Lecture 5. Average values
The concept of average in statistics
Arithmetic mean and its properties
Other types of power averages
Mode and median
Quartiles and deciles
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 extremely important, 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.
average value (in statistics) – a general indicator characterizing the typical size or level of social phenomena per unit of the population, all other things being equal.
Using the method of averages, the following can be solved: main goals:
1. Characteristics of the level of development of phenomena.
2. Comparison of two or more levels.
3. Study of the interrelations of socio-economic phenomena.
4. Analysis of the location of socio-economic phenomena in space.
Statistical averages are calculated on the basis of mass data from correctly statistically organized mass observation (continuous and selective). In this case, 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 output of a salesperson depends on many reasons: qualifications, length of service, age, form of service, health, etc.
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 changes caused by the action of basic factors. This allows the average to reflect the typical level of the trait and abstract from the individual characteristics inherent in individual units.
The average value is a reflection of the values of the characteristic being studied, therefore, it is measured in the same dimension as the given characteristic.
Each average value characterizes the population under study according to any one characteristic. In order to obtain a complete and comprehensive picture of the population being studied according to a number of essential characteristics, in general it is extremely important 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.
The concept of average in statistics - concept and types. Classification and features of the category "The concept of average value in statistics" 2017, 2018.
Statistical averages have several types, but all of them belong to the class of power averages, i.e. averages constructed from various degrees of options: arithmetic average, harmonic average, quadratic average, geometric average, etc.
The general form of the power average formula is as follows:
Where X - average of a certain degree (read “X with a line”); X - options (changing characteristic values); P - number option (number of units in total); T - exponent of average value; Z - summation sign.
When calculating various power averages, all the main indicators on the basis of which this calculation is carried out (x, P ), remain unchanged. Only the magnitude changes T and accordingly x.
If t = 2, then it turns out mean square. Its formula:
If T = 1, then it turns out arithmetic average. Its formula:
If t = - 1, then it turns out harmonic mean. Its formula:
If t = 0, then it turns out geometric mean. Its formula:
Different types of averages with the same initial indicators (value of option x and their number P ) have, due to different values of the degree, far from the same numerical values. Let's look at them using specific examples.
Let's assume that in village N in 1995 three motor vehicle crimes were registered, and in 1996 - six. In this case x x = 3, x 2 = 6, a P (number of options, years) in both cases is 2.
When the degree value T = 2 we get the root mean square value:
When the degree value t = 1 we get the arithmetic average:
When the degree value T = 0 we obtain the geometric mean value:
When the degree value t = - 1 we get the harmonic mean value:
The calculations showed that different averages form the following chain of inequality among themselves:
The pattern is simple: the lower the degree of the average (2; 1; 0; -1), the lower the value of the corresponding average. Thus, each average of the given series is majorant (from the French majeur - greater) in relation to the averages to the right of it. It is called the rule of majorance of averages.
In the given simplified examples, the values of option (x) were not repeated: the value 3 appeared once and the value 6 also. Statistical realities are more complex. Option values can be repeated several times. Let us recall the rationale for the sampling method based on the experimental extraction of cards numbered from 1 to 10. Some card numbers were extracted two, three, five, eight times. When calculating the average age of convicts, the average sentence, the average period of investigation or consideration of criminal cases, the same option (x), for example, age 20 years or a sentence of five years, can be repeated dozens and even hundreds of times, i.e. or another frequency (/). In this case, the symbol / - is introduced into the general and special formulas for calculating averages frequency. The frequencies are called statistical weights, or average weights, and the average itself is called weighted power average. This means that each option (age 25 years) is, as it were, weighed by frequency (40 people), i.e., multiplied by it.
So, the general formula for a weighted power average is:
Where X - weighted average t x - options (changing values of the characteristic); T - average degree index; I - summation sign; / - frequency option.
The formulas for other weighted averages will look like this:
mean square - 
arithmetic average - 
geometric mean -
harmonic mean - 
The choice of a regular average or a weighted one is determined by the statistical material, and the choice of the type of power (arithmetic, geometric, etc.) is determined by the purpose of the study. Let us remember that when we calculated the average annual increase in absolute indicators, we resorted to the arithmetic mean, and when we calculated the average annual growth (decrease) rates, we were forced to turn to the geometric mean, since the arithmetic mean could not perform this task, as it led to erroneous conclusions.
In legal statistics, the arithmetic mean is most widely used. It is used to assess the workload of operational workers, investigators, prosecutors, judges, lawyers, and other employees of legal institutions; calculating the absolute increase (decrease) in crime, criminal and civil cases and other units of measurement; justification for selective observation, etc.
The geometric mean value is used when calculating the average annual growth (decrease) rate of legally significant phenomena.
The mean square indicator (mean square deviation, standard deviation) plays an important role in measuring the relationships between the phenomena being studied and their causes, in substantiating the correlation dependence.
Some of these means, which are widely used in legal statistics, as well as the mode and median, will be discussed in more detail in subsequent paragraphs. The harmonic mean, the cubic mean, and the progressive mean (an invention of the Soviet era) are practically not used in legal statistics. The harmonic mean, for example, which previous forensic statistics textbooks have discussed in detail with abstract examples, is disputed by prominent economic statisticians. They consider the harmonic mean to be the reciprocal of the arithmetic mean, and therefore, in their opinion, it has no independent meaning, although other statisticians see certain advantages in it. Without delving into the theoretical disputes of economic statisticians, we will say that we do not describe the harmonic mean in detail due to its non-application in legal analysis.
In addition to ordinary and weighted power averages, to characterize the average value, options in the variation series can be taken not by calculated, but by descriptive averages: fashion(the most common option) and median(middle option in the variation series). They are widely used in legal statistics.
- See: Ostroumov S.S. Decree. op. pp. 177-180.
- See: Paskhaver I.S. Average values in statistics. M., 1979. S. 134-150; Ryauzov N. N. Decree. op. pp. 171-174.
General theory of statistics: lecture notes Konik Nina Vladimirovna
2. Types of averages
2. Types of averages
In statistics, various types of averages are used, which are divided into two large classes:
1) power means (harmonic mean, geometric mean, arithmetic mean, quadratic mean, cubic mean);
2) structural averages (mode, median). To calculate power averages, it is necessary to use all available characteristic values. The mode and median are determined only by the structure of the distribution. Therefore, they are called structural, positional averages. The median and mode are often used as an average characteristic in those populations where calculating the power mean is impossible or impractical.
The most common type of average is the arithmetic mean. The arithmetic mean is the value of a characteristic that each unit of the population would have if the total sum of all values of the characteristic were distributed evenly among all units of the population. In the general case, its calculation comes down to summing all the values of the varying characteristic and dividing the resulting amount by the total number of units in the population. For example, five workers fulfilled an order for the manufacture of parts, while the first produced 5 parts, the second - 7, the third - 4, the fourth - 10, the fifth - 12. Since in the source data the value of each option occurred only once to determine the average output of one worker , you should apply the simple arithmetic average formula:
i.e. in our example, the average output of one worker
Along with the simple arithmetic average, the weighted arithmetic average is studied. For example, let’s calculate the average age of students in a group of 20 people, whose ages vary from 18 to 22 years, where x i are the variants of the characteristic being averaged, f is the frequency, which shows how many times the i-th value occurs in the population.
Applying the weighted arithmetic mean formula, we get:
There is a certain rule for choosing a weighted arithmetic average: if there is a series of data on two interrelated indicators, for one of which it is necessary to calculate the average value, and the numerical values of the denominator of its logical formula are known, and the values of the numerator are not known, but can be found as a product these indicators, then the average value should be calculated using the weighted arithmetic average formula.
In some cases, the nature of the initial statistical data is such that the calculation of the arithmetic mean loses its meaning and the only generalizing indicator can only be another type of mean - the harmonic mean. Currently, the computational properties of the arithmetic mean have lost their relevance in the calculation of general statistical indicators due to the widespread introduction of electronic computing technology. The harmonic mean value, which can also be simple and weighted, has acquired great practical importance. If the numerical values of the numerator of a logical formula are known, but the values of the denominator are not known, then the average value is calculated using the harmonic weighted average formula.
If, when using the harmonic mean, the weights of all options (f ;) are equal, then instead of the weighted one, you can use a simple (unweighted) harmonic mean:
where x are individual options;
n – number of variants of the characteristic being averaged.
For example, simple harmonic mean can be applied to speed if the path segments covered at different speeds are equal.
Any average value must be calculated so that when it replaces each variant of the averaged characteristic, the value of some final, general indicator that is associated with the averaged indicator does not change. Thus, when replacing actual speeds on individual sections of the path with their average value (average speed), the total distance should not change.
The average formula is determined by the nature (mechanism) of the relationship between this final indicator and the averaged indicator. Therefore, the final indicator, the value of which should not change when replacing the options with their average value, is called the determining indicator. To derive the formula for the average, you need to create and solve an equation using the relationship between the averaged indicator and the determining one. This equation is constructed by replacing the variants of the characteristic (indicator) being averaged with their average value.
In addition to the arithmetic mean and harmonic mean, other types (forms) of the mean are used in statistics. All of them are special cases of power average. If we calculate all types of power averages for the same data, then their values will be the same; the rule of majority of averages applies here. As the exponent of the average increases, the average value itself increases.
The geometric mean is used when there are n growth coefficients, and the individual values of the characteristic are, as a rule, relative dynamics values, constructed in the form of chain values, as a ratio to the previous level of each level in the dynamics series. The average thus characterizes the average growth rate. The simple geometric mean is calculated using the formula:
The weighted geometric mean formula is as follows:
The above formulas are identical, but one is applied for current coefficients or growth rates, and the second is applied for absolute values of series levels.
The mean square is used when calculating with the values of quadratic functions, it is used to measure the degree of fluctuation of individual values of a characteristic around the arithmetic mean in the distribution series and is calculated by the formula:
The weighted mean square is calculated using another formula:
The cubic average is used when calculating with the values of cubic functions and is calculated using the formula:
and the average cubic weighted:
All average values discussed above can be presented as a general formula:
Where x- average value;
x – individual value;
n – number of units of the studied population;
k – exponent that determines the type of average.
When using the same initial data, the larger k in the general power average formula, the larger the average value. It follows from this that there is a natural relationship between the values of power averages:
The average values described above give a generalized idea of the population being studied, and from this point of view, their theoretical, applied and educational significance is indisputable. But it happens that the average value does not coincide with any of the actually existing options. Therefore, in addition to the considered averages, in statistical analysis it is advisable to use the values of specific options that occupy a very specific position in the ordered (ranked) series of attribute values. Among these quantities, the most commonly used are structural (or descriptive) averages– mode (Mo) and median (Me).
Fashion– the value of a characteristic that is most often found in a given population. In relation to a variational series, the mode is the most frequently occurring value of the ranked series, that is, the option with the highest frequency. Fashion can be used in determining the stores that are visited more often, the most common price for any product. It shows the size of a feature characteristic of a significant part of the population, and is determined by the formula:
Where x 0– lower limit of the interval;
h– interval size;
f m– interval frequency;
f m1– frequency of the previous interval;
f m+1– frequency of the next interval.
Median the option located in the center of the ranked row is called. The median divides the series into two equal parts in such a way that there are the same number of population units on either side of it. In this case, one half of the units in the population has a value of the varying characteristic that is less than the median, while the other half has a value greater than it. The median is used when studying an element whose value is greater than or equal to, or at the same time less than or equal to, half of the elements of a distribution series. The median gives a general idea of where the attribute values are concentrated, in other words, where their center is.
The descriptive nature of the median is manifested in the fact that it characterizes the quantitative limit of the values of a varying characteristic that half of the units in the population possess. The problem of finding the median for a discrete variation series is easily solved. If all units of the series are given ordinal numbers, then the ordinal number of the median option is defined as (n+1) /2 with an odd number of terms n. If the number of members of the series is an even number, then the median will be the average value of two options having ordinal numbers n / 2 and n/2+1.
When determining the median in interval variation series, first determine the interval in which it is located (median interval). This interval is characterized by the fact that its accumulated sum of frequencies is equal to or exceeds half the sum of all frequencies of the series. The median of an interval variation series is calculated using the formula:
Where x 0– lower limit of the interval;
h– interval size;
f m– interval frequency;
f – number of series members;
? m -1– the sum of the accumulated terms of the series preceding the given one.
Along with the median, to more fully characterize the structure of the population under study, other values of options that occupy a very specific position in the ranked series are also used. These include quartiles and deciles. Quartiles divide the series by the sum of frequencies into four equal parts, and deciles into ten equal parts. There are three quartiles and nine deciles.
The median and mode, unlike the arithmetic mean, do not eliminate individual differences in the values of a variable characteristic and therefore are additional and very important characteristics of the statistical population. In practice, they are often used instead of the average or along with it. It is especially advisable to calculate the median and mode in cases where the population under study contains a certain number of units with a very large or very small value of the varying characteristic. These values of the options, which are not very characteristic of the population, while affecting the value of the arithmetic mean, do not affect the values of the median and mode, which makes the latter very valuable indicators for economic and statistical analysis.
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A statistical population consists of a set of units, objects or phenomena that are homogeneous in some respects and at the same time have different characteristics. The magnitude of the characteristics of each object is determined both by those common to all units of the population and by its individual characteristics.
Analyzing the ordered series of the distribution (ranking, interval, etc.), one can notice that the elements of the statistical population are clearly concentrated around certain central values. Such a concentration of individual attribute values around certain central values, as a rule, occurs in all statistical distributions. The tendency of individual values of the characteristic under study to group around the center of the frequency distribution is called central tendency. To characterize the central tendency of the distribution, generalizing indicators are used, which are called average values.
Average size in statistics they call a general indicator that characterizes the typical size of a characteristic in a qualitatively homogeneous population under specific conditions of place and time and reflects the value of a varying characteristic per unit of population. The average value is calculated in most cases by dividing the total volume of the characteristic by the number of units possessing this characteristic. If, for example, the monthly wage fund and the number of workers per month are known, then the average monthly wage can be determined by dividing the wage fund by the number of workers.
The average values are indicators such as the average length of a working day, week, year, average wage category of workers, average level of labor productivity, average national income per capita, average grain yield in the country, average food consumption per capita, etc. .d.
Average values are calculated from both absolute and relative values, are named indicators and are measured in the same units of measurement as the averaged characteristic. They characterize the value of the population under study with one number. The average values reflect the objective and typical level of socio-economic phenomena and processes.
Each average characterizes the population under study according to one particular 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, it is used system of averages. For example, indicators of average wages are assessed together with indicators of labor productivity (average output per unit of working time), capital-labor ratio and energy production, level of mechanization and automation of work, etc.
In statistical science and practice, averages are extremely important. The method of averages is one of the most important statistical methods, and the average is one of the main categories of statistical science. The theory of averages occupies one of the central places in the theory of statistics. Average values are the basis for calculating measures of variation (section 5), sampling errors (section 6), variance analysis (section 8) and correlation analysis (section 9).
It is also impossible to imagine statistics without indices, and the latter essentially represent average values. The use of the statistical grouping method also leads to the use of average values.
As already noted, the grouping method is one of the main methods of statistics. The method of averages in combination with the grouping method is an integral part of a scientifically developed statistical methodology. Average indicators organically complement the method of statistical groupings.
Average values are used to characterize changes in phenomena over time, to calculate average growth rates and increments. For example, a comparison of the average growth rates of labor productivity and wages for a certain period (a number of years) reveals the nature of the development of the phenomenon over the period of time being studied, separately labor productivity and separately wages. A comparison of the growth rates of these two phenomena gives an idea of the nature and peculiarity of the relationship between the growth or decline of labor productivity relative to its payment for certain periods of time.
In all cases when it becomes necessary to characterize with one number a set of values of a characteristic that changes, its average value is used.
In a statistical aggregate, the value of a characteristic changes from object to object, that is, it varies. By averaging these values and providing the level value of the attribute to each member of the population, we abstract from the individual values of the attribute, thereby, as it were, replacing the series of distributions of attribute values with the same value equal to the average value. However, such an abstraction is legitimate only if the averaging does not change the basic property in relation to the given feature as a whole. This basic property of a statistical population, associated with individual values of a characteristic, and which, when averaging, must be kept unchanged, is called the defining property of the average in relation to the characteristic under study. In other words, the average, replacing the individual values of the attribute, should not change the overall volume of the phenomenon, i.e. This equality is mandatory: the volume of the phenomenon is equal to the product of the average value and the size of the population. For example, if from three barley yield values (x, = 20.0; 23.3; 23.6 c/ha), the average is calculated (20.0 + 23.3 + 23.6): 3 = 22.3 c/ha ha, then according to the defining property of the average the following equality must be observed:
As can be seen from the above example, the average barley yield does not coincide with any of the individual ones, since not a single farm yielded 22.3 c/ha. However, if we imagine that each farm received 22.3 c/ha, then the total yield will not change and will be equal to 66.9 c/ha. Consequently, the average, replacing the actual value of individual individual indicators, cannot change the size of the entire sum of values of the characteristic being studied.
The main significance of average values lies in their generalizing function, i.e. in replacing many different individual values of a characteristic with an average value that characterizes the entire set of phenomena. The ability of the average to characterize not individual units, but to express the level of a characteristic per each unit of the population is its distinctive ability. This feature makes the average a generalizing indicator of the level of varying characteristics, i.e. an indicator that abstracts from the individual values of the value of a characteristic in individual units of the population. But the fact that the average is abstract does not deprive it of scientific research. Abstraction is a necessary degree of any scientific research. In the average value, as in any abstraction, the dialectical unity of the individual and the general is realized. The relationship between the average and individual values of the averaged characteristic serves as an expression of the dialectical connection between the individual and the general.
The use of averages should be based on the understanding and interrelation of the dialectical categories of general and individual, mass and individual.
The average value reflects what is common in each individual, individual object. Thanks to this, the average becomes of great importance for identifying patterns inherent in mass social phenomena and not noticeable in individual phenomena.
In the development of phenomena, necessity is combined with chance. Therefore, average values are related to the law of large numbers. The essence of this connection is that when calculating the average value, random fluctuations that have different directions, due to the law of large numbers, are mutually balanced, canceled out, and the average value clearly displays the basic pattern, necessity, and influence of general conditions characteristic of a given population. The average reflects the typical, real level of the phenomena being studied. Estimating these levels and changing them in time and space is one of the main tasks of averages. Thus, through averages, for example, the pattern of increasing labor productivity, crop yields, and animal productivity is manifested. Consequently, average values represent general indicators in which the effect of general conditions and the pattern of the phenomenon being studied are expressed.
Using average values, we study changes in phenomena in time and space, trends in their development, connections and dependencies between characteristics, the effectiveness of various forms of organization of production, labor and technology, the introduction of scientific and technological progress, the identification of new, progressive in the development of certain social- economic phenomena and processes.
Average values are widely used in the statistical analysis of socio-economic phenomena, since it is in them that the patterns and trends in the development of mass social phenomena that vary both in time and space find their manifestation. So, for example, the pattern of increasing labor productivity in the economy is reflected in the growth of average production per worker employed in production, the increase in gross harvests - in the growth of average crop yields, etc.
The average value gives a generalized characteristic of the phenomenon under study based on only one characteristic, which reflects one of its most important aspects. In this regard, for a comprehensive analysis of the phenomenon under study, it is necessary to build a system of average values for a number of interrelated and complementary essential features.
In order for the average to reflect what is truly typical and natural in the social phenomena being studied, when calculating it, it is necessary to adhere to the following conditions.
1. The criterion by which the average is calculated must be significant. Otherwise, an insignificant or distorted average will be obtained.
2. The average must be calculated only for a qualitatively homogeneous population. Therefore, the direct calculation of averages must be preceded by statistical grouping, which makes it possible to divide the population under study into qualitatively homogeneous groups. In this regard, the scientific basis of the method of averages is the method of statistical groupings.
The question of the homogeneity of a population should not be decided formally by the form of its distribution. This, like the question of the typicality of the average, must be resolved based on the causes and conditions that form the totality. A set is also homogeneous, the units of which are formed under the influence of common main causes and conditions that determine the general level of a given characteristic, characteristic of the entire set.
3. The calculation of the average value should be based on the coverage of all units of a given type or a sufficiently large set of objects so that random fluctuations are mutually equal to each other and a pattern appears, typical and characteristic sizes of the characteristic being studied.
4. A general requirement when calculating any type of average values is the obligatory preservation of the total volume of the attribute in the aggregate when replacing its individual values with an average value (the so-called defining property of the average).
Topic 3. Method 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. The characteristics 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. Starting with W. Petty, averages began to be considered as the main technique of statistical analysis.
General principles for using averages:
1) a reasonable choice of the population unit for which the average value is calculated is necessary;
2) 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;
3) 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;
4) general 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” in this case 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.
If a population with qualitatively homogeneous characteristics is studied, then the average value acts here as typical average. For example, for groups of workers in a certain industry with a fixed income level, a typical average expenditure on basic necessities is determined.
When studying a population with qualitatively heterogeneous characteristics, the atypicality of average indicators may come to the fore. These, for example, are the average indicators of produced national income per capita (different age groups). Average values generalize qualitatively heterogeneous values of characteristics or systemic spatial aggregates (international community, continent, state, region, region, etc.) or dynamic aggregates extended in time (century, decade, year, season, etc.). ). Such average values are called system averages.
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.
Harmonic mean is called the inverse arithmetic mean, since this value is obtained at k = -1. Simple harmonic mean used when the weights of the characteristic values are the same. For example, you need to calculate the average speed of two cars that covered the same path, but at different speeds: the first at a speed of 100 km/h, the second at 90 km/h. Using the harmonic mean method, we calculate the average speed:
In statistical practice it is more often used weighted harmonic mean – for those cases when the weights (or volumes of phenomena) for each attribute are not equal, and in the initial ratio for calculating the average the numerator is known, but the denominator is unknown.
For example, when calculating the average price, we must use the ratio of the sales amount to the number of units sold. We do not know the number of units sold (we are talking about different products), but we know the sales amounts of these different products. Let's say you need to find out the average price of goods sold (Table 3.2).
Table 3.2 – Initial data
We get:
If you use the arithmetic average formula here, you can get an average price that will be unrealistic:
If, when calculating the average price by weight, we take the number of goods, then the correct result is given by the formula for the arithmetic weighted average. If we use the cost of the batches as weights, then the harmonic average gives the correct result.
That is, averageHarmonic is not a special type of average, but rather a special method of calculating the arithmetic average. In statistics, it is still customary to distinguish the harmonic mean as a separate type of mean, because with its help, the technique of calculating the arithmetic mean can be simplified and, more importantly, the nature of the available statistical material can be taken into account.
The correctness of the choice of the form of the mean (arithmetic or harmonic) can also be checked additional criterion: if absolute values are used as weights, any intermediate actions when calculating the average should give significant indicators. For example, to calculate the average price, multiply the price by the number of goods to obtain their cost. And dividing the cost of goods by their prices gives the quantity of goods.
Using the harmonic mean in statistics, the average percentage of plan completion is also determined (based on the actual implementation of the plan), the average time spent on performing operations (based on the average time spent on one operation and the total work time for individual employees), etc.
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.