In mathematics and statistics average arithmetic (or easy average) of a set of numbers is the sum of all the numbers in this set divided by their number. The arithmetic mean is a particularly universal and most common representation of an average.

You will need

• Knowledge of mathematics.

Instructions

1. Let a set of four numbers be given. Need to be discovered average meaning this kit. To do this, we first find the sum of all these numbers. Possible numbers are 1, 3, 8, 7. Their sum is S = 1 + 3 + 8 + 7 = 19. The set of numbers must consist of numbers of the same sign, otherwise the sense in calculating the average value is lost.

2. Average meaning set of numbers is equal to the sum of numbers S divided by the number of these numbers. That is, it turns out that average meaning equals: 19/4 = 4.75.

3. For a set of numbers it is also possible to detect not only average arithmetic, but also average geometric. The geometric mean of several regular real numbers is a number that can replace any of these numbers so that their product does not change. The geometric mean G is sought using the formula: the Nth root of the product of a set of numbers, where N is the number in the set. Let's look at the same set of numbers: 1, 3, 8, 7. Let's find them average geometric. To do this, let's calculate the product: 1*3*8*7 = 168. Now from the number 168 you need to extract the 4th root: G = (168)^1/4 = 3.61. Thus average the geometric set of numbers is 3.61.

Average The geometric average is generally used less often than the arithmetic average, however, it can be useful when calculating the average value of indicators that change over time (the salary of an individual employee, the dynamics of academic performance indicators, etc.).

You will need

• Engineering calculator

Instructions

1. In order to find the geometric mean of a series of numbers, you first need to multiply all these numbers. Let's say you are given a set of five indicators: 12, 3, 6, 9 and 4. Let's multiply all these numbers: 12x3x6x9x4=7776.

2. Now from the resulting number you need to extract the root of a power equal to the number of elements of the series. In our case, from the number 7776 it will be necessary to extract the fifth root using an engineering calculator. The number obtained after this operation - in this case the number 6 - will be the geometric mean for the initial group of numbers.

3. If you don’t have an engineering calculator at hand, then you can calculate the geometric mean of a series of numbers using the SRGEOM function in Excel or using one of the online calculators specifically designed for calculating geometric mean values.

Note!
If you need to find the geometric mean of each for 2 numbers, then you do not need an engineering calculator: you can extract the second root (square root) of any number using the most ordinary calculator.

Helpful advice
Unlike the arithmetic mean, the geometric mean is not so powerfully affected by huge deviations and fluctuations between individual values ​​in the set of indicators under study.

Average value is one of the collations of a set of numbers. Represents a number that cannot fall outside the range defined by the largest and smallest values ​​in that set of numbers. Average arithmetic value is a particularly commonly used type of average.

Instructions

1. Add up all the numbers in the set and divide them by the number of terms to get the arithmetic mean. Depending on certain calculation conditions, it is sometimes easier to divide each of the numbers by the number of values ​​in the set and sum the total.

2. Use, say, the calculator included with Windows OS if calculating the arithmetic average in your head is not possible. You can open it with support from the program launch dialog. To do this, press the “hot keys” WIN + R or click the “Start” button and select the “Run” command from the main menu. After that, type calc in the input field and press Enter on your keyboard or click the “OK” button. The same can be done through the main menu - open it, go to the “All programs” section and to the “Typical” segments and select the “Calculator” line.

3. Enter all the numbers of the set step by step by pressing the Plus key on the keyboard after all of them (besides the last one) or by clicking the corresponding button in the calculator interface. You can also enter numbers either from the keyboard or by clicking the corresponding interface buttons.

4. Press the slash key or click this icon in the calculator interface after entering the last value of the set and type the number of numbers in the sequence. After that, press the equal sign and the calculator will calculate and display the arithmetic mean.

5. You can use the Microsoft Excel spreadsheet editor for the same purpose. In this case, launch the editor and enter all the values ​​of the sequence of numbers into the adjacent cells. If, after entering the entire number, you press Enter or the down or right arrow key, the editor itself will move the input focus to the adjacent cell.

6. Select all entered values ​​and in the lower left corner of the editor window (in the status bar) you will see the arithmetic mean value for the selected cells.

7. Click the cell next to the last number entered if you just want to see the average. Expand the drop-down list with the image of the Greek letter sigma (Σ) in the Editing command group on the Main tab. Select the line " Average" and the editor will insert the necessary formula for calculating the arithmetic mean into the selected cell. Press the Enter key and the value will be calculated.

The arithmetic mean is one of the measures of central propensity, widely used in mathematics and statistical calculations. It is very easy to find the arithmetic average for several values, but every problem has its own nuances, which you need to know in order to perform correct calculations.

What is an arithmetic mean

The arithmetic mean defines the average value for each initial array of numbers. In other words, from a certain set of numbers a value that is universal for all elements is selected, the mathematical comparison of which with all elements is approximately equal. The arithmetic average is used preferably in the preparation of financial and statistical reports or for calculating the quantitative results of similar skills.

How to find the arithmetic mean

Finding the arithmetic mean for an array of numbers should begin by determining the algebraic sum of these values. For example, if the array contains the numbers 23, 43, 10, 74 and 34, then their algebraic sum will be equal to 184. When writing, the arithmetic mean is denoted by the letter? (mu) or x (x with a line). Next, the algebraic sum should be divided by the number of numbers in the array. In the example under consideration there were five numbers, therefore the arithmetic mean will be equal to 184/5 and will be 36.8.

Features of working with negative numbers

If the array contains negative numbers, then the arithmetic mean is found using a similar algorithm. The difference only exists when calculating in the programming environment, or if the problem contains additional data. In these cases, finding the arithmetic mean of numbers with different signs comes down to three steps: 1. Finding the universal arithmetic mean using the standard method;2. Finding the arithmetic mean of negative numbers.3. Calculation of the arithmetic mean of positive numbers. The results of each action are written separated by commas.

Natural and decimal fractions

If an array of numbers is represented by decimal fractions, the solution is carried out using the method of calculating the arithmetic mean of integers, but the reduction of the total is made according to the requirements of the problem for the accuracy of the result. When working with natural fractions, they should be reduced to a common denominator, the one that is multiplied by the number of numbers in the array. The numerator of the result will be the sum of the given numerators of the initial fractional elements.

The geometric mean of numbers depends not only on the absolute value of the numbers themselves, but also on their number. It is impossible to confuse the geometric mean and the arithmetic mean of numbers, since they are found using different methodologies. In this case, the geometric mean is invariably less than or equal to the arithmetic mean.

You will need

• Engineering calculator.

Instructions

1. Consider that in the general case the geometric mean of numbers is found by multiplying these numbers and taking from them the root of the power that corresponds to the number of numbers. For example, if you need to find the geometric mean of five numbers, then you will need to extract the fifth root from the product.

2. To find the geometric mean of 2 numbers, use the basic rule. Find their product, then take the square root of the number two, which corresponds to the degree of the root. Let's say, in order to find the geometric mean of the numbers 16 and 4, find their product 16 4 = 64. From the resulting number, take the square root?64=8. This will be the desired value. Please note that the arithmetic mean of these 2 numbers is larger and equal to 10. If the root is not extracted in its entirety, round the total to the required order.

3. To find the geometric mean of more than 2 numbers, also use the basic rule. To do this, find the product of all numbers for which you need to find the geometric mean. From the resulting product, extract the root of the power equal to the number of numbers. For example, in order to find the geometric mean of the numbers 2, 4 and 64, find their product. 2 4 64=512. Because it is necessary to find the result of the geometric mean of 3 numbers, extract the third root from the product. It is difficult to do this verbally, so use an engineering calculator. For this purpose it has a button “x^y”. Dial the number 512, press the “x^y” button, then dial the number 3 and press the “1/x” button to find the value 1/3, press the “=” button. We get the result of raising 512 to the power of 1/3, which corresponds to the third root. Get 512^1/3=8. This is the geometric mean of the numbers 2.4 and 64.

4. With the support of an engineering calculator, you can find the geometric mean using another method. Find the log button on your keyboard. After this, take the logarithm for all of the numbers, find their sum and divide it by the number of numbers. Take the antilogarithm from the resulting number. This will be the geometric mean of the numbers. Let's say, in order to find the geometric mean of the same numbers 2, 4 and 64, perform a set of operations on the calculator. Dial the number 2, then press the log button, press the “+” button, dial the number 4 and press log and “+” again, dial 64, press log and “=”. The result will be a number equal to the sum of the decimal logarithms of the numbers 2, 4 and 64. Divide the resulting number by 3, since this is the number of numbers by which the geometric mean is sought. From the total, take the antilogarithm by switching the register button and use the same log key. The result will be the number 8, this is the desired geometric mean.

Note!
The average value cannot be larger than the largest number in the set and smaller than the smallest.

Helpful advice
In mathematical statistics, the average value of a quantity is called the mathematical expectation.

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

Average - This is one of the common generalization techniques. A correct understanding of the essence of the average determines its special significance in a market economy, when the average, through the individual and random, allows us to identify the general and necessary, to identify the trend of patterns of economic development.

average value - these are generalizing indicators in which the effects of general conditions and patterns of the phenomenon being studied are expressed.

Statistical averages are calculated on the basis of mass data from 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 for a heterogeneous population, and such an average loses all meaning.

With the help of the average, differences in the value of a characteristic that arise for one reason or another in individual units of observation are smoothed out.

For example, the average productivity 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 characteristic being studied, therefore, it is measured in the same dimension as this characteristic.

Each average value characterizes the population under study according to any one characteristic. In order to obtain a complete and comprehensive understanding of the population under study according to a number of essential characteristics, in general it is necessary to have a system of average values ​​that can describe the phenomenon from different angles.

There are different averages:

arithmetic mean;

geometric mean;

harmonic mean;

mean square;

average chronological.

Let's look at 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 characteristic are called variants and are denoted by x(); the number of population units is denoted by n, the average value of the characteristic is denoted by . Therefore, the arithmetic simple mean is equal to:

According to the discrete distribution series data, it is clear that the same characteristic values ​​(variants) are repeated several times. Thus, option x occurs 2 times in total, and option x 16 times, etc.

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

Let's calculate the average salary of one worker in rub.:

The wage fund for each group of workers is equal to the product of options and frequency, and the sum of these products gives the total wage fund of all workers.

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

The resulting formula is called the weighted arithmetic mean.

As a result of processing, statistical material 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 average for grouped data is calculated using the weighted arithmetic average formula:

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

Basic properties of the arithmetic mean .

The arithmetic mean has a number of properties:

1. The value of the arithmetic mean will not change from decreasing or increasing the frequency of each value of the characteristic x by n times.

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

2. The common multiplier of individual values ​​of a characteristic can be taken beyond the sign of the average:

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

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

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

Harmonic mean.

Along with the arithmetic mean, statistics uses the harmonic mean, the inverse of the arithmetic mean of the inverse values ​​of the attribute. Like the arithmetic mean, it can be simple and weighted.

Characteristics of variation series, along with averages, are mode and median.

Fashion - this is the value of a characteristic (variant) that is most often repeated in the population under study. For discrete distribution series, the mode will be the value of the variant with the highest frequency.

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

Where
- initial value of the interval containing the mode;

- the value of the modal interval;

- frequency of the modal interval;

- frequency of the interval preceding the modal one;

- frequency of the interval following the modal one.

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

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

meaning and differences

Accurate indicators do not always provide an understanding of the situation. In order to assess a particular situation, it is sometimes necessary to analyze a huge number of figures. And then averages come to the rescue. They allow us to assess the situation as a whole.

Since school days, many adults remember the existence of the arithmetic mean. It is very simple to calculate - the sum of a sequence of n terms is divided 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 ​​are used in the calculations. In this case, the required value will be 30.

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

The harmonic mean is usually not a subject of study in secondary schools. However, it is used quite often. This value is the inverse of the arithmetic mean and is calculated as the quotient of n - the number of values ​​and the sum 1/a 1 +1/a 2 +...+1/a n. If we take the same one again 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, the “weight” of each number used in the calculations plays an important role. The results are more indicative and correct because they take into account more information. This group of quantities is generally called the “weighted average”. They are not taught in school, so it is worth looking at them in more detail.

First of all, it is worth telling what is meant by the “weight” of a particular value. The easiest way to explain this is with a specific example. Twice a day in the hospital the body temperature of each patient is measured. Out of 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 average, then this value in general for the hospital will be more than 38 degrees! But almost half of the patients have absolutely And here it would be more correct to use a weighted average value, and the “weight” of each value will be the number of people. In this case, the calculation result 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 is related to the arithmetic mean 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 variation used in number series. This is a weighted moving average. It is on this 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 ordinary weighted average is usually used.

Calculation methods

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

The easiest way is to consider the calculation using a specific example.

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

So, the weighted average is calculated 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 like this:

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.

The most common type of average is the arithmetic mean.

Simple arithmetic mean

A simple arithmetic mean is the average term, in determining which the total volume of a given attribute in the data is equally distributed among all units included in the given population. Thus, the average annual output per employee is the amount of output that would be produced by each employee if the entire volume of output were equally distributed among all employees of the organization. The arithmetic mean simple value is calculated using the formula:

Simple arithmetic average— Equal to the ratio of the sum of individual values ​​of a characteristic to the number of characteristics 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 average salary
Solution: (3 + 3.2 + 3.3 +3.5 + 3.8 + 3.1) / 6 = 3.32 thousand rubles.

Arithmetic average weighted

If the volume of the data set is large and represents a distribution series, then the 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 by the price of a unit of production) is divided by the total quantity of production.

Let's imagine this in the form of the following formula:

Weighted arithmetic average— equal to the ratio of (the sum of the products of the value of a feature to the frequency of repetition of this feature) to (the sum of the frequencies of all features). It is used when variants of the population under study occur an unequal number of times.

Example 2 . Find the average salary of workshop workers per month

Average wages can be obtained by dividing the total wages by the total number of workers:

Answer: 3.35 thousand rubles.

Arithmetic mean for interval series

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

Averages calculated from interval series are approximate.

Example 3. Determine the average age of evening students.

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 distribution.

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

The arithmetic mean has a number of properties that more fully reveal its essence and simplify calculations:

1. The product of the average by the sum of frequencies is always equal to the sum of the products of the variant by frequencies, i.e.

2. The arithmetic mean of the sum of varying quantities is equal to the sum of the arithmetic means of these quantities:

3. The algebraic sum of deviations of individual values ​​of a characteristic from the average is equal to zero:

4. The sum of squared deviations of options from the average is less than the sum of squared deviations from any other arbitrary value, i.e.

Discipline: Statistics

Option No. 2

Average values ​​used in statistics

Introduction………………………………………………………………………………….3

Theoretical task

Average value in statistics, its essence and conditions of application.

1.1. The essence of average size and conditions of use………….4

1.2. Types of averages………………………………………………………8

Practical task

Task 1,2,3…………………………………………………………………………………14

Conclusion………………………………………………………………………………….21

List of references………………………………………………………...23

Introduction

This test consists of two parts – theoretical and practical. In the theoretical part, such an important statistical category as the average value will be examined in detail in order to identify its essence and conditions of application, as well as highlight the types of averages and methods for their calculation.

Statistics, as we know, studies mass socio-economic phenomena. Each of these phenomena may have a different quantitative expression of the same characteristic. For example, wages of workers of the same profession or market prices for the same product, etc. Average values ​​characterize the qualitative indicators of commercial activity: distribution costs, profit, profitability, etc.

To study any population according to varying (quantitatively changing) characteristics, statistics uses average values.

Medium sized entity

The average value is a generalizing quantitative characteristic of a set of similar phenomena based on one varying characteristic. In economic practice, a wide range of indicators are used, calculated as average values.

The most important property of the average value is that it represents the value of a certain characteristic in the entire population with one number, despite its quantitative differences in individual units of the population, and expresses what is common to all units of the population under study. Thus, through the characteristics of a unit of a population, it characterizes the entire population as a whole.

Average values ​​are related to the law of large numbers. The essence of this connection is that during averaging, random deviations of individual values, due to the action of the law of large numbers, cancel each other out and the main development trend, necessity, and pattern are revealed in the average. Average values ​​allow you to compare indicators related to populations with different numbers of units.

In modern conditions of development of market relations in the economy, averages serve as a tool for studying the objective patterns of socio-economic phenomena. However, in economic analysis one cannot limit oneself only to average indicators, since general favorable averages may hide large serious shortcomings in the activities of individual economic entities, and the sprouts of a new, progressive one. For example, the distribution of the population by income makes it possible to identify the formation of new social groups. Therefore, along with average statistical data, it is necessary to take into account the characteristics of individual units of the population.

The average value is the resultant of all factors influencing the phenomenon under study. That is, when calculating average values, the influence of random (perturbation, individual) factors cancels out and, thus, it is possible to determine the pattern inherent in the phenomenon under study. Adolphe Quetelet emphasized that the significance of the method of averages is the possibility of transition from the individual to the general, from the random to the regular, and the existence of averages is a category of objective reality.

Statistics studies mass phenomena and processes. Each of these phenomena has both common to the entire set and special, individual properties. The difference between individual phenomena is called variation. Another property of mass phenomena is their inherent similarity of characteristics of individual phenomena. So, the interaction of elements of a set leads to a limitation of the variation of at least part of their properties. This trend exists objectively. It is in its objectivity that lies the reason for the widest use of average values ​​in practice and in theory.

The average value in statistics is a general indicator that characterizes the typical level of a phenomenon in specific conditions of place and time, reflecting the value of a varying characteristic per unit of a qualitatively homogeneous population.

In economic practice, a wide range of indicators are used, calculated as average values.

Using the method of averages, statistics solves many problems.

The main significance of averages lies in their generalizing function, that is, the replacement of many different individual values ​​of a characteristic with an average value that characterizes the entire set of phenomena.

If the average value generalizes qualitatively homogeneous values ​​of a characteristic, then it is a typical characteristic of the characteristic in a given population.

However, it is incorrect to reduce the role of average values ​​only to the characterization of typical values ​​of characteristics in populations homogeneous for a given characteristic. In practice, much more often modern statistics use average values ​​that generalize clearly homogeneous phenomena.

The average national income per capita, the average grain yield throughout the country, the average consumption of various food products - these are the characteristics of the state as a single economic system, these are the so-called system averages.

System averages can characterize both spatial or object systems that exist simultaneously (state, industry, region, planet Earth, etc.) and dynamic systems extended over time (year, decade, season, etc.).

The most important property of the average value is that it reflects what is common to all units of the population under study. The attribute values ​​of individual units of the population fluctuate in one direction or another under the influence of many factors, among which there may be both basic and random. For example, the stock price of a corporation as a whole is determined by its financial position. At the same time, on certain days and on certain exchanges, these shares, due to prevailing circumstances, may be sold at a higher or lower rate. The essence of the average lies in the fact that it cancels out the deviations of the characteristic values ​​of individual units of the population caused by the action of random factors, and takes into account the changes caused by the action of the main factors. This allows the average to reflect the typical level of the trait and abstract from the individual characteristics inherent in individual units.

Calculating the average is one of the most common generalization techniques; the average indicator reflects what is common (typical) for all units of the population being studied, while at the same time it ignores the differences of individual units. In every phenomenon and its development there is a combination of chance and necessity.

The average is a summary characteristic of the laws of the process in the conditions in which it occurs.

Each average characterizes the population under study according to any one characteristic, but to characterize any population, describe its typical features and qualitative features, a system of average indicators is needed. Therefore, in the practice of domestic statistics, to study socio-economic phenomena, as a rule, a system of average indicators is calculated. So, for example, the average wage indicator is assessed together with indicators of average output, capital-labor ratio and energy-labor ratio, the degree of mechanization and automation of work, etc.

The average should be calculated taking into account the economic content of the indicator under study. Therefore, for a specific indicator used in socio-economic analysis, only one true value of the average can be calculated based on the scientific method of calculation.

The average value is one of the most important generalizing statistical indicators, characterizing a set of similar phenomena according to some quantitatively varying characteristic. Averages in statistics are general indicators, numbers expressing the typical characteristic dimensions of social phenomena according to one quantitatively varying characteristic.

Types of averages

The types of average values ​​differ primarily in what property, what parameter of the initial varying mass of individual values ​​of the attribute must be kept unchanged.

Arithmetic mean

The arithmetic mean is the average value of a characteristic, during the calculation of which the total volume of the characteristic in the aggregate remains unchanged. Otherwise, we can say that the arithmetic mean is the average term. When calculating it, the total volume of the attribute is mentally distributed equally among all units of the population.

The arithmetic mean is used if the values ​​of the characteristic being averaged (x) and the number of population units with a certain characteristic value (f) are known.

The arithmetic average can be simple or weighted.

Simple arithmetic mean

Simple is used if each value of attribute x occurs once, i.e. for each x the value of the attribute is f=1, or if the source data is not ordered and it is unknown how many units have certain attribute values.

The formula for the arithmetic mean is simple:

where is the average value; x – the value of the averaged characteristic (variant), – the number of units of the population being studied.

Arithmetic average weighted

Unlike a simple average, a weighted arithmetic average is used if each value of attribute x occurs several times, i.e. for each value of the feature f≠1. This average is widely used in calculating the average based on a discrete distribution series:

where is the number of groups, x is the value of the characteristic being averaged, f is the weight of the characteristic value (frequency, if f is the number of units in the population; frequency, if f is the proportion of units with option x in the total volume of the population).

Harmonic mean

Along with the arithmetic mean, statistics uses the harmonic mean, the inverse of the arithmetic mean of the inverse values ​​of the attribute. Like the arithmetic mean, it can be simple and weighted. It is used when the necessary weights (f i) in the initial data are not specified directly, but are included as a factor in one of the available indicators (i.e., when the numerator of the initial ratio of the average is known, but its denominator is unknown).

Harmonic mean weighted

The product xf gives the volume of the averaged characteristic x for a set of units and is denoted w. If the source data contains values ​​of the characteristic x being averaged and the volume of the characteristic being averaged w, then the harmonic weighted method is used to calculate the average:

where x is the value of the averaged characteristic x (variant); w – weight of variants x, volume of the averaged characteristic.

Harmonic mean unweighted (simple)

This medium form, used much less frequently, has the following form:

where x is the value of the characteristic being averaged; n – number of x values.

Those. this is the reciprocal of the simple arithmetic mean of the reciprocal values ​​of the attribute.

In practice, the harmonic simple mean is rarely used in cases where the values ​​of w for population units are equal.

Mean square and mean cubic

In a number of cases in economic practice, there is a need to calculate the average size of a characteristic, expressed in square or cubic units of measurement. Then the mean square is used (for example, to calculate the average size of a side and square sections, the average diameters of pipes, trunks, etc.) and the average cubic (for example, when determining the average length of a side and cubes).

If, when replacing individual values ​​of a characteristic with an average value, it is necessary to keep the sum of the squares of the original values ​​unchanged, then the average will be a quadratic average value, simple or weighted.

Simple mean square

Simple is used if each value of the attribute x occurs once, in general it has the form:

where is the square of the values ​​of the characteristic being averaged; - the number of units in the population.

Weighted mean square

The weighted mean square is applied if each value of the averaged characteristic x occurs f times:

,

where f is the weight of options x.

Cubic average simple and weighted

The average cubic prime is the cube root of the quotient of dividing the sum of the cubes of individual attribute values ​​by their number:

where are the values ​​of the attribute, n is their number.

Average cubic weighted:

,

where f is the weight of the options x.

The square and cubic means have limited use in statistical practice. The mean square statistic is widely used, but not from the options themselves x , and from their deviations from the average when calculating variation indices.

The average can be calculated not for all, but for some part of the units in the population. An example of such an average could be the progressive average as one of the partial averages, calculated not for everyone, but only for the “best” (for example, for indicators above or below individual averages).

Geometric mean

If the values ​​of the characteristic being averaged are significantly different from each other or are specified by coefficients (growth rates, price indices), then the geometric mean is used for calculation.

The geometric mean is calculated by extracting the root of the degree and from the products of individual values ​​- variants of the characteristic X:

where n is the number of options; P - product sign.

The geometric mean is most widely used to determine the average rate of change in dynamics series, as well as in distribution series.

Average values ​​are general indicators in which the effect of general conditions and the pattern of the phenomenon being studied are expressed. Statistical averages are calculated on the basis of mass data from correctly statistically organized mass observation (continuous or sample). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). The use of averages should proceed from a dialectical understanding of the categories of general and individual, mass and individual.

The combination of general means with group means makes it possible to limit qualitatively homogeneous populations. By dividing the mass of objects that make up this or that complex phenomenon into internally homogeneous, but qualitatively different groups, characterizing each of the groups with its average, it is possible to reveal the reserves of the process of an emerging new quality. For example, the distribution of the population by income allows us to identify the formation of new social groups. In the analytical part, we looked at a particular example of using the average value. To summarize, we can say that the scope and use of averages in statistics is quite wide.

Practical task

Task No. 1

Determine the average purchase rate and average sale rate of one and $ US

Average purchase rate

Average selling rate

Task No. 2

The dynamics of the volume of own public catering products in the Chelyabinsk region for 1996-2004 are presented in the table in comparable prices (million rubles)

Close rows A and B. To analyze the series of dynamics of production of finished products, calculate:

1. Absolute growth, chain and base growth and growth rates

2. Average annual production of finished products

3. Average annual growth rate and increase in the company’s products

4. Perform analytical alignment of the dynamics series and calculate the forecast for 2005

5. Graphically depict a series of dynamics

6. Draw a conclusion based on the dynamics results

1) yi B = yi-y1 yi C = yi-y1

y2 B = 2.175 – 2.04 y2 C = 2.175 – 2.04 = 0.135

y3B = 2.505 – 2.04 y3 C = 2.505 – 2.175 = 0.33

y4 B = 2.73 – 2.04 y4 C = 2.73 – 2.505 = 0.225

y5 B = 1.5 – 2.04 y5 C = 1.5 – 2.73 = 1.23

y6 B = 3.34 – 2.04 y6 C = 3.34 – 1.5 = 1.84

y7 B = 3.6 3 – 2.04 y7 C = 3.6 3 – 3.34 = 0.29

y8 B = 3.96 – 2.04 y8 C = 3.96 – 3.63 = 0.33

y9 B = 4.41–2.04 y9 C = 4.41 – 3.96 = 0.45

Tr B2 Tr Ts2

Tr B3 Tr Ts3

Tr B4 Tr Ts4

Tr B5 Tr Ts5

Tr B6 Tr Ts6

Tr B7 Tr Ts7

Tr B8 Tr Ts8

Tr B9 Tr Ts9

Tr B = (TprB *100%) – 100%

Tr B2 = (1.066*100%) – 100% = 6.6%

Tr Ts3 = (1.151*100%) – 100% = 15.1%

2)y million rubles – average product productivity

2,921 + 0,294*(-4) = 2,921-1,176 = 1,745

2,921 + 0,294*(-3) = 2,921-0,882 = 2,039

(yt-y) = (1.745-2.04) = 0.087

(yt-yt) = (1.745-2.921) = 1.382

(y-yt) = (2.04-2.921) = 0.776

Tp

By

y2005=2.921+1.496*4=2.921+5.984=8.905

8,905+2,306*1,496=12,354

8,905-2,306*1,496=5,456

5,456 2005 12,354

Task No. 3

Statistical data on wholesale supplies of food and non-food items and the retail trade network of the region in 2003 and 2004 are presented in the corresponding graphs.

According to Tables 1 and 2, it is required

1. Find the general index of the wholesale supply of food products in actual prices;

2. Find the general index of the actual volume of food supply;

3. Compare general indices and draw the appropriate conclusion;

4. Find the general index of supply of non-food products in actual prices;

5. Find the general index of the physical volume of supply of non-food products;

6. Compare the obtained indices and draw conclusions on non-food products;

7. Find the consolidated general supply indexes of the entire commodity mass in actual prices;

8. Find the consolidated general index of physical volume (for the entire commodity mass of goods);

9. Compare the resulting summary indices and draw the appropriate conclusion.

	Base period			Reporting period (2004)			Supplies of the reporting period at prices of the base period
			1,291-0,681=0,61= - 39 Conclusion In conclusion, let's summarize. Average values ​​are general indicators in which the effect of general conditions and the pattern of the phenomenon being studied are expressed. Statistical averages are calculated on the basis of mass data from correctly statistically organized mass observation (continuous or sample). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). The use of averages should proceed from a dialectical understanding of the categories of general and individual, mass and individual. The average reflects what is common in each individual, individual object; therefore, the average becomes of great importance for identifying patterns inherent in mass social phenomena and invisible in individual phenomena. The deviation of the individual from the general is a manifestation of the development process. In some isolated cases, elements of the new, advanced may be laid down. In this case, it is specific factors, taken against the background of average values, that characterize the development process. Therefore, the average reflects the characteristic, typical, real level of the phenomena being studied. The characteristics of these levels and their changes in time and space are one of the main problems of averages. Thus, through the averages, for example, characteristic of enterprises at a certain stage of economic development is manifested; changes in the well-being of the population are reflected in average wages, family income in general and for individual social groups, and the level of consumption of products, goods and services. The average indicator is a typical value (ordinary, normal, prevailing as a whole), but it is such because it is formed in the normal, natural conditions of the existence of a specific mass phenomenon, considered as a whole. The average reflects the objective property of the phenomenon. In reality, often only deviant phenomena exist, and the average as a phenomenon may not exist, although the concept of typicality of a phenomenon is borrowed from reality. The average value is a reflection of the value of the characteristic being studied and, therefore, is measured in the same dimension as this characteristic. However, there are various ways to approximate the level of population distribution for comparing summary characteristics that are not directly comparable to each other, for example, the average population size in relation to the territory (average population density). Depending on which factor needs to be eliminated, the content of the average will also be determined. The combination of general means with group means makes it possible to limit qualitatively homogeneous populations. By dividing the mass of objects that make up this or that complex phenomenon into internally homogeneous, but qualitatively different groups, characterizing each of the groups with its average, it is possible to reveal the reserves of the process of an emerging new quality. For example, the distribution of the population by income allows us to identify the formation of new social groups. In the analytical part, we looked at a particular example of using the average value. To summarize, we can say that the scope and use of averages in statistics is quite wide. Bibliography 1. Gusarov, V.M. Theory of statistics by quality [Text]: textbook. allowance / V.M. Gusarov manual for universities. - M., 1998 2. Edronova, N.N. General theory of statistics [Text]: textbook / Ed. N.N. Edronova - M.: Finance and Statistics 2001 - 648 p. 3. Eliseeva I.I., Yuzbashev M.M. General theory of statistics [Text]: Textbook / Ed. Corresponding member RAS I.I. Eliseeva. – 4th ed., revised. and additional - M.: Finance and Statistics, 1999. - 480 pp.: ill. 4. Efimova M.R., Petrova E.V., Rumyantsev V.N. General theory of statistics: [Text]: Textbook. - M.: INFRA-M, 1996. - 416 p. 5. Ryauzova, N.N. General theory of statistics [Text]: textbook / Ed. N.N. Ryauzova - M.: Finance and Statistics, 1984. Gusarov V.M. Theory of statistics: Textbook. A manual for universities. - M., 1998.-P.60. Eliseeva I.I., Yuzbashev M.M. General theory of statistics. - M., 1999.-P.76. Gusarov V.M. Theory of statistics: Textbook. A manual for universities. -M., 1998.-P.61.