How to find the average value of the volume. Averages in ordinal scale

In mathematics, the arithmetic mean of numbers (or simply the average) is the sum of all the numbers in a given set divided by their number. This is the most generalized and widespread concept of the average value. As you already understood, in order to find you need to sum up all the numbers given to you, and divide the result by the number of terms.

What is the arithmetic mean?

Let's look at an example.

Example 1. Numbers are given: 6, 7, 11. You need to find their average value.

Decision.

First, let's find the sum of all given numbers.

Now we divide the resulting sum by the number of terms. Since we have three terms, respectively, we will divide by three.

Therefore, the average of 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 is clearly seen in the illustration.

The average value is somewhat reminiscent of the "alignment" of a series of numbers. As you can see, the piles of pencils have become one level.

Consider another example to consolidate the knowledge gained.

Example 2 Numbers are given: 3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29. You need to find their arithmetic mean.

Decision.

We find the sum.

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 consider negative numbers. Let's remember how to sum them up. For example, you have two numbers 1 and -4. Let's find their sum.

1 + (-4) = 1 - 4 = -3

Knowing this, consider another example.

Example 3 Find the average value of a series of numbers: 3, -7, 5, 13, -2.

Decision.

Finding the sum of numbers.

3 + (-7) + 5 + 13 + (-2) = 12

Since there are 5 terms, we 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 software package from Microsoft Office. Let's consider a brief instruction, 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 mean ranges and arrays).

To make it clearer, let's test the knowledge 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 that, 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, in cell C7 you should have the answer - 13.7. When you click on cell C7, the function (=Average(C1:C6)) will be displayed in the formula bar.

It is very useful to use this function for accounting, invoices, or when you just need to find the average of a very long range of numbers. Therefore, it is often used in offices and large companies. This allows you to keep the records in order and makes it possible to quickly calculate something (for example, the average income per month). You can also use Excel to find the mean of a function.

Signs of units of statistical aggregates are different in their meaning, for example, the wages of workers of one profession of an enterprise are not the same for the same period of time, market prices for the same products are different, crop yields in the farms of the region, etc. Therefore, in order to determine the value of a feature characteristic of the entire population of units under study, average values are calculated.
average value – it is a generalizing characteristic of the set of individual values of some quantitative trait.

The population studied by a quantitative attribute consists of individual values; they are influenced by both general causes and individual conditions. In the average value, the deviations characteristic of the individual values are canceled out. The average, being a function of a set of individual values, represents the entire set with one value and reflects the common thing that is inherent in all its units.

The average calculated for populations consisting of qualitatively homogeneous units is called typical average. For example, you can calculate the average monthly salary of an employee of one or another professional group (miner, doctor, librarian). Of course, the levels of monthly wages of miners, due to the difference in their qualifications, length of service, hours worked per month and many other factors, differ from each other, and from the level of average wages. However, the average level reflects the main factors that affect the level of wages, and mutually offset the differences that arise due to the individual characteristics of the employee. The average wage reflects the typical level of wages for this type of worker. Obtaining a typical average should be preceded by an analysis of how this population is qualitatively homogeneous. If the population consists of separate parts, it should be divided into typical groups (average temperature in the hospital).

Average values used as characteristics for heterogeneous populations are called system averages. For example, the average value of gross domestic product (GDP) per capita, the average consumption of various groups of goods per person and other similar values that represent the general characteristics of the state as a single economic system.

The average should be calculated for populations consisting of a sufficiently large number of units. Compliance with this condition is necessary in order for the law of large numbers to come into force, as a result of which random deviations of individual quantities from the general trend cancel each other out.

Types of averages and methods for calculating them

The choice of the type of average is determined by the economic content of a certain indicator and the initial data. However, any average value must be calculated so that when it replaces each variant of the averaged feature, the final, generalizing, or, as it is commonly called, does not change. defining indicator, which is related to the average. For example, when replacing the actual speeds on separate sections of the path, their average speed should not change the total distance traveled by the vehicle in the same time; when replacing the actual wages of individual employees of the enterprise with the average wage, the wage fund should not change. Consequently, in each specific case, depending on the nature of the available data, there is only one true average value of the indicator that is adequate to the properties and essence of the socio-economic phenomenon under study.
The most commonly used are the arithmetic mean, harmonic mean, geometric mean, mean square, and mean cubic.
The listed averages belong to the class power average and are combined by the general formula:
,
where is the average value of the studied trait;
m is the exponent of the mean;
– current value (variant) of the averaged feature;
n is the number of features.
Depending on the value of the exponent m, the following types of power averages are distinguished:
at m = -1 – mean harmonic ;
at m = 0 – geometric mean ;
at m = 1 – arithmetic mean;
at m = 2 – root mean square ;
at m = 3 - average cubic.
When using the same initial data, the larger the exponent m in the above formula, the larger the value of the average value:
.
This property of power-law means to increase with an increase in the exponent of the defining function is called the rule of majorance of means.
Each of the marked averages can take two forms: simple and weighted.
The simple form of the middle applies when the average is calculated on primary (ungrouped) data. weighted form– when calculating the average for secondary (grouped) data.

Arithmetic mean

The arithmetic mean is used when the volume of the population is the sum of all individual values of the varying attribute. It should be noted that if the type of mean value is not specified, the arithmetic mean is assumed. Its logical formula is:

simple arithmetic mean calculated by ungrouped data according to the formula:
or ,
where are the individual values of the feature;
j is the serial number of the unit of observation, which is characterized by the value ;
N is the number of observation units (set size).
Example. In the lecture “Summary and grouping of statistical data”, the results of observing the work experience of a team of 10 people were considered. Calculate the average work experience of the workers of the brigade. 5, 3, 5, 4, 3, 4, 5, 4, 2, 4.

According to the formula of the arithmetic mean simple, one also calculates chronological averages, if the time intervals for which the characteristic values are presented are equal.
Example. The volume of products sold for the first quarter amounted to 47 den. units, for the second 54, for the third 65 and for the fourth 58 den. units The average quarterly turnover is (47+54+65+58)/4 = 56 den. units
If momentary indicators are given in the chronological series, then when calculating the average, they are replaced by half-sums of values at the beginning and end of the period.
If there are more than two moments and the intervals between them are equal, then the average is calculated using the formula for the average chronological

,
where n is the number of time points
When the data is grouped by attribute values (i.e., a discrete variational distribution series is constructed) with weighted arithmetic mean is calculated using either frequencies , or frequencies of observation of specific values of the feature , the number of which (k) is significantly less than the number of observations (N) .
,
,
where k is the number of groups of the variation series,
i is the number of the group of the variation series.
Since , and , we obtain the formulas used for practical calculations:
and
Example. Let's calculate the average length of service of the working teams for the grouped series.
a) using frequencies:

b) using frequencies:

When the data is grouped by intervals , i.e. are presented in the form of interval distribution series; when calculating the arithmetic mean, the middle of the interval is taken as the value of the feature, based on the assumption of a uniform distribution of population units in this interval. The calculation is carried out according to the formulas:
and
where is the middle of the interval: ,
where and are the lower and upper boundaries of the intervals (provided that the upper boundary of this interval coincides with the lower boundary of the next interval).

Example. Let us calculate the arithmetic mean of the interval variation series constructed from the results of a study of the annual wages of 30 workers (see the lecture "Summary and grouping of statistical data").
Table 1 - Interval variation series of distribution.

Intervals, UAH	Frequency, pers.	frequency,	The middle of the interval
600-700 700-800 800-900 900-1000 1000-1100 1100-1200	3 6 8 9 3 1	0,10 0,20 0,267 0,30 0,10 0,033	(600+700):2=650 (700+800):2=750 850 950 1050 1150	1950 4500 6800 8550 3150 1150	65 150 226,95 285 105 37,95

UAH or UAH
The arithmetic means calculated on the basis of the initial data and interval variation series may not coincide due to the uneven distribution of the attribute values within the intervals. In this case, for a more accurate calculation of the arithmetic weighted average, one should use not the middle of the intervals, but the arithmetic simple averages calculated for each group ( group averages). The average calculated from group means using a weighted calculation formula is called general average.
The arithmetic mean has a number of properties.
1. The sum of deviations of the variant from the mean is zero:
.
2. If all values of the option increase or decrease by the value A, then the average value increases or decreases by the same value A:

3. If each option is increased or decreased by B times, then the average value will also increase or decrease by the same number of times:
or
4. The sum of the products of the variant by the frequencies is equal to the product of the average value by the sum of the frequencies:

5. If all frequencies are divided or multiplied by any number, then the arithmetic mean will not change:

6) if in all intervals the frequencies are equal to each other, then the arithmetic weighted average is equal to the simple arithmetic average:
,
where k is the number of groups in the variation series.

Using the properties of the average allows you to simplify its calculation.
Suppose that all options (x) are first reduced by the same number A, and then reduced by a factor of B. The greatest simplification is achieved when the value of the middle of the interval with the highest frequency is chosen as A, and the value of the interval as B (for rows with the same intervals). The quantity A is called the origin, so this method of calculating the average is called way b ohm reference from conditional zero or way of moments.
After such a transformation, we obtain a new variational distribution series, the variants of which are equal to . Their arithmetic mean, called moment of the first order, is expressed by the formula and according to the second and third properties, the arithmetic mean is equal to the mean of the original version, reduced first by A, and then by B times, i.e. .
To receive real average(middle of the original row) you need to multiply the moment of the first order by B and add A:

The calculation of the arithmetic mean by the method of moments is illustrated by the data in Table. 2.
Table 2 - Distribution of employees of the enterprise shop by length of service

Work experience, years	Amount of workers	Interval midpoint
0 – 5 5 – 10 10 – 15 15 – 20 20 – 25 25 – 30	12 16 23 28 17 14	2,5 7,5 12,7 17,5 22,5 27,5	15 -10 -5 0 5 10	3 -2 -1 0 1 2	36 -32 -23 0 17 28

Finding the moment of the first order . Then, knowing that A = 17.5, and B = 5, we calculate the average work experience of the shop workers:
years

Average harmonic
As shown above, the arithmetic mean is used to calculate the average value of a feature in cases where its variants x and their frequencies f are known.
If the statistical information does not contain frequencies f for individual options x of the population, but is presented as their product , the formula is applied average harmonic weighted. To calculate the average, denote , whence . Substituting these expressions into the weighted arithmetic mean formula, we obtain the weighted harmonic mean formula:
,
where is the volume (weight) of the indicator attribute values in the interval with number i (i=1,2, …, k).

Thus, the harmonic mean is used in cases where it is not the options themselves that are subject to summation, but their reciprocals: .
In cases where the weight of each option is equal to one, i.e. individual values of the inverse feature occur once, apply simple harmonic mean:
,
where are individual variants of the inverse trait that occur once;
N is the number of options.
If there are harmonic averages for two parts of the population with a number of and, then the total average for the entire population is calculated by the formula:

and called weighted harmonic mean of the group means.

Example. Three deals were made during the first hour of trading on the currency exchange. Data on the amount of hryvnia sales and the hryvnia exchange rate against the US dollar are given in Table. 3 (columns 2 and 3). Determine the average exchange rate of the hryvnia against the US dollar for the first hour of trading.
Table 3 - Data on the course of trading on the currency exchange

The average dollar exchange rate is determined by the ratio of the amount of hryvnias sold in the course of all transactions to the amount of dollars acquired as a result of the same transactions. The total amount of the hryvnia sale is known from column 2 of the table, and the amount of dollars purchased in each transaction is determined by dividing the hryvnia sale amount by its exchange rate (column 4). A total of $22 million was purchased during three transactions. This means that the average hryvnia exchange rate for one dollar was
.
The resulting value is real, because his substitution of the actual hryvnia exchange rates in transactions will not change the total amount of sales of the hryvnia, which acts as defining indicator: mln. UAH
If the arithmetic mean was used for the calculation, i.e. hryvnia, then at the exchange rate for the purchase of 22 million dollars. UAH 110.66 million would have to be spent, which is not true.

Geometric mean
The geometric mean is used to analyze the dynamics of phenomena and allows you to determine the average growth rate. When calculating the geometric mean, the individual values of the attribute are relative indicators of dynamics, built in the form of chain values, as the ratio of each level to the previous one.
The geometric simple mean is calculated by the formula:
,
where is the sign of the product,
N is the number of averaged values.
Example. The number of registered crimes over 4 years increased by 1.57 times, including for the 1st - by 1.08 times, for the 2nd - by 1.1 times, for the 3rd - by 1.18 and for the 4th - 1.12 times. Then the average annual growth rate of the number of crimes is: , i.e. The number of registered crimes has grown by an average of 12% annually.

1,8
-0,8
0,2
1,0
1,4

1
3
4
1
1

3,24
0,64
0,04
1
1,96

3,24
1,92
0,16
1
1,96

To calculate the mean square weighted, we determine and enter in the table and. Then the average value of deviations of the length of products from a given norm is equal to:

The arithmetic mean in this case would be unsuitable, because as a result, we would get zero deviation.
The use of the root mean square will be discussed later in the exponents of variation.

In the process of various calculations and work with data, it is often necessary to calculate their average value. It is calculated by adding the numbers and dividing the total by their number. Let's find out how to calculate the average of a set of numbers using Microsoft Excel in various ways.

The easiest and most well-known way to find the arithmetic mean of a set of numbers is to use the special button on the Microsoft Excel ribbon. We select a range of numbers located in a column or line of a document. Being in the "Home" tab, click on the "Autosum" button, which is located on the ribbon in the "Editing" tool block. Select "Average" from the drop-down list.

After that, using the "AVERAGE" function, the calculation is made. In the cell under the selected column, or to the right of the selected row, the arithmetic mean of the given set of numbers is displayed.

This method is good for simplicity and convenience. But, it also has significant drawbacks. Using this method, you can calculate the average value of only those numbers that are arranged in a row in one column, or in one row. But, with an array of cells, or with scattered cells on a sheet, you cannot work using this method.

For example, if you select two columns and calculate the arithmetic mean using the above method, then the answer will be given for each column separately, and not for the entire array of cells.

Calculation with the Function Wizard

For cases where you need to calculate the arithmetic mean of an array of cells, or scattered cells, you can use the Function Wizard. It still uses the same AVERAGE function we know from the first calculation method, but it does it in a slightly different way.

We click on the cell where we want the result of calculating the average value to be displayed. Click on the "Insert Function" button, which is located to the left of the formula bar. Or, we type the combination Shift + F3 on the keyboard.

The Function Wizard starts. In the list of functions presented, we are looking for "AVERAGE". Select it and click on the "OK" button.

The arguments window for this function opens. Function arguments are entered into the "Number" fields. These can be both ordinary numbers and cell addresses where these numbers are located. If it is inconvenient for you to enter cell addresses manually, then you should click on the button located to the right of the data entry field.

After that, the function arguments window will collapse, and you can select the group of cells on the sheet that you take for calculation. Then, again click on the button to the left of the data entry field to return to the function arguments window.

If you want to calculate the arithmetic mean between the numbers in disparate groups of cells, then do the same steps as mentioned above in the "Number 2" field. And so on until all the desired groups of cells are selected.

After that, click on the "OK" button.

The result of calculating the arithmetic mean will be highlighted in the cell that you selected before starting the Function Wizard.

Formula bar

There is a third way to run the "AVERAGE" function. To do this, go to the Formulas tab. Select the cell in which the result will be displayed. After that, in the group of tools "Library of functions" on the ribbon, click on the button "Other functions". A list appears in which you need to sequentially go through the items "Statistical" and "AVERAGE".

Then, exactly the same function arguments window is launched, as when using the Function Wizard, the work in which we described in detail above.

The next steps are exactly the same.

Manual function entry

But, do not forget that you can always enter the "AVERAGE" function manually if you wish. It will have the following pattern: "=AVERAGE(cell_range_address(number); cell_range_address(number)).

Of course, this method is not as convenient as the previous ones, and requires certain formulas to be kept in the user's head, but it is more flexible.

Calculation of the average value by condition

In addition to the usual calculation of the average value, it is possible to calculate the average value by condition. In this case, only those numbers from the selected range that meet a certain condition will be taken into account. For example, if these numbers are greater or less than a specific value.

For these purposes, the AVERAGEIF function is used. Like the AVERAGE function, you can run it through the Function Wizard, from the formula bar, or by manually entering it into a cell. After the function arguments window has opened, you need to enter its parameters. In the "Range" field, enter the range of cells whose values will be used to determine the arithmetic mean. We do this in the same way as with the AVERAGE function.

And here, in the "Condition" field, we must specify a specific value, numbers greater or less than which will be involved in the calculation. This can be done using comparison signs. For example, we took the expression ">=15000". That is, only cells in the range containing numbers greater than or equal to 15000 will be taken for calculation. If necessary, instead of a specific number, you can specify the address of the cell in which the corresponding number is located.

The field "Averaging range" is optional. Entering data into it is required only when using cells with text content.

When all the data is entered, click on the "OK" button.

After that, the result of the calculation of the arithmetic average for the selected range is displayed in the pre-selected cell, with the exception of cells whose data do not meet the conditions.

As you can see, in Microsoft Excel there are a number of tools with which you can calculate the average value of a selected series of numbers. Moreover, there is a function that automatically selects numbers from a range that do not meet a user-defined criteria. This makes calculations in Microsoft Excel even more user-friendly.

The USE in mathematics is one of the most difficult tests for graduates. Many years of practice has shown that very often students make inaccuracies when calculating the last digit of a natural number. This topic itself is quite complex, as it requires special accuracy, attentiveness and developed logical thinking. To cope with such tasks without any problems, we recommend using the convenient Shkolkovo online service. On our site you will find everything you need to solve equations for finding the last non-zero digit of a number and improve your knowledge in related topics.

Pass the Unified State Exam with excellent marks together with Shkolkovo!

Our educational portal is built in such a way that it is most convenient for the graduate to prepare for the final certification. First, the student turns to the "Theoretical Reference" section: he remembers the rules for solving equations, refreshes important formulas in his memory that help find the last digit of a number. After that, he goes to the "Catalogs", where he finds many tasks of various levels of complexity. If there are any difficulties with any exercise, you can move it to "Favorites" in order to return to it later and solve it yourself or with the help of a teacher.

Shkolkovo specialists collected, systematized and presented materials on the topic in the most simple and understandable way. Thus, a large amount of information is absorbed in a short time. Students will be able to complete even those tasks that recently caused them great difficulties, including those where it is necessary to indicate several solutions.

To make the lessons as effective as possible, we recommend starting with the easiest examples. If they did not cause difficulties, do not waste time - move on to the tasks of the middle level, so you will identify your weaknesses, focus on the tasks that are most difficult for you and achieve great results. After daily classes for 1-2 weeks, you can even bring out the last digit of Pi in a couple of minutes. This task is quite common in the exam in mathematics.

The database of exercises on our portal is constantly updated and supplemented by teachers with great experience. Schoolchildren have a great opportunity to receive completely new tasks every day, and not get hung up on the same examples, as you often have to do when repeating from a school textbook.

Start classes on the Shkolkovo website today, and the result will not keep you waiting!

Education on our portal is available to everyone. In order for you to track your progress and receive new tasks created personally for you, register in the system. We wish you successful preparation!

In mathematics, the arithmetic mean of numbers (or simply the average) is the sum of all the numbers in a given set divided by their number. This is the most generalized and widespread concept of the average value. As you already understood, in order to find the average value, you need to sum up all the numbers given to you, and divide the result by the number of terms.

What is the arithmetic mean?

Let's look at an example.

Example 1. Numbers are given: 6, 7, 11. You need to find their average value.

Decision.

First, let's find the sum of all given numbers.

Now we divide the resulting sum by the number of terms. Since we have three terms, respectively, we will 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 is clearly seen in the illustration.

The average value is somewhat reminiscent of the "alignment" of a series of numbers. As you can see, the piles of pencils have become one level.

Consider another example to consolidate the knowledge gained.

Example 2 Numbers are given: 3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29. You need to find their arithmetic mean.

Decision.

We find the sum.

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 consider negative numbers. Let's remember how to sum them up. For example, you have two numbers 1 and -4. Let's find their sum.

1 + (-4) = 1 – 4 = -3

Knowing this, consider another example.

Example 3 Find the average value of a series of numbers: 3, -7, 5, 13, -2.

Decision.

Finding the sum of numbers.

3 + (-7) + 5 + 13 + (-2) = 12

Since there are 5 terms, we 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 software package from Microsoft Office. Consider a brief instruction on how to find the arithmetic mean using this program.

To make it clearer, let's test the knowledge 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 the drop down list.
Select AVERAGE. After that, 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, in cell C7 you should have the answer - 13.7. When you click on cell C7, the function (=Average(C1:C6)) will be displayed in the formula bar.

Average

This term has other meanings, see the average meaning.

Average(in mathematics and statistics) sets of numbers - the sum of all numbers divided by their number. It is one of the most common measures of central tendency.

It was proposed (along with the geometric mean and harmonic mean) by the Pythagoreans.

Special cases of the arithmetic mean are the mean (of the general population) and the sample mean (of samples).

Introduction

Denote the set of data X = (x 1 , x 2 , …, x n), then the sample mean is usually denoted by a horizontal bar over the variable (x ¯ (\displaystyle (\bar (x))) , pronounced " x with a dash").

The Greek letter μ is used to denote the arithmetic mean of the entire population. For a random variable for which a mean value is defined, μ is probability mean or the mathematical expectation of a random variable. If the set X is a collection of random numbers with a probability mean μ, then for any sample x i from this collection μ = E( x i) is the expectation of this sample.

In practice, the difference between μ and x ¯ (\displaystyle (\bar (x))) is that μ is a typical variable because you can see the sample rather than the entire population. Therefore, if the sample is represented randomly (in terms of probability theory), then x ¯ (\displaystyle (\bar (x))) (but not μ) 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:

X ¯ = 1 n ∑ i = 1 n x i = 1 n (x 1 + ⋯ + x n) . (\displaystyle (\bar (x))=(\frac (1)(n))\sum _(i=1)^(n)x_(i)=(\frac (1)(n))(x_ (1)+\cdots +x_(n)).)

If a X is a random variable, then the mathematical expectation X can be considered as the arithmetic mean of the values in repeated measurements of the quantity 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 n+ 1 numbers above average 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 number is equal to the average. The more n, the smaller the difference between the new and old averages.

Note that there are several other "means" available, including power-law mean, Kolmogorov mean, harmonic mean, arithmetic-geometric mean, and various weighted means (e.g., arithmetic-weighted mean, geometric-weighted mean, harmonic-weighted mean).

Examples

For three numbers, you need to add them and divide by 3:

x 1 + x 2 + x 3 3 . (\displaystyle (\frac (x_(1)+x_(2)+x_(3))(3)).)

For four numbers, you need to add them and divide by 4:

x 1 + x 2 + x 3 + x 4 4 . (\displaystyle (\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

For a continuously distributed value f (x) (\displaystyle f(x)) the arithmetic mean on the interval [ a ; b ] (\displaystyle ) is defined via a definite integral:

F (x) ¯ [ a ; b ] = 1 b − a ∫ a b f (x) d x (\displaystyle (\overline (f(x)))_()=(\frac (1)(b-a))\int _(a)^(b) f(x)dx)

Some problems of using the average

Lack of robustness

Main article: Robustness in statistics

Although the arithmetic mean is often used as means or central trends, this concept does not apply to robust statistics, which means that the arithmetic mean is heavily influenced by "large deviations". It is noteworthy that for distributions with a large skewness, the arithmetic mean may not correspond to the concept of “average”, and the values of the mean from robust statistics (for example, the median) may better describe the central trend.

The classic example is the calculation of the average income. The arithmetic mean can be misinterpreted as a median, which can lead to the conclusion that there are more people with more income than there really are. "Mean" income is interpreted in such a way that most people's incomes are close to this number. This "average" (in the sense of the arithmetic mean) income is higher than the income of most people, since a high income with a large deviation from the average makes the arithmetic mean strongly skewed (in contrast, the median income "resists" such a skew). However, this "average" income says nothing about the number of people near the median income (and says nothing about the number of people near the modal income). However, if the concepts of "average" and "majority" are taken lightly, then one can incorrectly conclude that most people have incomes higher than they actually are. For example, a report on the "average" net income in Medina, Washington, calculated as the arithmetic average of all annual net incomes of residents, will give a surprisingly high number due to Bill Gates. Consider the sample (1, 2, 2, 2, 3, 9). The arithmetic mean is 3.17, but five of the six values are below this mean.

Compound interest

Main article: ROI

If numbers multiply, but not fold, you need to use the geometric mean, not the arithmetic mean. Most often, this incident happens when calculating the return on investment in finance.

For example, if stocks fell 10% in the first year and rose 30% in the second year, then it is incorrect to calculate the "average" increase over these two years as the arithmetic mean (−10% + 30%) / 2 = 10%; the correct average in this case is given by the compound annual growth rate, from which the annual growth is only about 8.16653826392% ≈ 8.2%.

The reason for this is that percentages have a new starting point each time: 30% is 30% from a number less than the price at the beginning of the first year: if the stock started at $30 and fell 10%, it is worth $27 at the start of the second year. If the stock is up 30%, it is worth $35.1 at the end of the second year. The arithmetic average of this growth is 10%, but since the stock has only grown by $5.1 in 2 years, an average increase of 8.2% gives a final result of $35.1:

[$30 (1 - 0.1) (1 + 0.3) = $30 (1 + 0.082) (1 + 0.082) = $35.1]. If we use the arithmetic mean of 10% in the same way, we will not get the actual value: [$30 (1 + 0.1) (1 + 0.1) = $36.3].

Compound interest at the end of year 2: 90% * 130% = 117% , i.e. a total increase of 17%, and the average annual compound interest is 117% ≈ 108.2% (\displaystyle (\sqrt (117\%))\approx 108.2\%) , that is, an average annual increase of 8.2%.

Directions

Main article: Destination statistics

When calculating the arithmetic mean of some variable that changes cyclically (for example, phase or angle), special care should be taken. For example, the average of 1° and 359° would be 1 ∘ + 359 ∘ 2 = (\displaystyle (\frac (1^(\circ )+359^(\circ ))(2))=) 180°. This number is incorrect for two reasons.

First, angular measures are only defined for the range from 0° to 360° (or from 0 to 2π when measured in radians). Thus, the same pair of numbers could be written as (1° and −1°) or as (1° and 719°). The averages of each pair will be different: 1 ∘ + (− 1 ∘) 2 = 0 ∘ (\displaystyle (\frac (1^(\circ )+(-1^(\circ )))(2))=0 ^(\circ )) , 1 ∘ + 719 ∘ 2 = 360 ∘ (\displaystyle (\frac (1^(\circ )+719^(\circ ))(2))=360^(\circ )) .
Second, in this case, a value of 0° (equivalent to 360°) would be the geometrically best mean, since the numbers deviate less from 0° than from any other value (value 0° has the smallest variance). Compare:
- the number 1° deviates from 0° by only 1°;
- the number 1° deviates from the calculated average of 180° by 179°.

The average value for a cyclic variable, calculated according to the above formula, will be artificially shifted relative to the real average to the middle of the numerical range. Because of this, the average is calculated in a different way, namely, the number with the smallest variance (center point) is chosen as the average value. Also, instead of subtracting, modulo distance (i.e., circumferential distance) is used. For example, the modular distance between 1° and 359° is 2°, not 358° (on a circle between 359° and 360°==0° - one degree, between 0° and 1° - also 1°, in total - 2 °).

Weighted average - what is it and how to calculate it?

In the process of studying mathematics, students get acquainted with the concept of the arithmetic mean. In the future, in statistics and some other sciences, students are also faced with the calculation of other averages. What can they be and how do they differ from each other?

Averages: Meaning and Differences

Not always accurate indicators give an understanding of the situation. In order to assess this or that situation, it is sometimes necessary to analyze a huge number of figures. And then averages come to the rescue. They allow you to assess the situation in general.

Since school days, many adults remember the existence of the arithmetic mean. It is very easy to calculate - the sum of a sequence of n terms is divisible by n. That is, if you need to calculate the arithmetic mean in the sequence of values 27, 22, 34 and 37, then you need to solve the expression (27 + 22 + 34 + 37) / 4, since 4 values \u200b\u200bare used in the calculations. In this case, the desired value will be equal to 30.

Often, as part of the school course, the geometric mean is also studied. The calculation of this value is based on extracting the root of the nth degree from the product of n terms. If we take the same numbers: 27, 22, 34 and 37, then the result of the calculations will be 29.4.

The harmonic mean in a general education school is usually not the subject of study. However, it is used quite often. This value is the reciprocal of the arithmetic mean and is calculated as a quotient of n - the number of values and the sum 1/a 1 +1/a 2 +...+1/a n . If we again take the same series of numbers for calculation, then the harmonic will be 29.6.

Weighted Average: Features

However, all of the above values may not be used everywhere. For example, in statistics, when calculating some average values, the "weight" of each number used in the calculation plays an important role. The results are more revealing and correct because they take into account more information. This group of values is collectively referred to as the "weighted average". They are not passed at school, so it is worth dwelling on them in more detail.

First of all, it is worth explaining what is meant by the "weight" of a particular value. The easiest way to explain this is with a specific example. The body temperature of each patient is measured twice a day in the hospital. Of the 100 patients in different departments of the hospital, 44 will have a normal temperature - 36.6 degrees. Another 30 will have an increased value - 37.2, 14 - 38, 7 - 38.5, 3 - 39, and the remaining two - 40. And if we take the arithmetic mean, then this value in general for the hospital will be over 38 degrees! But almost half of the patients have a completely normal temperature. And here it would be more correct to use the weighted average, and the "weight" of each value will be the number of people. In this case, the result of the calculation will be 37.25 degrees. The difference is obvious.

In the case of weighted average calculations, the "weight" can be taken as the number of shipments, the number of people working on a given day, in general, anything that can be measured and affect the final result.

Varieties

The weighted average corresponds to the arithmetic average discussed at the beginning of the article. However, the first value, as already mentioned, also takes into account the weight of each number used in the calculations. In addition, there are also weighted geometric and harmonic values.

There is another interesting variety used in series of numbers. This is a weighted moving average. It is on its basis that trends are calculated. In addition to the values themselves and their weight, periodicity is also used there. And when calculating the average value at some point in time, values for previous time periods are also taken into account.

Calculating all these values is not that difficult, but in practice, only the usual weighted average is usually used.

Calculation methods

In the age of computerization, there is no need to manually calculate the weighted average. However, it would be useful to know the calculation formula so that you can check and, if necessary, correct the results obtained.

It will be easiest to consider the calculation on a specific example.

It is necessary to find out what is the average wage at this enterprise, taking into account the number of workers receiving a particular salary.

So, the calculation of the weighted average is carried out using the following formula:

x = (a 1 *w 1 +a 2 *w 2 +...+a n *w n)/(w 1 +w 2 +...+w n)

For example, the calculation would be:

x = (32*20+33*35+34*14+40*6)/(20+35+14+6) = (640+1155+476+240)/75 = 33.48

Obviously, there is no particular difficulty in manually calculating the weighted average. The formula for calculating this value in one of the most popular applications with formulas - Excel - looks like the SUMPRODUCT (series of numbers; series of weights) / SUM (series of weights) function.

How to find average value in excel?

how to find arithmetic mean in excel?

Vladimir09854

As easy as pie. In order to find the average value in excel, you only need 3 cells. In the first we write one number, in the second - another. And in the third cell, we will score a formula that will give us the average value between these two numbers from the first and second cells. If cell No. 1 is called A1, cell No. 2 is called B1, then in the cell with the formula you need to write like this:

This formula calculates the arithmetic mean of two numbers.

For the beauty of our calculations, we can highlight the cells with lines, in the form of a plate.

There is also a function in Excel itself to determine the average value, but I use the old-fashioned method and enter the formula I need. Thus, I am sure that Excel will calculate exactly as I need, and will not come up with some kind of rounding of its own.

M3sergey

This is very easy if the data is already entered into the cells. If you are just interested in a number, just select the desired range / ranges, and the value of the sum of these numbers, their arithmetic mean and their number will appear in the status bar at the bottom right.

You can select an empty cell, click on the triangle (drop-down list) "Autosum" and select "Average" there, after which you will agree with the proposed range for calculation, or choose your own.

Finally, you can use the formulas directly - click "Insert Function" next to the formula bar and cell address. The AVERAGE function is in the "Statistical" category, and takes as arguments both numbers and cell references, etc. There you can also choose more complex options, for example, AVERAGEIF - calculation of the average by condition.

Find average in excel is a fairly simple task. Here you need to understand whether you want to use this average value in some formulas or not.

If you need to get only the value, then it is enough to select the required range of numbers, after which excel will automatically calculate the average value - it will be displayed in the status bar, the heading "Average".

In the case when you want to use the result in formulas, you can do this:

1) Sum the cells using the SUM function and divide it all by the number of numbers.

2) A more correct option is to use a special function called AVERAGE. The arguments to this function can be numbers given sequentially, or a range of numbers.

Vladimir Tikhonov

circle the values that will be used in the calculation, click the "Formulas" tab, there you will see "AutoSum" on the left and next to it a downward-pointing triangle. click on this triangle and choose "Average". Voila, done) at the bottom of the column you will see the average value :)

Ekaterina Mutalapova

Let's start at the beginning and in order. What does average mean?

The mean value is the value that is the arithmetic mean, i.e. is calculated by adding a set of numbers and then dividing the total sum of numbers by their number. For example, for the numbers 2, 3, 6, 7, 2 it will be 4 (the sum of the numbers 20 is divided by their number 5)

In an Excel spreadsheet, for me personally, the easiest way was to use the formula =AVERAGE. To calculate the average value, you need to enter data into the table, write the function =AVERAGE() under the data column, and in brackets indicate the range of numbers in the cells, highlighting the column with the data. After that, press ENTER, or simply left-click on any cell. The result will be displayed in the cell below the column. On the face of it, the description is incomprehensible, but in fact it is a matter of minutes.

Adventurer 2000

The Excel program is multi-faceted, so there are several options that will allow you to find the average:

First option. You simply sum all the cells and divide by their number;

Second option. Use a special command, write in the required cell the formula "=AVERAGE (and here specify the range of cells)";

Third option. If you select the required range, then note that on the page below, the average value in these cells is also displayed.

Thus, there are a lot of ways to find the average value, you just need to choose the best one for you and use it constantly.

In Excel, using the AVERAGE function, you can calculate the simple arithmetic mean. To do this, you need to enter a number of values. Press equals and select in the Statistical category, among which select the AVERAGE function

Also, using statistical formulas, you can calculate the arithmetic weighted average, which is considered more accurate. To calculate it, we need the values of the indicator and the frequency.

How to find the average in Excel?

The situation is this. There is the following table:

The columns shaded in red contain the numerical values of the grades for the subjects. In the "Average" column, you need to calculate their average value.
The problem is this: there are 60-70 objects in total and some of them are on another sheet.
I looked in another document, the average has already been calculated, and in the cell there is a formula like
="sheet name"!|E12
but this was done by some programmer who got fired.
Tell me, please, who understands this.

Hector

In the line of functions, you insert "AVERAGE" from the proposed functions and choose from where they need to be calculated (B6: N6) for Ivanov, for example. I don’t know for sure about neighboring sheets, but for sure this is contained in the standard Windows help

Tell me how to calculate the average value in Word

Please tell me how to calculate the average value in Word. Namely, the average value of the ratings, and not the number of people who received ratings.

Yulia pavlova

Word can do a lot with macros. Press ALT+F11 and write a macro program..
In addition, Insert-Object... will allow you to use other programs, even Excel, to create a sheet with a table inside a Word document.
But in this case, you need to write down your numbers in the table column, and put the average in the bottom cell of the same column, right?
To do this, insert a field into the bottom cell.
Insert-Field...-Formula
Field content
[=AVERAGE(ABOVE)]
returns the average of the sum of the cells above.
If the field is selected and the right mouse button is pressed, then it can be Updated if the numbers have changed,
view the code or field value, change the code directly in the field.
If something goes wrong, delete the entire field in the cell and re-create it.
AVERAGE means average, ABOVE - about, that is, a row of cells above.
I did not know all this myself, but I easily found it in HELP, of course, thinking a little.