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
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 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.
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: .
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: .
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.
Solution
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?
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: (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?
The topic of arithmetic and geometric mean is included in the mathematics program for grades 6-7. Since the paragraph is quite simple to understand, it is quickly passed, and by the end of the school year, students forget it. But knowledge in basic statistics is needed to pass the exam, as well as for international SAT exams. And for everyday life, developed analytical thinking never hurts.
How to calculate the arithmetic and geometric mean of numbers
Suppose 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 numbers 11, 4, 3, the answer will be 6. How is 6 obtained?
Solution: (11 + 4 + 3) / 3 = 6
The denominator must contain a number equal to the number of numbers whose average is to be found. The sum is divisible by 3, since there are three terms.
Now we need to deal with the geometric mean. Let's say there is a series of numbers: 4, 2 and 8.
The geometric mean is the product of all given numbers, which is under a root with a degree equal to the number of given numbers. That is, in the case of numbers 4, 2 and 8, the answer is 4. Here's how it happened:
Solution: ∛(4 × 2 × 8) = 4
In both options, whole answers were obtained, since special numbers were taken as an example. This is not always the case. 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, respectively, will be 5.5 and √30.
Can 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 there cannot be.
The concept of arithmetic mean means the result of a simple sequence of calculations of the average value for a series of numbers determined in advance. It should be noted that this value is currently widely used by specialists in a number of industries. For example, formulas are known when performing calculations by economists or employees of 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 calculating this value is the simplicity of the procedure. Carry out calculations anyone can. You don't need any special education for this. Often there is no need to use computer technology.
As an answer to the question of how to find the arithmetic mean, consider a number of situations.
The simplest way to calculate this value is to calculate it for two numbers. The calculation procedure in this case is very simple:
- Initially, it is required to carry out the operation of adding the selected numbers. This can often be done, as they say, manually, without using electronic equipment.
- After the addition is made and its result is obtained, it is necessary to divide. 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 get 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 a value for three
The calculation of 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 is found, it is required 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 required when calculating the arithmetic three will look like this:
(A+B+C)/3
In this formula the following notation has been adopted:
A, B and C are the numbers to which it will be necessary to find the arithmetic mean.
Calculating the arithmetic mean of four
As already seen by analogy with the previous options, the calculation of this value for a quantity equal to four will be of the following order:
- Four digits are selected for which the arithmetic mean is to be calculated. Next, the summation and finding the final result of this procedure is carried out.
- 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 get the following formula:
(A+B+C+E)/4
In this formula variables have the following meaning:
A, B, C and E are those for which you need 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, you just need to add up 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 have the following notation:
A, B, C, E and P are the numbers for which you want to get the arithmetic mean.
Universal Calculation Formula
Carrying out consideration of various variants of 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 apply the general formula for finding the arithmetic mean. After all, there are situations when the number and size of calculations can be very large. Therefore, it would be wiser to use a universal formula and not deduce an individual technology each time to calculate this value.
The main thing in determining the formula is the principle of calculating the arithmetic mean about.
This principle, as it was seen from the above examples, looks like this:
- The number of numbers that are specified to obtain the required value is counted. This operation can be carried out both manually with a small number of numbers, and with the help of 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:
(А+В+…+N)/N
This formula contains the following variables:
A and B are numbers that are chosen in advance to calculate their arithmetic mean.
N is the number of numbers that were taken in order to calculate the required value.
Substituting the selected numbers into this formula each time, we can always get the required value of the arithmetic mean.
As seen, finding the arithmetic mean is an easy procedure. However, one must be attentive to the calculations and check the result obtained. This approach is explained by the fact that even in the simplest situations, there is a possibility of getting an error, which can then affect further calculations. In this regard, it is recommended to use computer technology that is capable of making calculations of any complexity.
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 up 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 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 attendance, 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.
Suppose 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:
In the calculation of the average value is lost.
Average meaning set of numbers is equal to the sum of the numbers S divided by the number of these numbers. That is, it turns out that average meaning equals: 19/4 = 4.75.
note
If you need to find the geometric mean for just two numbers, then you will not need an engineering calculator: you can extract the second degree root (square root) of any number using the most common calculator.
Useful advice
Unlike the arithmetic mean, the geometric mean is not so strongly influenced by large deviations and fluctuations between individual values in the studied set of indicators.
Sources:
- Online calculator that calculates the geometric mean
- geometric mean formula
Average value is one of the characteristics of a set of numbers. Represents a number that cannot be outside the range defined by the largest and smallest values in this set of numbers. Average arithmetic value - the most commonly used variety of averages.
Instruction
Add all the numbers in the set and divide them by the number of terms to get the arithmetic mean. Depending on the specific conditions of the calculation, it is sometimes easier to divide each of the numbers by the number of values in the set and sum the result.
Use, for example, included in the Windows operating system, if it is not possible to calculate the arithmetic mean in your mind. You can open it using the program launcher dialog. To do this, press the "hot keys" WIN + R or click the "Start" button and select the "Run" command from the main menu. Then type calc into the input field and press Enter or click the OK button. The same can be done through the main menu - open it, go to the "All Programs" section and in the "Standard" section and select the "Calculator" line.
Enter all the numbers in the set in succession by pressing the Plus key after each of them (except the last one) or by clicking the corresponding button in the calculator interface. You can also enter numbers both from the keyboard and by clicking the corresponding interface buttons.
Press the slash key or click this in the calculator interface after entering the last set value and print the number of numbers in the sequence. Then press the equal sign and the calculator will calculate and display the arithmetic mean.
You can use the spreadsheet editor Microsoft Excel for the same purpose. In this case, start the editor and enter all the values of the sequence of numbers into adjacent cells. If after entering each number you press Enter or the down or right arrow key, the editor itself will move the input focus to the adjacent cell.
Click the cell next to the last number you entered, if you don't want to just see the arithmetic mean. Expand the Greek sigma (Σ) dropdown of the Editing commands on the Home tab. Select the line " Average” and the editor will insert the desired formula for calculating the arithmetic mean in the selected cell. Press the Enter key and the value will be calculated.
The arithmetic mean is one of the measures of central tendency, widely used in mathematics and statistical calculations. Finding the arithmetic average of several values is very simple, but each task has its own nuances, which are simply necessary to know in order to perform correct calculations.
What is the arithmetic mean
The arithmetic mean determines the average value for the entire original array of numbers. In other words, from a certain set of numbers, a value common to all elements is selected, the mathematical comparison of which with all elements is approximately equal. The arithmetic mean is used primarily in the preparation of financial and statistical reports or for calculating the results of similar experiments.How to find the arithmetic mean
The search for the arithmetic mean for an array of numbers should begin with determining the algebraic sum of these values. For example, if the array contains the numbers 23, 43, 10, 74 and 34, then their algebraic sum will be 184. When writing, the arithmetic mean is denoted by the letter μ (mu) or x (x with a bar). Next, the algebraic sum should be divided by the number of numbers in the array. In this example, there were five numbers, so the arithmetic mean will be 184/5 and will be 36.8.Features of working with negative numbers
If there are negative numbers in the array, then the arithmetic mean is found using a similar algorithm. There is a difference only when calculating in the programming environment, or if there are additional conditions in the task. In these cases, finding the arithmetic mean of numbers with different signs comes down to three steps:1. Finding the common arithmetic mean by the standard method;
2. Finding the arithmetic mean of negative numbers.
3. Calculation of the arithmetic mean of positive numbers.
The responses of each of the actions are written separated by commas.
Natural and decimal fractions
If the array of numbers is represented by decimal fractions, the solution occurs according to the method of calculating the arithmetic mean of integers, but the result is reduced according to the requirements of the problem for the accuracy of the answer.When working with natural fractions, they should be reduced to a common denominator, which is multiplied by the number of numbers in the array. The numerator of the answer will be the sum of the given numerators of the original fractional elements.
- Engineering calculator.
Instruction
Keep in mind that in the general case, the geometric mean of numbers is found by multiplying these numbers and extracting from them the root of the degree that corresponds to the number of numbers. For example, if you need to find the geometric mean of five numbers, then you will need to extract the root of the degree from the product.
To find the geometric mean of two numbers, use the basic rule. Find their product, and then extract the square root from it, since the numbers are two, which corresponds to the degree of the root. For example, in order to find the geometric mean of the numbers 16 and 4, find their product 16 4=64. From the resulting number, extract the square root √64=8. This will be the desired value. Please note that the arithmetic mean of these two numbers is greater than and equal to 10. If the root is not taken completely, round the result to the desired order.
To find the geometric mean of more than two numbers, also use the basic rule. To do this, find the product of all the numbers for which you want to find the geometric mean. From the resulting product, extract the root of the degree equal to the number of numbers. For example, to find the geometric mean of the numbers 2, 4, and 64, find their product. 2 4 64=512. Since you need to find the result of the geometric mean of three numbers, extract the root of the third degree from the product. It is difficult to do this verbally, so use an engineering calculator. To do this, it has a button "x ^ y". Dial the number 512, press the "x^y" button, then dial the number 3 and press the "1/x" button, to find the value 1/3, press the "=" button. We get the result of raising 512 to the power of 1/3, which corresponds to the root of the third degree. Get 512^1/3=8. This is the geometric mean of the numbers 2.4 and 64.
Using an engineering calculator, you can find the geometric mean in another way. Find the log button on your keyboard. After that, take the logarithm for each of the numbers, find their sum and divide it by the number of numbers. From the resulting number, take the antilogarithm. This will be the geometric mean of the numbers. For example, in order to find the geometric mean of the same numbers 2, 4 and 64, make a set of operations on the calculator. Type the number 2, then press the log button, press the "+" button, type the number 4 and press log and "+" again, type 64, press log and "=". The result will be a number equal to the sum of the decimal logarithms of the numbers 2, 4 and 64. Divide the resulting number by 3, since this is the number of numbers by which the geometric mean is sought. From the result, take the antilogarithm by toggling the register key and use the same log key. The result is the number 8, this is the desired geometric mean.
