefore, the sample mean is used to estimate the unknown expected value.In elementary algebra, it is proven 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 equal to the average. The larger n, the smaller the difference between the new and old averages.
Note that there are several other averages, including the power average, the Kolmogorov average, the harmonic average, the arithmetic-geometric average, and various weighted averages.
Examples edit 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 simpler: 5+5=10, 10:2. Because we were adding 2 numbers, which means how many numbers we add, we divide by that many.Continuous random value edit edit wiki text
For a continuously distributed quantity f(x), the arithmetic mean on the segment a;b is determined through a definite integral: Some problems of using the mean Lack of robustness edit Main article: Robustness in statistics Although the arithmetic mean is often used as average values or central tendencies, this concept does not apply to Robust statistics, which means that the arithmetic mean is strongly influenced by large deviations. It is noteworthy that for distributions with a large skewness coefficient, the arithmetic mean - This is adding up the numbers and dividing them, how many were like this 33+66+99= adding up 33+66+99= 198 and dividing how many were read out, we have 3 numbers that are 33 66 and 99 and we need to divide what we got like this: 33+ 66+99=198:3=66 is the average orethmetic
- 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 divided by the number of numbers in this set.
The simplest case is to find the arithmetic mean of two numbers x1 and x2. Then their arithmetic mean is 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 in the form: X = (1/n)xi, where the summation is carried out over 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
The situation of interest is when the set of numbers represents the terms arithmetic progression. As is known, the terms of an arithmetic progression are equal to a1+(n-1)d, where d is the progression step, and n is the number of the progression term.Let a1, a1+d, a1+2d,...a1+(n-1)d be terms of an arithmetic progression. Their arithmetic mean is equal to 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
- this is when you add everything up and divide it
- If I'm not mistaken, this is when you add up 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!!! !
In your own words!
The arithmetic mean is the average value between two values.... It is found as the sum of numbers divided by the number... Or simply, if two numbers are around someone’s number (or rather, there is some number in order between them), then this number will be the average. ar. !6 + 8... av ar = 7
- divider gygygygygygyggy
- The average between maximum and minimum (all numerical indicators are added up and divided by their number
) - this is when you add up numbers and divide by the number of numbers