In mathematics, the arithmetic mean of numbers (or simply the mean) is the sum of all the numbers in a given set divided by the number of numbers. This is the most generalized and widespread concept of average value. As you already understood, to find you need to sum up all the numbers given to you, and divide the resulting result by the number of terms.
What is the arithmetic mean?
Let's look at an example.
Example 1. Given numbers: 6, 7, 11. You need to find their average value.
Solution.
First, let's find the sum of all these numbers.
Now divide the resulting sum by the number of terms. Since we have three terms, we will therefore divide by three.
Therefore, the average of the numbers 6, 7 and 11 is 8. Why 8? Yes, because the sum of 6, 7 and 11 will be the same as three eights. This can be clearly seen in the illustration.
The average is a bit like “evening out” a series of numbers. As you can see, the piles of pencils have become the same level.
Let's look at another example to consolidate the knowledge gained.
Example 2. Given numbers: 3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29. You need to find their arithmetic mean.
Solution.
Find the amount.
3 + 7 + 5 + 13 + 20 + 23 + 39 + 23 + 40 + 23 + 14 + 12 + 56 + 23 + 29 = 330
Divide by the number of terms (in this case - 15).
Therefore, the average value of this series of numbers is 22.
Now let's look at negative numbers. Let's remember how to summarize them. For example, you have two numbers 1 and -4. Let's find their sum.
1 + (-4) = 1 - 4 = -3
Knowing this, let's look at another example.
Example 3. Find the average value of a series of numbers: 3, -7, 5, 13, -2.
Solution.
Find the sum of numbers.
3 + (-7) + 5 + 13 + (-2) = 12
Since there are 5 terms, divide the resulting sum by 5.
Therefore, the arithmetic mean of the numbers 3, -7, 5, 13, -2 is 2.4.
In our time of technological progress, it is much more convenient to use computer programs to find the average value. Microsoft Office Excel is one of them. Finding the average in Excel is quick and easy. Moreover, this program is included in the Microsoft Office software package. Let's consider brief instructions, value using this program.
In order to calculate the average value of a series of numbers, you must use the AVERAGE function. The syntax for this function is:
= Average(argument1, argument2, ... argument255)
where argument1, argument2, ... argument255 are either numbers or cell references (cells refer to ranges and arrays).
To make it more clear, let’s try out the knowledge we have gained.
- Enter the numbers 11, 12, 13, 14, 15, 16 in cells C1 - C6.
- Select cell C7 by clicking on it. In this cell we will display the average value.
- Click on the Formulas tab.
- Select More Functions > Statistical to open
- Select AVERAGE. After this, a dialog box should open.
- Select and drag cells C1-C6 there to set the range in the dialog box.
- Confirm your actions with the "OK" button.
- If you did everything correctly, you should have the answer in cell C7 - 13.7. When you click on cell C7, the function (=Average(C1:C6)) will appear in the formula bar.
This feature is very useful for accounting, invoices, or when you just need to find the average of a very long series of numbers. Therefore, it is often used in offices and large companies. This allows you to maintain order in your records and makes it possible to quickly calculate something (for example, average monthly income). You can also use Excel to find the average value of a function.
The most common type of average is the arithmetic mean.
Simple arithmetic mean
A 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 the given population. Thus, the average annual output per employee is the amount of output that would be produced by each employee if the entire volume of output were equally distributed among all employees of the organization. The arithmetic mean simple value is calculated using the formula:
Example 1 . A team of 6 workers receives 3 3.2 3.3 3.5 3.8 3.1 thousand rubles per month.Simple arithmetic average— Equal to the ratio of the sum of individual values of a characteristic to the number of characteristics in the aggregate
Find average salary
Solution: (3 + 3.2 + 3.3 +3.5 + 3.8 + 3.1) / 6 = 3.32 thousand rubles.
Arithmetic average weighted
If the volume of the data set is large and represents a distribution series, then the 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 by the price of a unit of production) is divided by the total quantity of production.
Let's imagine this in the form of the following formula:
Example 2 . Find the average salary of workshop workers per monthWeighted arithmetic average— equal to the ratio of (the sum of the products of the value of a feature to the frequency of repetition of this feature) to (the sum of the frequencies of all features). It is used when variants of the population under study occur an unequal number of times.
The average salary can be obtained by dividing the total wages on total number workers:
Answer: 3.35 thousand rubles.
Arithmetic mean for interval series
When calculating the arithmetic mean for an interval variation series, first determine the mean for each interval as the half-sum of the upper and lower limits, and then the mean of the entire series. In the case of open intervals, the value of the lower or upper interval is determined by the size of the intervals adjacent to them.
Averages calculated from interval series are approximate.
Example 3. Define average age evening students.
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 uniformity.
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 calculations:
1. The product of the average by the sum of frequencies is always equal to the sum of the products of the variant by frequencies, i.e.
2. The arithmetic mean of the sum of varying quantities is equal to the sum of the arithmetic means of these quantities:
3. The algebraic sum of deviations of individual values of a characteristic from the average is zero:
4. The sum of squared deviations of options from the average is less than the sum of squared deviations from any other arbitrary value, i.e.
The concept of arithmetic average of numbers means the result of a simple sequence of calculations of the average value for a number 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 sum of three has been found, the division procedure must be performed 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 variables have next value:
A, B, C and E are those for which it is necessary to find the value of the arithmetic mean.
Using this formula, it will always be possible to calculate the required value for a given number of 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 the fact that they have a common 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 more reasonable to use a universal formula and not 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.
- first add these numbers (1+3+5+7) and get 16
- We need to divide the resulting result by 4 (quantity): 16/4 and get the result 4.
- we are looking for the total weight of all apples (the sum of all indicators) - it is equal to 1080 grams,
- divide the total weight by the number of apples 1080:5 = 216 grams. This is the arithmetic mean.
- 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?
The arithmetic mean is the sum of numbers divided by the number of these same numbers. And finding the arithmetic mean is very simple.
As follows from the definition, we must take the numbers, add them and divide by their number.
Let's give an example: we are given the numbers 1, 3, 5, 7 and we need to find the arithmetic mean of these numbers.
So the average arithmetic numbers 1, 3, 5 and 7 are 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.
We find the average weight of 1 apple as follows:
This is the most commonly used indicator in statistics.
The arithmetic mean is numbers added together and divided by their number, the resulting 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 arithmetically average, each of them put in the piggy bank.
There is such a simple example: one person eats meat, another person eats cabbage, and the arithmetically average they both eat cabbage rolls.
The average salary is calculated in the same way...
The arithmetic mean is the sum of all values and divided by their number.
For example the numbers 2, 3, 5, 6. You need to add them 2+ 3+ 5 + 6 = 16
We 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 avg 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 sum by 4:
(3 + 4 + 5 + 6) : 4 = 18: 4 = 4,5.
I remember taking the final test in mathematics
So there it was necessary to find the arithmetic mean.
Good that good people They told me what to do, otherwise there would be trouble.
For example, we have 4 numbers.
Add the numbers and divide by their number (in 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. Anyone who had a good math teacher could remember this simple action the first time.
When finding the arithmetic mean, you need to add up 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 at the store. How many kilograms of fruit did I buy on average?
7/4= 1.8 kilograms. This will be the arithmetic mean.
The arithmetic mean is the average number between 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 up 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 application of the arithmetic mean is quite wide.
For example, knowing the coordinates of two points on 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).
Let us denote the middle of this segment by coordinates X3,Y3,Z3.
We separately find the midpoint for each coordinate:
The arithmetic mean is the average of the given...
Those. Simply, we have a number of sticks of different lengths and want to find out 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 after several successive operations of the same type. For example, when running a hundred meters, a person shows every time different time, but the average value will be within, for example, 12 seconds. Finding the arithmetic mean in this way comes down to sequentially summing all the numbers in a certain series (race results) and dividing this sum by the number of these races (attempts, numbers). In formula form it looks like this:
Sarif = (Х1+Х2+..+Хn)/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, geometric mean, harmonic mean. These concepts are proposed in ancient Greece 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 sum by the number of terms.
The formula is:
Example. Find the arithmetic mean of the 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 up all 3 numbers and divide the resulting sum by their number, i.e. by 3). We have: x=(100+175+325)/3=600/3=200.
Three children went into the forest to pick berries. Eldest daughter found 18 berries, the middle one - 15, and the 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 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 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: 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.
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 clothes, we look at the temperature in this country in this moment. However, we won't know big picture weather. 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 players football team, that is, in order to establish 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 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 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.