Method of averages

3.1 The essence and meaning of averages in statistics. Types of averages

Average size in statistics is a generalized characteristic of qualitatively homogeneous phenomena and processes according to some varying characteristic, which shows the level of the characteristic related to a unit of the population. average value abstract, because characterizes the value of a characteristic in some impersonal unit of the population.Essence average value is that through the individual and random the general and necessary are revealed, that is, the tendency and pattern in the development of mass phenomena. Signs that are generalized in average values ​​are inherent in all units of the population. Due to this, the average value is of great importance for identifying patterns inherent in mass phenomena and not noticeable in individual units of the population

General principles for using averages:

a reasonable choice of the population unit for which the average value is calculated is necessary;

when determining the average value, one must proceed from the qualitative content of the characteristic being averaged, take into account the relationship of the characteristics being studied, as well as the data available for calculation;

average values ​​should be calculated based on qualitatively homogeneous populations, which are obtained by the grouping method, which involves the calculation of a system of generalizing indicators;

overall averages must be supported by group averages.

Depending on the nature of the primary data, the scope of application and the method of calculation in statistics, the following are distinguished: main types of medium:

1) power averages(arithmetic mean, harmonic, geometric, mean square and cubic);

2) structural (nonparametric) means(mode and median).

In statistics, the correct characterization of the population being studied according to a varying characteristic in each individual case is provided only by a very specific type of average. The question of what type of average needs to be applied in a particular case is resolved through a specific analysis of the population being studied, as well as based on the principle of meaningfulness of the results when summing or when weighing. These and other principles are expressed in statistics theory of averages.

For example, the arithmetic mean and the harmonic mean are used to characterize the average value of a varying characteristic in the population being studied. The geometric mean is used only when calculating average rates of dynamics, and the quadratic mean is used only when calculating variation indices.

Formulas for calculating average values ​​are presented in Table 3.1.

Table 3.1 – Formulas for calculating average values

Types of averages	Calculation formulas
	simple	weighted
1. Arithmetic mean
2. Harmonic mean
3. Geometric mean
4. Mean square

Designations:- quantities for which the average is calculated; - average, where the bar above indicates that averaging of individual values ​​takes place; - frequency (repeatability of individual values ​​of a characteristic).

Obviously, the various averages are derived from general formula for power average (3.1) :

, (3.1)

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

Average values ​​can be simple or weighted. Weighted averages values ​​are called that take into account that some variants of attribute values ​​may have different numbers; in this regard, each option has to be multiplied by this number. The “scales” are the numbers of aggregate units in different groups, i.e. Each option is “weighted” by its frequency. The frequency f is called statistical weight or average weight.

Eventually correct choice of average assumes the following sequence:

a) establishing a general indicator of the population;

b) determination of a mathematical relationship of quantities for a given general indicator;

c) replacing individual values ​​with average values;

d) calculation of the average using the appropriate equation.

3.2 Arithmetic mean and its properties and calculus techniques. Harmonic mean

Arithmetic mean– the most common type of medium size; it is calculated in cases where the volume of the averaged characteristic is formed as the sum of its values ​​for individual units of the statistical population being studied.

The most important properties of the arithmetic mean:

1. The product of the average by the sum of frequencies is always equal to the sum of the products of variants (individual values) by frequencies.

2. If you subtract (add) any arbitrary number from each option, then the new average will decrease (increase) by the same number.

3. If each option is multiplied (divided) by some arbitrary number, then the new average will increase (decrease) by the same amount

4. If all frequencies (weights) are divided or multiplied by any number, then the arithmetic average will not change.

5. The sum of deviations of individual options from the arithmetic mean is always zero.

You can subtract an arbitrary constant value from all the values ​​of the attribute (preferably the value of the middle option or options with the highest frequency), reduce the resulting differences by a common factor (preferably by the value of the interval), and express the frequencies in particulars (in percentages) and multiply the calculated average by the common factor and add an arbitrary constant value. This method of calculating the arithmetic mean is called method of calculation from conditional zero .

Geometric mean finds its application in determining average growth rates (average growth coefficients), when individual values ​​of a characteristic are presented in the form of relative values. It is also used if it is necessary to find the average between the minimum and maximum values ​​of a characteristic (for example, between 100 and 1000000).

Mean square used to measure the variation of a characteristic in the aggregate (calculation of the standard deviation).

Valid in statistics rule of majority of averages:

X harm.< Х геом. < Х арифм. < Х квадр. < Х куб.

3.3 Structural averages (mode and median)

To determine the structure of a population, special average indicators are used, which include the median and mode, or the so-called structural averages. If the arithmetic mean is calculated based on the use of all variants of attribute values, then the median and mode characterize the value of the variant that occupies a certain average position in the ranked variation series

Fashion- the most typical, most frequently encountered value of the attribute. For discrete series The fashion will be the option with the highest frequency. To determine fashion interval series First, the modal interval (the interval having the highest frequency) is determined. Then, within this interval, the value of the feature is found, which can be a mode.

To find a specific value of the mode of an interval series, you must use formula (3.2)

(3.2)

where XMo is the lower limit of the modal interval; i Mo - the value of the modal interval; f Mo - frequency of the modal interval; f Mo-1 - frequency of the interval preceding the modal one; f Mo+1 is the frequency of the interval following the modal one.

Fashion is widespread in marketing activities when studying consumer demand, especially when determining the most popular sizes of clothing and shoes, and when regulating pricing policies.

Median - the value of a varying characteristic falling in the middle of the ranked population. For ranked series with an odd number individual values ​​(for example, 1, 2, 3, 6, 7, 9, 10) the median will be the value that is located in the center of the series, i.e. the fourth value is 6. For ranked series with an even number individual values ​​(for example, 1, 5, 7, 10, 11, 14) the median will be the arithmetic mean value, which is calculated from two adjacent values. For our case, the median is (7+10)/2= 8.5.

Thus, to find the median, you first need to determine its serial number (its position in the ranked series) using formulas (3.3):

(if there are no frequencies)

N Me =
(if there are frequencies) (3.3)

where n is the number of units in the aggregate.

Numerical value of the median interval series determined by accumulated frequencies in a discrete variation series. To do this, you must first indicate the interval where the median is found in the interval series of the distribution. The median is the first interval where the sum of accumulated frequencies exceeds half of the observations from total number all observations.

The numerical value of the median is usually determined by formula (3.4)

(3.4)

where x Ме is the lower limit of the median interval; iMe - interval value; SМе -1 is the accumulated frequency of the interval that precedes the median; fMe - frequency of the median interval.

Within the found interval, the median is also calculated using the formula Me = xl e, where the second factor on the right side of the equality shows the location of the median within the median interval, and x is the length of this interval. The median divides the variation series in half by frequency. Still being determined quartiles , which divide the variation series into 4 parts of equal size in probability, and deciles , dividing the row into 10 equal parts.

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 the conditions 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 the characteristic, which arise for one reason or another in individual units of observation.

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 being studied 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.:

Fund wages for each group of workers equal to the product variants by 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 series distributions 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 odd number 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).

Most of all in eq. In practice, we have to use the arithmetic mean, which can be calculated as the simple and weighted arithmetic mean.

Arithmetic average (SA)-n The most common type of average. It is used in cases where the volume of a varying characteristic for the entire population is the sum of the values ​​of the characteristics of its individual units. Social phenomena are characterized by the additivity (totality) of the volumes of a varying characteristic; this determines the scope of application of SA and explains its prevalence as a general indicator, for example: the general salary fund is the sum of the salaries of all employees.

To calculate SA, you need to divide the sum of all feature values ​​by their number. SA is used in 2 forms.

Let's first consider a simple arithmetic average.

1-CA simple (initial, defining form) is equal to the simple sum of the individual values ​​of the characteristic being averaged, divided by the total number of these values ​​(used when there are ungrouped index values ​​of the characteristic):

The calculations made can be generalized into the following formula:

(1)

Where - the average value of the varying characteristic, i.e., the simple arithmetic average;

means summation, i.e. the addition of individual characteristics;

x- individual values ​​of a varying characteristic, which are called variants;

n - number of units of the population

Example 1, it is required to find the average output of one worker (mechanic), if it is known how many parts each of 15 workers produced, i.e. given a series of ind. attribute values, pcs.: 21; 20; 20; 19; 21; 19; 18; 22; 19; 20; 21; 20; 18; 19; 20.

Simple SA is calculated using formula (1), pcs.:

Example2. Let's calculate SA based on conditional data for 20 stores included in the trading company (Table 1). Table 1

Distribution of stores of the trading company "Vesna" by sales area, sq. M

Store no.		Store no.

To calculate the average store area ( ) it is necessary to add up the areas of all stores and divide the resulting result by the number of stores:

Thus, the average store area for this group of retail enterprises is 71 sq.m.

Therefore, to determine a simple SA, you need to divide the sum of all values ​​of a given attribute by the number of units possessing this attribute.

2

Where f 1 , f 2 , … ,f n – weight (frequency of repetition of identical signs);

– the sum of the products of the magnitude of features and their frequencies;

– the total number of population units.

- SA weighted - With The middle of options that are repeated a different number of times, or, as they say, have different weights. The weights are the numbers of units in different groups of the population (identical options are combined into a group). SA weighted – average of grouped values x 1 , x 2 , .., x n, – calculated: (2)

Where X- options;

f- frequency (weight).

Weighted SA is the quotient of dividing the sum of the products of options and their corresponding frequencies by the sum of all frequencies. Frequencies ( f) appearing in the SA formula are usually called scales, as a result of which the SA calculated taking into account the weights is called weighted.

We will illustrate the technique of calculating weighted SA using example 1 discussed above. To do this, we will group the initial data and place them in the table.

The average of the grouped data is determined as follows: first, the options are multiplied by the frequencies, then the products are added and the resulting sum is divided by the sum of the frequencies.

According to formula (2), the weighted SA is equal, pcs.:

Distribution of workers for parts production

P

The data presented in the previous example 2 can be combined into homogeneous groups, which are presented in table. Table

Distribution of Vesna stores by sales area, sq. m

Thus, the result was the same. However, this will already be a weighted arithmetic mean value.

In the previous example, we calculated the arithmetic average provided that the absolute frequencies (number of stores) are known. However, in a number of cases, absolute frequencies are absent, but relative frequencies are known, or, as they are commonly called, frequencies that show the proportion or the proportion of frequencies in the entire set.

When calculating SA weighted use frequencies allows you to simplify calculations when the frequency is expressed in large, multi-digit numbers. The calculation is made in the same way, however, since the average value turns out to be increased by 100 times, the result should be divided by 100.

Then the formula for the arithmetic weighted average will look like:

Where d– frequency, i.e. the share of each frequency in the total sum of all frequencies.

(3)

In our example 2, we first determine the share of stores by group in the total number of stores of the Vesna company. So, for the first group the specific gravity corresponds to 10%
. We get the following data Table3

Every person in modern world When planning to take out a loan or stocking up on vegetables for the winter, you periodically come across such a concept as “average value”. Let's find out: what it is, what types and classes exist, and why it is used in statistics and other disciplines.

Average value - what is it?

A similar name (SV) is a generalized characteristic of a set of homogeneous phenomena, determined by any one quantitative variable characteristic.

However, people who are far from such abstruse definitions understand this concept as an average amount of something. For example, before taking out a loan, a bank employee will definitely ask a potential client to provide data on average income for the year, that is, the total amount of money a person earns. It is calculated by summing up the earnings for the entire year and dividing by the number of months. Thus, the bank will be able to determine whether its client will be able to repay the debt on time.

Why is it used?

As a rule, average values ​​are widely used to give a summary description of certain social phenomena of a mass nature. They can also be used for smaller scale calculations, as in the case of a loan in the example above.

However, most often average values ​​are still used for global purposes. An example of one of them is the calculation of the amount of electricity consumed by citizens during one calendar month. Based on the data obtained, maximum standards are subsequently established for categories of the population enjoying benefits from the state.

Also, using average values, the warranty service life of certain household appliances, cars, buildings, etc. is developed. Based on the data collected in this way, modern standards of work and rest were once developed.

Virtually any phenomenon modern life, which is of a mass nature, is in one way or another necessarily connected with the concept under consideration.

Areas of application

This phenomenon is widely used in almost all exact sciences, especially those of an experimental nature.

Finding the average is of great importance in medicine, engineering, cooking, economics, politics, etc.

Based on the data obtained from such generalizations, therapeutic drugs are developed, learning programs, set minimum living wages and salaries, build educational schedules, produce furniture, clothing and shoes, hygiene products and much more.

In mathematics, this term is called the “average value” and is used to solve various examples and problems. The simplest ones are addition and subtraction with ordinary fractions. After all, as you know, to solve such examples it is necessary to bring both fractions to a common denominator.

Also in the queen of exact sciences the term “average value”, which is similar in meaning, is often used random variable" It is more familiar to most as “mathematical expectation”, more often considered in probability theory. It is worth noting that a similar phenomenon also applies when performing statistical calculations.

Average value in statistics

However, the concept being studied is most often used in statistics. As is known, this science itself specializes in the calculation and analysis of the quantitative characteristics of mass social phenomena. Therefore, the average value in statistics is used as a specialized method for achieving its main objectives - collecting and analyzing information.

The essence of this statistical method consists in replacing individual unique values ​​of the characteristic under consideration with a certain balanced average value.

An example is the famous food joke. So, at a certain factory on Tuesdays for lunch, its bosses usually eat meat casserole, and ordinary workers eat stewed cabbage. Based on these data, we can conclude that, on average, the plant staff dine on cabbage rolls on Tuesdays.

Although this example is slightly exaggerated, it illustrates the main drawback of the method of finding the average value - leveling individual characteristics objects or persons.

In average values ​​they are used not only for analyzing the collected information, but also for planning and predicting further actions.

It is also used to evaluate the results achieved (for example, the implementation of the plan for growing and harvesting wheat for the spring-summer season).

How to calculate correctly

Although depending on the type of SV there are different formulas for calculating it, in general theory statistics, as a rule, only one method is used to calculate the average value of a characteristic. To do this, you first need to add together the values ​​of all phenomena, and then divide the resulting sum by their number.

When making such calculations, it is worth remembering that the average value always has the same dimension (or units) as the individual unit of the population.

Conditions for correct calculation

The formula discussed above is very simple and universal, so it is almost impossible to make a mistake with it. However, it is always worth considering two aspects, otherwise the data obtained will not reflect the real situation.

SV classes

Having found answers to the basic questions: “What is the average value?”, “Where is it used?” and “How can you calculate it?”, it is worth finding out what classes and types of SVs exist.

First of all, this phenomenon is divided into 2 classes. These are structural and power averages.

Types of power SVs

Each of the above classes, in turn, is divided into types. The sedate class has four.

• The arithmetic average is the most common type of SV. It is the average term, in determining which the total volume of the characteristic under consideration in a set of data is equally distributed among all units of this set.

  This type is divided into subtypes: simple and weighted arithmetic SV.

• The harmonic mean is an indicator that is the inverse of the simple arithmetic mean, calculated from the reciprocal values ​​of the characteristic under consideration.

  It is used in cases where the individual values ​​of the attribute and the product are known, but the frequency data are not.

• The geometric average is most often used when analyzing the growth rates of economic phenomena. It makes it possible to preserve unchanged the product of the individual values ​​of a given quantity, and not the sum.

  It can also be simple and balanced.

• The mean square value is used when calculating individual indicators, such as the coefficient of variation, characterizing the rhythm of product output, etc.

  It is also used to calculate the average diameters of pipes, wheels, average sides of a square and similar figures.

  Like all other types of averages, the root mean square can be simple and weighted.

Types of structural quantities

In addition to average SVs, they are often used in statistics structural views. They are better suited for calculating the relative characteristics of the values ​​of a varying characteristic and internal structure distribution rows.

There are two such types.

The average value is the most valuable from an analytical point of view and a universal form of expression for statistical indicators. The most common average - the arithmetic average - has a number of mathematical properties that can be used in its calculation. At the same time, when calculating a specific average, it is always advisable to rely on its logical formula, which is the ratio of the volume of the attribute to the volume of the population. For each average there is only one true initial relationship, the implementation of which, depending on the available data, may require various shapes average. However, in all cases where the nature of the value being averaged implies the presence of weights, it is impossible to use their unweighted formulas instead of weighted average formulas.

The average value is the most characteristic value of the attribute for the population and the size of the attribute of the population distributed in equal shares between units of the population.

The characteristic for which the average value is calculated is called averaged .

The average value is an indicator calculated by comparing absolute or relative values. The average value is denoted

The average value reflects the influence of all factors influencing the phenomenon under study and is the resultant for them. In other words, extinguishing individual deviations and eliminating the influence of cases, the average value, reflecting the general measure of the results of this action, acts general pattern the phenomenon being studied.

Conditions for using average values:

Ø homogeneity of the population under study. If some elements of a population subject to the influence of a random factor have values ​​of the characteristic being studied that are significantly different from the rest, then these elements will affect the size of the average for this population. In this case, the average will not express the most typical value of the attribute for the population. If the phenomenon under study is heterogeneous, it requires its division into groups containing homogeneous elements. In this case, group averages are calculated - group averages, expressing the most characteristic value of the phenomenon in each group, and then the overall average value is calculated for all elements, characterizing the phenomenon as a whole. It is calculated as the average of group averages, weighted by the number of population elements included in each group;

Ø a sufficient number of units in total;

Ø the maximum and minimum values ​​of the characteristic in the population being studied.

Average value (indicator)is a generalized quantitative characteristic of a characteristic in a systematic aggregate under specific conditions of place and time.

In statistics, the following forms (types) of averages, called power and structural, are used:

Ø arithmetic mean(simple and weighted);

simple