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 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.
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.
In order to find the average value in Excel (whether it is a numerical, textual, percentage or other value), there are many functions. And each of them has its own characteristics and advantages. After all, certain conditions can be set in this task.
For example, the average values of a series of numbers in Excel are calculated using statistical functions. You can also manually enter your own formula. Let's consider various options.
How to find the arithmetic mean of numbers?
To find the arithmetic mean, you add all the numbers in the set and divide the sum by the number. For example, a student's grades in computer science: 3, 4, 3, 5, 5. What goes for a quarter: 4. We found the arithmetic mean using the formula: \u003d (3 + 4 + 3 + 5 + 5) / 5.
How to do it quickly using Excel functions? Take for example a series of random numbers in a string:
Or: make the cell active and simply manually enter the formula: =AVERAGE(A1:A8).
Now let's see what else the AVERAGE function can do.
Find the arithmetic mean of the first two and last three numbers. Formula: =AVERAGE(A1:B1;F1:H1). Result:
Average by condition
The condition for finding the arithmetic mean can be a numerical criterion or a text one. We will use the function: =AVERAGEIF().
Find the arithmetic mean of numbers that are greater than or equal to 10.
Function: =AVERAGEIF(A1:A8,">=10")
The result of using the AVERAGEIF function on the condition ">=10": The third argument - "Averaging range" - is omitted. First, it is not required. Secondly, the range parsed by the program contains ONLY numeric values. In the cells specified in the first argument, the search will be performed according to the condition specified in the second argument.
Attention! The search criterion can be specified in a cell. And in the formula to make a reference to it.
Let's find the average value of the numbers by the text criterion. For example, the average sales of the product "tables".
The function will look like this: =AVERAGEIF($A$2:$A$12;A7;$B$2:$B$12). Range - a column with product names. The search criterion is a link to a cell with the word "tables" (you can insert the word "tables" instead of the link A7). Averaging range - those cells from which data will be taken to calculate the average value.
As a result of calculating the function, we obtain the following value:
Attention! For a text criterion (condition), the averaging range must be specified.
How to calculate the weighted average price in Excel?
How do we know the weighted average price?
Formula: =SUMPRODUCT(C2:C12,B2:B12)/SUM(C2:C12).
Using the SUMPRODUCT formula, we find out the total revenue after the sale of the entire quantity of goods. And the SUM function - sums up the quantity of goods. By dividing the total revenue from the sale of goods by the total number of units of goods, we found the weighted average price. This indicator takes into account the "weight" of each price. Its share in the total mass of values.
Standard deviation: formula in Excel
Distinguish between the standard deviation for the general population and for the sample. In the first case, this is the root of the general variance. In the second, from the sample variance.
To calculate this statistical indicator, a dispersion formula is compiled. The root is taken from it. But in Excel there is a ready-made function for finding the standard deviation.
The standard deviation is linked to the scale of the source data. This is not enough for a figurative representation of the variation of the analyzed range. To get the relative level of scatter in the data, the coefficient of variation is calculated:
standard deviation / arithmetic mean
The formula in Excel looks like this:
STDEV (range of values) / AVERAGE (range of values).
The coefficient of variation is calculated as a percentage. Therefore, we set the percentage format in the cell.
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.
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 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 a 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 recorded 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 recorded 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 mean of n + 1 numbers is greater than the mean of n numbers if and only if the new number is greater than the old mean, less if and only if the new number is less than the mean, 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 by 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
