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.

1. Enter the numbers 11, 12, 13, 14, 15, 16 in cells C1 - C6.
2. Select cell C7 by clicking on it. In this cell, we will display the average value.
3. Click on the "Formulas" tab.
4. Select More Functions > Statistical to open
5. Select AVERAGE. After that, a dialog box should open.
6. Select and drag cells C1-C6 there to set the range in the dialog box.
7. Confirm your actions with the "OK" button.
8. 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.

How to calculate the average of numbers in Excel

You can find the arithmetic mean of numbers in Excel using the function.

Syntax AVERAGE

=AVERAGE(number1,[number2],…) - Russian version

Arguments AVERAGE

• number1- the first number or range of numbers, for calculating the arithmetic mean;
• number2(Optional) – second number or range of numbers to calculate the arithmetic mean. The maximum number of function arguments is 255.

To calculate, do the following steps:

• Select any cell;
• Write a formula in it =AVERAGE(
• Select the range of cells for which you want to make a calculation;
• Press the "Enter" key on the keyboard

The function will calculate the average value in the specified range among those cells that contain numbers.

How to find the average value given text

If there are empty lines or text in the data range, then the function treats them as "zero". If there are logical expressions FALSE or TRUE among the data, then the function perceives FALSE as “zero”, and TRUE as “1”.

How to find the arithmetic mean by condition

The function is used to calculate the average by a condition or criterion. For example, let's say we have product sales data:

Our task is to calculate the average sales of pens. To do this, we will take the following steps:

• In a cell A13 write the name of the product “Pens”;
• In a cell B13 let's enter the formula:

=AVERAGEIF(A2:A10,A13,B2:B10)

Cell range “ A2:A10” points to the list of products in which we will search for the word “Pens”. Argument A13 this is a link to a cell with text that we will search for among the entire list of products. Cell range “ B2:B10” is a range with product sales data, among which the function will find “Pens” and calculate the average value.

The average value is a generalizing indicator that characterizes the typical level of the phenomenon. It expresses the value of the attribute, related to the unit of the population.

The average value is:

1) the most typical value of the attribute for the population;

2) the volume of the sign of the population, distributed equally among the units of the population.

The characteristic for which the average value is calculated is called “averaged” in statistics.

The average always generalizes the quantitative variation of the trait, i.e. in average values, individual differences in the units of the population due to random circumstances are canceled out. In contrast to the average, the absolute value that characterizes the level of a feature of an individual unit of the population does not allow comparing the values ​​of the feature for units belonging to different populations. So, if you need to compare the levels of remuneration of workers at two enterprises, then you cannot compare two employees of different enterprises on this basis. The wages of the workers selected for comparison may not be typical for these enterprises. If we compare the size of wage funds at the enterprises under consideration, then the number of employees is not taken into account and, therefore, it is impossible to determine where the level of wages is higher. Ultimately, only averages can be compared, i.e. How much does one worker earn on average in each company? Thus, there is a need to calculate the average value as a generalizing characteristic of the population.

It is important to note that in the process of averaging, the aggregate value of the attribute levels or its final value (in the case of calculating average levels in a time series) must remain unchanged. In other words, when calculating the average value, the volume of the trait under study should not be distorted, and the expressions made when calculating the average must necessarily make sense.

Calculating the average is one common generalization technique; the average indicator denies the general that is typical (typical) for all units of the studied population, at the same time it ignores the differences between individual units. In every phenomenon and its development there is a combination of chance and necessity. When calculating averages, due to the operation of the law of large numbers, randomness cancels each other out, balances out, so you can abstract from the insignificant features of the phenomenon, from the quantitative values ​​of the attribute in each specific case. In the ability to abstract from the randomness of individual values, fluctuations, lies the scientific value of averages as generalizing characteristics of aggregates.

In order for the average to be truly typifying, it must be calculated taking into account certain principles.

Let us dwell on some general principles for the application of averages.

1. The average should be determined for populations consisting of qualitatively homogeneous units.

2. The average should be calculated for a population consisting of a sufficiently large number of units.

3. The average should be calculated for the population, the units of which are in a normal, natural state.

4. The average should be calculated taking into account the economic content of the indicator under study.

5.2. Types of averages and methods for calculating them

Let us now consider the types of averages, the features of their calculation and areas of application. Average values ​​are divided into two large classes: power averages, structural averages.

Power-law averages include the most well-known and commonly used types, such as geometric mean, arithmetic mean, and mean square.

The mode and median are considered as structural averages.

Let us dwell on power averages. Power averages, depending on the presentation of the initial data, can be simple and weighted. simple average is calculated from ungrouped data and has the following general form:

,

where X i is the variant (value) of the averaged feature;

n is the number of options.

Weighted average is calculated by grouped data and has a general form

,

where X i is the variant (value) of the averaged feature or the middle value of the interval in which the variant is measured;

m is the exponent of the mean;

f i - frequency showing how many times the i-e value of the averaged feature occurs.

If we calculate all types of averages for the same initial data, then their values ​​will not be the same. Here the rule of majorance of averages applies: with an increase in the exponent m, the corresponding average value also increases:

In statistical practice, more often than other types of weighted averages, arithmetic and harmonic weighted averages are used.

Types of Power Means

Type of power middle	Indicator degrees (m)	Calculation formula
		Simple	weighted
harmonic
Geometric
Arithmetic
quadratic
cubic

The harmonic mean has a more complex structure than the arithmetic mean. The harmonic mean is used for calculations when the weights are not the units of the population - the carriers of the trait, but the products of these units and the values ​​of the trait (i.e. m = Xf). The average harmonic downtime should be used in cases of determining, for example, the average costs of labor, time, materials per unit of output, per part for two (three, four, etc.) enterprises, workers engaged in the manufacture of the same type of product , the same part, product.

The main requirement for the formula for calculating the average value is that all stages of the calculation have a real meaningful justification; the resulting average value should replace the individual values ​​of the attribute for each object without breaking the connection between individual and summary indicators. In other words, the average value should be calculated in such a way that when each individual value of the averaged indicator is replaced by its average value, some final summary indicator connected in one way or another with the averaged indicator remains unchanged. This result is called determining since the nature of its relationship with individual values ​​determines the specific formula for calculating the average value. Let's show this rule on the example of the geometric mean.

Geometric mean formula

most often used when calculating the average value of individual relative values ​​of the dynamics.

The geometric mean is used if a sequence of chain relative values ​​of dynamics is given, indicating, for example, an increase in production compared to the level of the previous year: i 1 , i 2 , i 3 ,…, i n . Obviously, the volume of production in the last year is determined by its initial level (q 0) and subsequent growth over the years:

q n =q 0 × i 1 × i 2 ×…×i n .

Taking q n as a defining indicator and replacing the individual values ​​of the dynamics indicators with average ones, we arrive at the relation

From here

A special type of average values ​​- structural averages - is used to study the internal structure of the series of distribution of attribute values, as well as to estimate the average value (power type), if, according to the available statistical data, its calculation cannot be performed (for example, if there were no data in the considered example). and on the volume of production, and on the amount of costs by groups of enterprises).

Indicators are most often used as structural averages. fashion - the most frequently repeated feature value - and median - the value of a feature that divides the ordered sequence of its values ​​into two parts equal in number. As a result, in one half of the population units, the value of the attribute does not exceed the median level, and in the other half it is not less than it.

If the feature under study has discrete values, then there are no particular difficulties in calculating the mode and median. If the data on the values ​​of the attribute X are presented in the form of ordered intervals of its change (interval series), the calculation of the mode and median becomes somewhat more complicated. Since the median value divides the entire population into two parts equal in number, it ends up in one of the intervals of the feature X. Using interpolation, the median value is found in this median interval:

,

where X Me is the lower limit of the median interval;

h Me is its value;

(Sum m) / 2 - half of the total number of observations or half of the volume of the indicator that is used as a weighting in the formulas for calculating the average value (in absolute or relative terms);

S Me-1 is the sum of observations (or the volume of the weighting feature) accumulated before the beginning of the median interval;

m Me is the number of observations or the volume of the weighting feature in the median interval (also in absolute or relative terms).

When calculating the modal value of a feature according to the data of the interval series, it is necessary to pay attention to the fact that the intervals are the same, since the indicator of the frequency of feature values ​​X depends on this. For an interval series with equal intervals, the mode value is determined as

,

where X Mo is the lower value of the modal interval;

m Mo is the number of observations or the volume of the weighting feature in the modal interval (in absolute or relative terms);

m Mo-1 - the same for the interval preceding the modal;

m Mo+1 - the same for the interval following the modal;

h is the value of the interval of change of the trait in groups.

TASK 1

The following data are available for the group of industrial enterprises for the reporting year

№ enterprises	Production volume, million rubles		Average number of employees, pers.	Profit, thousand rubles
	197,7	10,0		13,5
		22,8	1500	136,2
	465,5	18,4	1412	97,6
	296,2	12,6	1200	44,4
	584,1	22,0	1485	146,0
	480,0	119,0	1420	110,4
	57805	21,6	1390	138,7
	204,7			30,6
	466,8	19,4	1375	111,8
	292,2	113,6	1200	49,6
	423,1	17,6	1365	105,8
	192,6			30,7
	360,5	14,0	1290	64,8
	280,3	10,2		33,3

It is required to perform a grouping of enterprises for the exchange of products, taking the following intervals:

up to 200 million rubles


from 200 to 400 million rubles


1. from 400 to 600 million rubles
   

   For each group and for all together, determine the number of enterprises, the volume of production, the average number of employees, the average output per employee. The grouping results should be presented in the form of a statistical table. Formulate a conclusion.
   

   DECISION
   

   Let's make a grouping of enterprises for the exchange of products, the calculation of the number of enterprises, the volume of production, the average number of employees according to the formula of a simple average. The results of grouping and calculations are summarized in a table.
   

   Groups by production volume	№ enterprises	Production volume, million rubles	Average annual cost of fixed assets, million rubles	average sleep juicy number of employees, pers.	Profit, thousand rubles	Average output per worker
   1 group up to 200 million rubles	1,8,12	197,7 204,7 192,6	10,0 9,4 8,8	900 817	13,5 30,6 30,7
   			28,2	2567	74,8	0,23
   Middle level		198,3			24,9
   2 group from 200 to 400 million rubles	4,10,13,14	196,2 292,2 360,5 280,3	12,6 113,6 14,0 10,2	1200 1200 1290	44,4 49,6 64,8 33,3
   		1129,2	150,4	4590	192,1	0,25
   Middle level		282,3	37,6	1530	64,0
   3 group from 400 to 600 million	2,3,5,6,7,9,11	592 465,5 584,1 480,0 578,5 466,8 423,1	22,8 18,4 22,0 119,0 21,6 19,4 17,6	1500 1412 1485 1420 1390 1375 1365	136,2 97,6 146,0 110,4 138,7 111,8 105,8
   		3590	240,8	9974	846,5	0,36
   Middle level		512,9	34,4	1421	120,9
   Total in aggregate		5314,2	419,4	17131	1113,4	0,31
   Aggregate average		379,6	59,9	1223,6	79,5

   Conclusion. Thus, in the aggregate under consideration, the largest number of enterprises in terms of output fell into the third group - seven, or half of the enterprises. The value of the average annual value of fixed assets is also in this group, as well as the large value of the average number of employees - 9974 people, the enterprises of the first group are the least profitable.
   

   TASK 2
   

   We have the following data on the enterprises of the company
   

   Number of the enterprise belonging to the company	I quarter		II quarter
   	Output, thousand rubles		Worked by working man-days	Average output per worker per day, rub.
   	59390,13

Topic 5. Averages as statistical indicators

The concept of average. Scope of average values ​​in a statistical study

Average values ​​are used at the stage of processing and summarizing the obtained primary statistical data. The need to determine the average values ​​is due to the fact that for different units of the studied populations, the individual values ​​of the same trait, as a rule, are not the same.

Average value call an indicator that characterizes the generalized value of a feature or a group of features in the study population.

If a population with qualitatively homogeneous characteristics is being studied, then the average value appears here as typical average. For example, for groups of workers in a particular industry with a fixed level of income, a typical average spending on basic necessities is determined, i.e. the typical average generalizes the qualitatively homogeneous values ​​of the attribute in the given population, which is the share of expenses of workers in this group on essential goods.

In the study of a population with qualitatively heterogeneous characteristics, the atypical average indicators may come to the fore. Such, for example, are the average indicators of the produced national income per capita (different age groups), the average yields of grain crops throughout Russia (areas of different climatic zones and different grain crops), the average birth rates of the population in all regions of the country, the average temperature for a certain period, etc. Here, average values ​​generalize qualitatively heterogeneous values ​​of features or systemic spatial aggregates (international community, continent, state, region, district, etc.) or dynamic aggregates extended in time (century, decade, year, season, etc.) . These averages are called system averages.

Thus, the meaning of average values ​​consists in their generalizing function. The average value replaces a large number of individual values ​​of a trait, revealing common properties inherent in all units of the population. This, in turn, makes it possible to avoid random causes and to identify common patterns due to common causes.

Types of average values ​​and methods for their calculation

At the stage of statistical processing, a variety of research tasks can be set, for the solution of which it is necessary to choose the appropriate average. In this case, it is necessary to be guided by the following rule: the values ​​\u200b\u200bthat represent the numerator and denominator of the average must be logically related to each other.

power averages;

structural averages.

Let us introduce the following notation:

The values ​​for which the average is calculated;

Average, where the line above indicates that the averaging of individual values ​​takes place;

Frequency (repeatability of individual trait values).

Various means are derived from the general power mean formula:

(5.1)

for k = 1 - arithmetic mean; k = -1 - harmonic mean; k = 0 - geometric mean; k = -2 - root mean square.

Averages are either simple or weighted. weighted averages are called quantities that take into account that some variants of the values ​​of the attribute may have different numbers, and therefore each variant has to be multiplied by this number. In other words, the "weights" are the numbers of population units in different groups, i.e. each option is "weighted" by its frequency. The frequency f is called statistical weight or weight average.

Arithmetic mean- the most common type of medium. It is used when the calculation is carried out on ungrouped statistical data, where you want to get the average summand. The arithmetic mean is such an average value of a feature, upon receipt of which the total volume of the feature in the population remains unchanged.

The arithmetic mean formula (simple) has the form

where n is the population size.

For example, the average salary of employees of an enterprise is calculated as the arithmetic average:

The determining indicators here are the wages of each employee and the number of employees of the enterprise. When calculating the average, the total amount of wages remained the same, but distributed, as it were, equally among all workers. For example, it is necessary to calculate the average salary of employees of a small company where 8 people are employed:

When calculating averages, individual values ​​of the attribute that is averaged can be repeated, so the average is calculated using grouped data. In this case, we are talking about using arithmetic mean weighted, which looks like

(5.3)

So, we need to calculate the average share price of a joint-stock company at the stock exchange. It is known that transactions were carried out within 5 days (5 transactions), the number of shares sold at the sales rate was distributed as follows:

1 - 800 ac. - 1010 rubles

2 - 650 ac. - 990 rub.

3 - 700 ak. - 1015 rubles.

4 - 550 ac. - 900 rub.

5 - 850 ak. - 1150 rubles.

The initial ratio for determining the average share price is the ratio of the total amount of transactions (TCA) to the number of shares sold (KPA):

OSS = 1010 800+990 650+1015 700+900 550+1150 850= 3 634 500;

CPA = 800+650+700+550+850=3550.

In this case, the average share price was equal to

It is necessary to know the properties of the arithmetic mean, which is very important both for its use and for its calculation. There are three main properties that most of all led to the widespread use of the arithmetic mean in statistical and economic calculations.

Property one (zero): the sum of positive deviations of the individual values ​​of a feature from its mean value is equal to the sum of negative deviations. This is a very important property, because it shows that any deviations (both with + and with -) due to random causes will be mutually canceled.

Proof:

The second property (minimum): the sum of the squared deviations of the individual values ​​of the attribute from the arithmetic mean is less than from any other number (a), i.e. is the minimum number.

Proof.

Compose the sum of the squared deviations from the variable a:

(5.4)

To find the extremum of this function, it is necessary to equate its derivative with respect to a to zero:

From here we get:

(5.5)

Therefore, the extremum of the sum of squared deviations is reached at . This extremum is the minimum, since the function cannot have a maximum.

Third property: the arithmetic mean of a constant is equal to this constant: at a = const.

In addition to these three most important properties of the arithmetic mean, there are so-called design properties, which are gradually losing their significance due to the use of electronic computers:

if the individual value of the sign of each unit is multiplied or divided by a constant number, then the arithmetic mean will increase or decrease by the same amount;

the arithmetic mean will not change if the weight (frequency) of each feature value is divided by a constant number;

if the individual values ​​of the attribute of each unit are reduced or increased by the same amount, then the arithmetic mean will decrease or increase by the same amount.

Average harmonic. This average is called the reciprocal arithmetic average, since this value is used when k = -1.

Simple harmonic mean is used when the weights of the characteristic values ​​are the same. Its formula can be derived from the base formula by substituting k = -1:

For example, we need to calculate the average speed of two cars that have traveled the same path, but at different speeds: the first at a speed of 100 km/h, the second at 90 km/h. Using the harmonic mean method, we calculate the average speed:

In statistical practice, harmonic weighted is more often used, the formula of which has the form

This formula is used in cases where the weights (or volumes of phenomena) for each attribute are not equal. In the original ratio, the numerator is known to calculate the average, but the denominator is unknown.

A simple arithmetic mean is the average term, in determining which the total volume of a given attribute in aggregates data is equally distributed among all units included in this set. So, the average annual production output per worker is such a value of the volume of production that would fall on each employee if the entire volume of output was equally distributed among all employees of the organization. The arithmetic mean simple value is calculated by the formula:

simple arithmetic mean- Equal to the ratio of the sum of individual values ​​of a feature to the number of features in the aggregate

Example 1. A team of 6 workers receives 3 3.2 3.3 3.5 3.8 3.1 thousand rubles per month.

Find the average wage Solution: (3 + 3.2 + 3.3 +3.5 + 3.8 + 3.1) / 6 = 3.32 thousand rubles.

Arithmetic weighted average

If the volume of the data set is large and represents a distribution series, then a weighted arithmetic mean is calculated. This is how the weighted average price per unit of production is determined: the total cost of production (the sum of the products of its quantity and the price of a unit of production) is divided by the total quantity of production.

We represent this in the form of the following formula:

Weighted arithmetic mean- is equal to the ratio (the sum of the products of the attribute value to the frequency of repetition of this attribute) to (the sum of the frequencies of all the attributes). It is used when the variants of the studied population occur an unequal number of times.

Example 2. Find the average monthly salary of shop workers

Salary of one worker thousand rubles; X	Number of workers F

The average wage can be obtained by dividing the total wage by the total number of workers:

Answer: 3.35 thousand rubles.

Arithmetic mean for an interval series

When calculating the arithmetic mean for an interval variation series, the average for each interval is first determined as the half-sum of the upper and lower boundaries, and then the average of the entire series. In the case of open intervals, the value of the lower or upper interval is determined by the value of the intervals adjacent to them.

Averages calculated from interval series are approximate.

Example 3. Determine the average age of students in the evening department.

Age in years!!x??	Number of students	Interval mean	The product of the middle of the interval (age) and the number of students
		(18 + 20) / 2 =19 18 in this case, the boundary of the lower interval. Calculated as 20 - (22-20)
		(20 + 22) / 2 = 21
		(22 + 26) / 2 = 24
		(26 + 30) / 2 = 28
30 or more		(30 + 34) / 2 = 32

Averages calculated from interval series are approximate. The degree of their approximation depends on the extent to which the actual distribution of population units within the interval approaches uniform.

When calculating averages, not only absolute, but also relative values ​​(frequency) can be used as weights.