Average values ​​are widespread in statistics. Average values ​​characterize the qualitative indicators of commercial activity: distribution costs, profit, profitability, etc.

Average is one of the common generalization techniques. A correct understanding of the essence of the average determines its special significance in the conditions of a market economy, when the average, through the single and random, makes it possible to identify the general and necessary, to reveal the tendency of the laws of economic development.

average value are generalizing indicators in which actions are expressed general conditions, patterns of the studied phenomenon.

Statistical averages are calculated on the basis of mass data of correctly statistically organized mass observation (continuous and selective). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). For example, if you calculate the average wage in cooperatives and state-owned enterprises, and extend the result to the entire population, then the average is fictitious, since it is calculated over a heterogeneous population, and such an average loses all meaning.

With the help of the average, there is, as it were, smoothing out the differences in the value of the attribute, which arise for one reason or another in individual units of observation.

For example, the average output of a salesperson depends on many reasons: qualifications, length of service, age, form of service, health, etc.

Average output reflects the general property of the entire population.

The average value is a reflection of the values ​​of the trait under study, therefore, it is measured in the same dimension as this trait.

Each average value characterizes the studied population for any one criterion. To get a complete and comprehensive picture of the studied population for a number of essential features, in general, it is necessary to have a system of average values ​​that can describe the phenomenon from different angles.

There are various averages:

arithmetic mean;

geometric mean;

average harmonic;

root mean square;

average chronological.

Let's consider some types of averages that are most often used in statistics.

Arithmetic mean

The simple arithmetic mean (unweighted) is equal to the sum of the individual values ​​of the attribute, divided by the number of these values.

Individual values ​​of a feature are called variants and are denoted by x (); the number of units in the population is denoted by n, the average value of the feature is denoted by ... Therefore, the simple arithmetic mean is:

According to discrete series distribution, it can be seen that the same values ​​of the characteristic (variants) are repeated several times. So, option x occurs in aggregate 2 times, and option x - 16 times, etc.

The number of identical values ​​of a feature in the distribution series is called the frequency or weight and is denoted by the symbol n.

Let's calculate the average wage of one worker in rubles:

Fund wages for each group of workers is equal to the product of the options by the frequency, and the sum of these products gives the total wages of all workers.

In accordance with this, the calculations can be presented in general form:

The resulting formula is called the weighted arithmetic mean.

The statistical material as a result of processing can be presented not only in the form of discrete distribution series, but also in the form of interval variation series with closed or open intervals.

The calculation of the average for the grouped data is made according to the formula of the arithmetic weighted average:

In the practice of economic statistics, sometimes it is necessary to calculate the average by group means or by means of individual parts of the population (private means). In such cases, group or partial averages are taken as options (x), on the basis of which the total average is calculated as the usual weighted arithmetic mean.

Basic properties of the arithmetic mean .

The arithmetic mean has a number of properties:

1. From a decrease or increase in the frequencies of each value of the attribute x in n times, the value of the arithmetic mean will not change.

If all frequencies are divided or multiplied by any number, then the value of the average will not change.

2. The common factor of individual values ​​of the attribute can be taken out of the mean sign:

3. The average of the sum (difference) of two or more values ​​is equal to the sum (difference) of their average:

4. If x = c, where c is a constant, then
.

5. The sum of the deviations of the values ​​of the attribute X from the arithmetic mean x is equal to zero:

Average harmonic.

Along with the arithmetic mean, in statistics, the harmonic mean is used, the reciprocal of the arithmetic mean of the reciprocal values ​​of the attribute. Like the arithmetic mean, it can be simple and weighted.

The characteristics of the variation series, along with the mean, are the mode and the median.

Fashion - This is the value of a feature (option), which is most often repeated in the studied population. For discrete distribution series, the mode will be the value of the variant with the highest frequency.

For interval series of distribution with equal intervals, the mode is determined by the formula:

where
- the initial value of the interval containing the mode;

- the value of the modal interval;

- the frequency of the modal interval;

- the frequency of the interval preceding the modal;

is the frequency of the interval following the modal.

Median - this is a variant located in the middle of the variation series. If the distribution series is discrete and has odd number members, then the median will be the option located in the middle of the ordered row (an ordered row is the arrangement of the units of the population in ascending or descending order).

Suppose you want to find the average number of days to complete tasks by different employees. Or you want to calculate a time interval of 10 years Average temperature on a specific day. Calculating the average value of a series of numbers in several ways.

Average is a measure function of the central tendency at which the center of a series of numbers in a statistical distribution is located. The three most common criteria for the central trend are.

The average The arithmetic mean is calculated by adding a series of numbers and then dividing the number of those numbers. For example, the mean of 2, 3, 3, 5, 7, and 10 has 30 divided by 6, 5;

Median The middle number of a row of numbers. Half of the numbers have values ​​that are greater than the Median, and half of the numbers have values ​​that are less than the Median. For example, the median of 2, 3, 3, 5, 7 and 10 is 4.

Mode The most common number in a group of numbers. For example mode 2, 3, 3, 5, 7 and 10 - 3.

These three measures of the central tendency of the symmetrical distribution of a number of numbers are the same. In an asymmetric distribution of a number of numbers, they can be different.

Calculate the average of cells located continuously in one row or one column

Follow the steps below.

Calculating the average of scattered cells

To accomplish this task, use the function AVERAGE... Copy the table below onto a blank sheet.

Calculating the weighted average

SUMPRODUCT and sums... Example vThis calculates the average unit price paid over three purchases, where each purchase resides for a different number of units at different unit prices.

Copy the table below onto a blank sheet.

Calculating the average of numbers, excluding zero values

To accomplish this task, use the functions AVERAGE and if... Copy the table below and keep in mind that in this example, to make it easier to understand, copy it onto a blank sheet.

Now let's talk about how to calculate the average.
In its classical form, the general theory of statistics offers us one version of the selection rules average size.
First, it is necessary to draw up the correct logical formula for calculating the average value (LFS). For each average value, there is always only one logical formula for its calculation, so it is difficult to be mistaken here. But you must always remember that in the numerator (this is what is on top of the fraction) is the sum of all phenomena, and in the denominator (what is below the fraction) total amount elements.

After the logical formula has been drawn up, you can use the rules (for ease of understanding, we will simplify and shorten them):
1. If the denominator of the logical formula is presented in the initial data (determined by frequency), then the calculation is carried out according to the formula of the arithmetic weighted average.
2. If the numerator of the logical formula is presented in the initial data, then the calculation is carried out according to the formula of the average harmonic weighted.
3. If the problem presents both the numerator and the denominator of a logical formula (this happens rarely), then the calculation is carried out according to this formula or according to the formula of the arithmetic average simple.
it classic presentation on the choice of the correct formula for calculating the average. Next, we present the sequence of actions when solving problems for calculating the average.

Algorithm for solving problems for calculating the average

A. Determine the method for calculating the average - simple or balanced ... If the data is presented in the table, then we use a weighted method, if the data is represented by a simple listing, then we use a simple calculation method.

B. Define or arrange legend – x - options, f - frequency ... Variant is for which phenomenon you want to find the average. The rest of the data in the table will be the frequency.

B. Determine the form of calculating the average - arithmetic or harmonic ... The definition is carried out by the frequency column. The arithmetic form is used if the frequencies are given by an explicit quantity (conventionally, they can be substituted with the word pieces, the number of elements "pieces"). The harmonic form is used if the frequencies are specified not by an explicit amount, but by a complex indicator (the product of the averaged value and the frequency).

The most difficult thing is to guess where and how much is given, especially to a student who is inexperienced in such matters. In such a situation, you can use one of the following methods. For some tasks (economic), a statement developed over years of practice is suitable (clause B.1). In other situations, you will have to use paragraph B.2.

B.1 If the frequency is set in monetary units (in rubles), then the harmonic mean is used for calculating, this statement is always true if the revealed frequency is set in money, in other situations this rule does not apply.

B.2 Use the rules for choosing the average indicated above in this article. If the frequency is set by the denominator of the logical formula for calculating the average, then we calculate by the arithmetic mean form, if the frequency is set by the numerator of the logical formula for calculating the mean, then we calculate by the mean harmonic form.

Let's look at examples of the use of this algorithm.

A. Since the data is presented in a line, we use a simple calculation method.

BV We only have data on the amount of pensions, and they will be our option - x. The data are presented in simple numbers (12 people), for the calculation we use the simple arithmetic mean.

The average pension of a pensioner is 9208.3 rubles.

B. Since it is required to find the average size of the payment for one child, the options are in the first column, we put the designation x there, the second column automatically becomes the frequency f.

B. The frequency (the number of children) is set by an explicit number (you can substitute the word for the number of children, from the point of view of the Russian language, it is an incorrect phrase, but, in fact, it is very convenient to check), which means that the arithmetic weighted average is used for the calculation.

It is fashionable to solve the same problem not in a formulaic way, but in a tabular way, that is, to enter all the data of intermediate calculations into a table.

As a result, all you have to do now is split the two totals in the correct order.

The average payment for one child per month was 1910 rubles.

A. Since the data are presented in the table, we use a weighted form for the calculation.

B. The frequency (production cost) is set by an implicit quantity (the frequency is set in rubles algorithm item B1), which means that the average harmonic weighted is used for the calculation. In general, in fact, the cost of production is a complex indicator, which is obtained by multiplying the cost of a unit of a product by the number of such products, this is the essence of the average harmonic value.

In order for this problem to be solved according to the arithmetic mean formula, it is necessary that instead of the production cost price there should be a number of products with the corresponding cost price.

Please note that the amount in the denominator obtained after calculating 410 (120 + 80 + 210) is the total number of products produced.

The average unit cost of the product was 314.4 rubles.

A. Since the data are presented in the table, we use a weighted form for the calculation.

B. Since it is required to find the average cost of a unit of product, the options are in the first column, we put the designation x there, the second column automatically becomes the frequency f.

B. The frequency (the total number of gaps) is set by an implicit amount (this is the product of two indicators of the number of gaps and the number of students with such a number of gaps), which means that the harmonic weighted average is used for the calculation. We will use algorithm item B2.

In order for this problem to be solved by the arithmetic mean formula, it is necessary that instead of the total passes stood the number of students.

We draw up a logical formula for calculating the average number of absences per student.

Frequency by task condition The total number of skips. In the logical formula, this indicator is in the numerator, which means we use the harmonic average formula.

Please note that the sum in the denominator resulting from the calculations of 31 (18 + 8 + 5) is the total number of students.

The average number of absences per student is 13.8 days.

When starting to talk about average values, they most often remember how they graduated from school and entered educational institution... Then, according to the certificate, average score: all grades (both good and not so) were added, the resulting sum was divided by their number. This is how the simplest form of the average is calculated, which is called the simple arithmetic mean. In practice, statistics are used different kinds averages: arithmetic, harmonic, geometric, quadratic, structural averages. One or another of their types is used depending on the nature of the data and the objectives of the study.

average value is the most common statistical indicator, with the help of which a generalizing characteristic of a set of phenomena of the same type is given according to one of the varying signs. It shows the level of the trait per unit of the population. With the help of average values, a comparison is made of various aggregates according to varying characteristics, the patterns of development of phenomena and processes of social life are studied.

In statistics, two classes of means are used: power (analytical) and structural. The latter are used to characterize the structure of the variational series and will be discussed further in Ch. eight.

The group of power averages includes the arithmetic mean, harmonic, geometric, quadratic. Individual formulas for their calculation can be reduced to a form common to all power averages, namely

where m is the exponent of the power-law mean: for m = 1 we obtain the formula for calculating the arithmetic mean, for m = 0 - the geometric mean, m = -1 - the harmonic mean, with m = 2 - the mean square;

x i - options (values ​​that the attribute takes);

f i - frequencies.

The main condition under which power averages can be used in statistical analysis is the homogeneity of the population, which should not contain initial data that differ sharply in their quantitative value (in the literature, they are called anomalous observations).

Let us demonstrate the importance of this condition with the following example.

Example 6.1. Let's calculate the average salary of employees of a small enterprise.

To calculate the average wage, it is necessary to sum up the wages accrued to all employees of the enterprise (i.e. find the payroll), and divide by the number of employees:

And now we will add to our totality only one person (the director of this enterprise), but with a salary of 50,000 rubles. In this case, the calculated average will be completely different:

As you can see, it exceeds 7,000 rubles, etc. it is greater than all the values ​​of the characteristic, with the exception of one single observation.

In order for such cases not to occur in practice, and the average would not lose its meaning (in Example 6.1 it no longer fulfills the role of a generalizing characteristic of the population, which should be), when calculating the average, anomalous, sharply distinguished observations should be excluded or excluded from the analysis and topics by doing so, make the population homogeneous, or divide the population into homogeneous groups and calculate the average values ​​for each group and analyze not the overall average, but the group averages.

6.1. Arithmetic mean and its properties

The arithmetic mean is calculated either as a simple or as a weighted value.

When calculating the average wage according to the table of example 6.1, we added up all the values ​​of the attribute and divided by their number. We write down the course of our calculations in the form of a formula for the arithmetic mean of a simple

where x i - options (individual values ​​of the feature);

n is the number of units in the aggregate.

Example 6.2. Now let's group our data from the table in Example 6.1, etc. Let us construct a discrete variational series of distribution of workers according to the level of wages. The grouping results are presented in the table.

Let's write an expression for calculating the average wage level in a more compact form:

In example 6.2, the formula for the arithmetic weighted average was applied

where f i - frequencies showing how many times the value of the attribute x i y occurs in the units of the population.

It is convenient to calculate the arithmetic weighted average in the table, as shown below (Table 6.3):

It should be noted that the simple arithmetic mean is used in cases where the data is not grouped or grouped, but all frequencies are equal.

Often the observation results are presented in the form of an interval distribution series (see table in example 6.4). Then, when calculating the average, the midpoints of the intervals are taken as x i. If the first and last intervals are open (do not have one of the boundaries), then they are conventionally "closed", taking the value of the adjacent interval as the value of this interval, and so on. the first is closed based on the value of the second, and the last - according to the value of the penultimate one.

Example 6.3. Based on the results of a sample survey of one of the population groups, we will calculate the size of the average per capita money income.

In the above table, the middle of the first interval is 500. Indeed, the value of the second interval is 1000 (2000-1000); then the lower boundary of the first is 0 (1000-1000), and its middle is 500. We do the same with the last interval. We take 25,000 as its middle: the value of the penultimate interval is 10,000 (20,000-10,000), then its upper limit is 30,000 (20,000 + 10,000), and the middle, respectively, is 25,000.

Then the average per capita monthly income will be

The average is a generalized indicator that characterizes the typical level of the phenomenon. It expresses the value of a feature per unit of the population.

The average is:

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

2) the volume of the attribute of the population, distributed equally between the units of the population.

The attribute 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 ​​are repaid individual differences units of the population due to random circumstances. Unlike average absolute value characterizing the level of a trait of an individual unit of a population does not allow comparing the values ​​of a trait in units belonging to different populations. So, if it is necessary to compare the levels of remuneration of workers at two enterprises, then it is impossible to compare on this basis two workers of different enterprises. The salaries 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 average indicators can be compared, i.e. how much one worker receives on average at each enterprise. Thus, it becomes necessary 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 levels of the attribute or its final value (in the case of calculating the average levels in the series of dynamics) should remain unchanged. In other words, when calculating the average value, the volume of the trait under study should not be distorted, and the expressions compiled when calculating the average must necessarily make sense.

Calculating the average is one of the common generalization techniques; average denies that common, which is characteristic (typical) for all units of the studied population, at the same time, he ignores the differences of individual units. In every phenomenon and its development, there is a combination of chance and necessity. When calculating averages by virtue of the operation of the law large numbers accidents are mutually canceled out, balanced, therefore, one 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 randomness individual values, fluctuations and concluded the scientific value of averages as generalizing characteristics of aggregates.

In order for the average to be truly typifying, it must be calculated based on certain principles.

Let's dwell on some general principles the use 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 how to calculate them

Let us now consider the types of averages, features of their calculation and scope. Averages are divided into two large classes: power averages, structural averages.

Power-law means include the most famous and commonly used types such as geometric mean, arithmetic mean and root-mean-square.

Mode and median are considered as structural means.

Let's dwell on power averages. Power averages, depending on the presentation of the initial data, can be simple and weighted. Simple average is calculated from not grouped data and has the following general form:

,

where X i - options (value) of the averaged feature;

n is the number of options.

Weighted average is calculated from 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 - indicator of the degree of the average;

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

If we calculate all types of averages for the same initial data, then their values ​​will turn out to be unequal. Here the rule of majorantness 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 averages and harmonic weighted averages are used.

Types of power averages

The harmonic mean has a more complex construction than the arithmetic mean. The harmonic mean is used for calculations when not the aggregate units - the carriers of the feature - are used as weights, but the product of these units by the feature values ​​(i.e., m = Xf). The average harmonic downtime should be resorted to in cases of determining, for example, the average cost of labor, time, materials per unit of production, per one 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 is that all stages of the calculation have a real substantive justification; the resulting average value should replace the individual values ​​of the attribute for each object without disrupting the connection between individual and summary indicators. In other words, the average value should be calculated so that when replacing each individual value of the averaged indicator with its average value, some final summary indicator remains unchanged, related to or in some other way with the averaged. This bottom line is called defining, since the nature of its relationship with individual values ​​determines the specific formula for calculating the average. Let us show this rule using the example of a geometric mean.

Geometric mean formula

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

Geometric mean is used if a chain sequence is specified. relative values dynamics indicating, for example, an increase in the volume of production in comparison with the level of the previous year: i 1, i 2, i 3,…, i n. It is obvious that the volume of production in 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 indicators of dynamics with averages, we arrive at the relation

From here

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

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

If the studied feature has discrete values, then there are no special 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 appears in some of the intervals of the attribute X. Using interpolation, the median value is found in this median interval:

,

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

h Me - 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 (in absolute or relative terms);

S Me-1 - the sum of observations (or the volume of the weighing trait) accumulated before the beginning of the median interval;

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

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

,

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

m Mo - the number of observations or the volume of the weighing 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 - the value of the interval of changes in the trait in groups.

PROBLEM 1

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

It is required to group the 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 are presented in the form of a statistical table. Formulate a conclusion.
   

   SOLUTION
   

   Let us group the enterprises for the exchange of products, calculate the number of enterprises, the volume of production, the average number of employees using the simple average formula. The results of grouping and calculations are summarized in a table.
   

   Groups by product volume	№ enterprises	Production volume, mln. Rub.	Average annual cost of fixed assets, mln. Rub.	Srednespi juicy number of employees, people	Profit, thousand rubles	Average output per employee
   1st 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
   Average level		198,3			24,9
   Group 2 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
   Average level		282,3	37,6	1530	64,0
   Group 3 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
   Average level		512,9	34,4	1421	120,9
   Total in aggregate		5314,2	419,4	17131	1113,4	0,31
   Population average		379,6	59,9	1223,6	79,5

   Output. Thus, in the considered population greatest number enterprises in terms of production fell into the third group - seven, or half of the enterprises. The quantity average annual cost fixed assets also in this group, as well as a large value of the average number of employees - 9974 people, the least profitable enterprises of the first group.
   

   OBJECTIVE 2
   

   There is the following data on the enterprises of the company
   

   Company number of the company	I quarter		II quarter
   	Production output, thousand rubles		Worked by workers man-days	Average output per worker per day, rubles
   	59390,13