Topic 4
Main questions: 1. Absolute statistical values.
2. Types of absolute statistical quantities.
3. Relative values.
4. Types of relative quantities.
5. Average value. Types of averages.
6. Arithmetic mean.
7. Harmonic mean.
8. Geometric mean.
9. Mean square and mean cubic.
10. Structural averages.
11. Relationships between the arithmetic mean, median and mode in statistical distributions.
1.Absolute statistical values. To reflect the size and volume of phenomena, absolute values are used in statistics. The absolute value (A.V.) is obtained as a result of a summary of statistical material. A.V. are expressed in various units of measurement - natural, cost (monetary), conditional, labor.
1) Natural units of measurement characterize the magnitude and size of the phenomena being studied. They are expressed in meters, tons, liters, etc. Natural units can only be summed up for homogeneous products; you cannot add up tons of steel with meters of fabric.
2) Cost units are used to evaluate many statistical indicators in monetary terms: the size of retail trade turnover, GDP, personal income, etc.
3) Conditional. In some cases, not all types of homogeneous products can be summarized. You cannot add up soap (since it has a different percentage of fat content), fuel (different calorie content), etc. U.e.i. used to account for homogeneous products different varieties. For example, canned food is produced in jars of different capacities. Therefore, they are counted in thousands of conventional jars. The net weight of the product is 400 grams for one conventional can.
4) Labor units of measurement – man-hours, man-days, etc. Used to measure labor resources, labor costs.
2.Types of absolute statistical quantities. By way of expression:
1) Individual - A.V., characterizing the size of a characteristic in individual units of the population (for example, the salary of an individual employee, the size of the sown area of a particular farm). They are obtained directly in the process of statistical observation and are recorded in primary accounting documents.
2) Total A.V. – express the value of one or another characteristic of all units of the population being studied or its individual groups and are obtained as a result of summing up individual A.V. (salary according to the enterprise).
A.V. are always named numbers. They are expressed in certain units of measurement (kg, pcs., tons, ha, m, etc.).
In practical activities in the absence necessary information absolute values are obtained by calculation, for example, based on balance sheet linkage:
where is the stock at the beginning of the period; – receipts for the period; – expense for the period; – stock at the end of the period.
From here 
.
Absolute statistical values are widely used in the analysis and forecasting of the state and development of social life phenomena.
Based on A.V. calculate relative quantities.
3.Relative values (R.V.). They are obtained by dividing one quantity by another. The numerator of the ratio is the value being compared, it is called current or reporting quantity, the denominator of the ratio is called the basis of comparison or the basis of comparison.
If the comparison base is 100, then O.V. expressed in (%), if the comparison base is 1,000 – ppm (‰), 10,000 – in prodecimille (‰0).
The compared quantities can be of the same name or different. If values of the same name are compared, they are expressed in coefficients, percentages, ppm. When comparing different values, the names of the relative values are formed from the names of the compared values: population density - people/km 2, yield - c/ha, etc.
4.Types of relative values (indicators).
1) plan target - GPZ;
2) implementation of the plan - OPVP;
3) speakers (OPD);
4) structures (d);
5) intensity and level of development;
6) coordination (OPK);
7) comparisons (OPS).
1) OPZ- serves for planning. It is calculated by the ratio of the level planned for the upcoming period (P) to the level of the indicator achieved in the previous period ():
2) OPVP– serves to compare the results actually achieved with those previously planned.
,
– the level achieved in the current period; - plan for the same period.
3) OPD– characterizes the change in the level of an economic phenomenon over time and is obtained by dividing the level of an attribute for a certain period or point in time by the level of the same indicator in the previous period or point in time. In another way, they are called growth rates. Calculated in coefficients or %.
4) d– characterize the composition of the population under study, shares, specific gravity elements of the population in the overall total and represent the ratio of a part of the units of the population () to the total number of units of the population ():
5) Intensity and level of development– characterize the degree of saturation or development of a given phenomenon in a certain environment, are named and can be expressed in multiple ratios, %, ‰ and other forms.
6) defense industry– characterizes the relationship of parts of the population being studied to one of them, taken as the basis of comparison. They show how many times one part of a population is larger than another, or how many units of one part are equal to 1, 10, 100, 1000 units of another part. These relative values can be calculated both by absolute indicators and by structural indicators.
7) OPS– characterize the relationships of the same absolute or relative indicators corresponding to the same period or point in time, but relating to different objects or territories.
5.Average value. Types of averages.
Definition: 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.
Types of averages: 1) arithmetic;
2) harmonic;
3) geometric;
4) quadratic;
5) cubic.
All these averages belong to the class of power averages and are united by the general formula (for different meanings m):
,
where is the average value of the phenomenon under study;
– average degree indicator;
– current value of the characteristic being averaged;
– number of signs.
Depending on the value of the exponent m, there are the following types power averages:
at – harmonic mean;
at – geometric mean;
at – arithmetic mean;
at – root mean square;
at – average cubic .
When using the same data, the larger m, the more value average size:
– the rule of majorance of averages.
The type of average is selected in each case through a specific analysis of the population being studied; it is determined by the material content of the phenomenon being studied.
6.Arithmetic mean.
a) Simple arithmetic mean 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 (the most common).
Often it is necessary to calculate the average using group averages or averages individual parts population (partial average), i.e. the average of the averages. For example, the average life expectancy of a country’s citizens is the average of the average life expectancies for individual regions of a given country.
The average of the average values is calculated using the following formula, counting:
,
where is the number of units in each group.
Properties of average values:
1. If all individual values of a characteristic are reduced (increased) by a factor, then the average value of the new characteristic will correspondingly decrease (increase) by a factor.
; 
2. If the variants of the characteristic being averaged are reduced (increased) by , then the arithmetic mean will correspondingly decrease (increase) by the same number.
3. If the weights of all averaged options decrease (increase) by a factor, then the arithmetic average will not change.
4. The sum of deviations from the average is zero.
7.Harmonic mean. Used in cases where frequencies for individual options are not known x aggregates, and their work is presented. Let us denote this product by , then we obtain the formula for the harmonic weighted average:
.
is a transformed form and is identical to it. You can always calculate instead, but to do this you need to determine the weights individual values signs hidden in the scales of the harmonic mean.
In cases where the weight of each option is equal to one, the mean harmonic simple:
,
where are individual variants of the inverse characteristic, occurring once,
– number of options.
If harmonic averages are given for two parts of the population (number and ), then the overall harmonic average for the entire population can be represented as a weighted harmonic average of the group averages:
.
8.Geometric mean. It is used when the individual values of the attribute are characterized by the average growth coefficient (they 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). Calculated by the formula:
– number of options; - sign of the work.
It is most widely used to determine the average rate of change in time series, as well as in distribution series (we will consider its use later).
9.Mean square and mean cubic.
– used to calculate the average side size of n square sections, pipe diameters, etc.
Definition:Mode () – value random variable meeting with most likely in a discrete variation series – the option with the highest frequency.
Widely used in studying customer demand, recording prices, etc.
Formula for calculation:
,
where is the lower limit of the modal interval;
– frequencies in the modal, previous and following modal interval (respectively).
The modal interval is determined by the highest frequency.
Definition:Median is an option that is in the middle of the variation series.
Divides the series into two equal (by the number of units) parts - with attribute values less than the median and with attribute values greater than the median.
The mode and median, as a rule, differ from the mean value, coinciding with it only in the case of a symmetrical frequency distribution of the variation series. Therefore, the ratio of the mode, median and arithmetic mean allows us to evaluate the asymmetry of the distribution series.
Mode and median are usually complementary to the population mean and are used in mathematical statistics to analyze the shape of distribution series.
Similarly to the median, the values of a characteristic are calculated, dividing the population into four equal (by the number of units) parts - quartiles, into five - quintiles, into ten - deciles, into one hundred - percentiles.
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” 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.
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 the total number of 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.
General theory of statistics: lecture notes Konik Nina Vladimirovna
2. Types of averages
In statistics they use different kinds average values, 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 it occurs i-th value In total.
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. Big practical significance acquired an average harmonic value, which can also be simple and weighted. 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. So, when replacing actual speeds on individual sections of the route 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 root mean square is used when calculating with quantities quadratic functions, 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 general idea about where the values of the attribute are concentrated, in other words, where their center is located.
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 cancel out individual differences in the values of a varying characteristic and are therefore additional and very important characteristics 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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Average values are widely used in statistics. average value- this is a general indicator that reflects actions general conditions and patterns of the phenomenon being studied.
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 tendency of patterns economic development. Average values characterize qualitative indicators commercial activities: distribution costs, profit, profitability, etc.
Statistical averages are calculated on the basis of data from properly 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 wages in cooperatives and state-owned enterprises, and the result is extended to the entire population, then the average is fictitious, since it was calculated based on 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. At the same time, generalizing general property aggregate, the average obscures (understates) some indicators and overestimates others.
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 based on a number of essential characteristics as a whole, it is necessary to have a system of average values that can describe the phenomenon from different angles.
The most important condition for the scientific use of average values in the statistical analysis of social phenomena is population homogeneity, for which the average is calculated. Identical in form and calculation technique, the average is fictitious in some conditions (for a heterogeneous population), while in others (for a homogeneous population) it corresponds to reality. The qualitative homogeneity of the population is determined on the basis of comprehensive theoretical analysis essence of the phenomenon.
There are different types of averages in simple or weighted form:
(clickable)
Majority rule average: the higher the exponent m, the greater the average value.
The simultaneous use of certain properties makes it possible to simplify the calculation of the arithmetic mean:
you can subtract a constant value from all characteristic valuesA,reduce the differences by a common factorK, and all the weights fdivide by the same number and, using the changed data, calculate the average. Then, if the resulting average value is multiplied byK, and add to the productA, then we obtain the desired value of the arithmetic mean using the formula:
The resulting transformed average is called first order moment, and the above method for calculating the average is way of moments, or counting from a conditional zero.
If, during grouping, the values of the characteristic being averaged are specified in intervals, then when calculating the arithmetic mean, the midpoints of these intervals are taken as the value of the characteristic in groups, that is, they are based on the assumption of a uniform distribution of population units over the interval of characteristic values. For open intervals in the first and last group, if there are any, the values of the attribute must be determined expertly, based on the essence of the properties of the attribute and the aggregate. If it is not possible expert assessment, attribute values in open intervals, to find the missing boundary of an open interval, use the range (the difference between the values of the end and beginning of the interval) of the adjacent interval (the “neighbor” principle). In other words, the width (step) of an open interval is determined by the size of the adjacent interval.
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 average (2; 1; 0; -1), the less value 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's remember the rationale sampling method based on experimental retrieval of cards numbered from 1 to 10. Some card numbers were retrieved 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, general formula weighted power average has the form:
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 most wide application finds the arithmetic mean. 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.
