The most common type of average is the arithmetic mean.
Simple arithmetic mean
A simple arithmetic mean is an average term, in determining which the total volume of a given attribute in the data is equally distributed among all the units included in this population. So, the average annual output per employee is the amount of output that would fall on each employee if the entire volume of output was equally distributed among all employees of the organization. The arithmetic average simple value is calculated by the formula:
Example 1 ... A team of 6 workers receives 3 3.2 3.3 3.5 3.8 3.1 thousand rubles a month.Simple arithmetic mean - Equal to the ratio of the sum of individual values \u200b\u200bof a feature to the number of features in the aggregate
Find Average Salary
Solution: (3 + 3.2 + 3.3 +3.5 + 3.8 + 3.1) / 6 \u003d 3.32 thousand rubles.
Weighted arithmetic mean
If the volume of the data set is large and represents a distribution series, then a weighted arithmetic mean is calculated. This is how the weighted average price per unit of production is determined: the total cost of production (the sum of the products of its quantity by the price of a unit of production) is divided by the total amount of production.
We represent this in the form of the following formula:
Example 2 ... Find the average monthly wage of a workshop workerWeighted arithmetic mean - is equal to the ratio (the sum of the products of the value of a feature to the frequency of repetition of a given feature) to (the sum of the frequencies of all features) .It is used when the variants of the studied population occur an unequal number of times.
Average wages can be obtained by dividing the total wages by the total number of workers:
Answer: 3.35 thousand rubles.
Arithmetic mean for interval series
When calculating the arithmetic mean for an interval variation series, first determine the average for each interval, as the half-sum of the upper and lower boundaries, and then - the average of the entire series. In the case of open intervals, the value of the lower or upper interval is determined by the size of the intervals adjacent to them.
Averages calculated from interval series are approximate.
Example 3... Determine the average age of evening students.
Averages calculated from interval series are approximate. The degree of their approximation depends on the extent to which the actual distribution of the population units within the interval approaches uniform.
When calculating averages, not only absolute, but also relative values \u200b\u200b(frequency) can be used as weights:
The arithmetic mean has a number of properties that more fully reveal its essence and simplify the calculation:
1. The product of the average by the sum of frequencies is always equal to the sum of the products of the variant by the frequencies, i.e.
2. The arithmetic mean of the sum of varying quantities is equal to the sum of the arithmetic means of these quantities:
3. The algebraic sum of deviations of the individual values \u200b\u200bof the attribute from the mean is equal to zero:
4. The sum of the squares of the deviations of the options from the mean is less than the sum of the squares of the deviations from any other arbitrary value, i.e.
In the process of studying mathematics, students get acquainted with the concept of the arithmetic mean. In the future, in statistics and some other sciences, students are faced with the calculation of others. What can they be and how do they differ from each other?
meaning and differences
Not always accurate indicators give an understanding of the situation. In order to assess a particular situation, it is sometimes necessary to analyze a huge number of figures. And then averages come to the rescue. They make it possible to assess the situation as a whole.
Since school days, many adults remember the existence of the arithmetic mean. It is very easy to calculate - the sum of a sequence of n members is divisible by n. That is, if you need to calculate the arithmetic mean in a sequence of values \u200b\u200b27, 22, 34 and 37, then you need to solve the expression (27 + 22 + 34 + 37) / 4, since 4 values \u200b\u200bare used in the calculations. In this case, the desired value will be 30.
Often, within the framework of the school course, geometric mean is also studied. The calculation of this value is based on extracting the nth root of the product of n terms. If we take the same numbers: 27, 22, 34 and 37, the result of the calculations will be 29.4.
Harmonic mean in general education school is usually not a subject of study. Nevertheless, it is used quite often. This value is the reciprocal of the arithmetic mean and is calculated as a quotient of n - the number of values \u200b\u200band the sum 1 / a 1 + 1 / a 2 + ... + 1 / a n. If we again take the same for the calculation, then the harmonic will be 29.6.
Weighted average: features
However, all of the above values \u200b\u200bmay not be used everywhere. For example, in statistics, when calculating some, an important role is played by the "weight" of each number used in the calculations. The results are more indicative and correct because they take into account more information. This group of values \u200b\u200bis collectively referred to as "weighted average". They do not pass at school, so it is worth dwelling on them in more detail.
First of all, it is worth telling what is meant by "weight" of this or that value. The easiest way to explain this is with a specific example. Every patient's body temperature is measured twice a day in the hospital. Out of 100 patients in different departments of the hospital, 44 will have a normal temperature - 36.6 degrees. Another 30 will have an increased value - 37.2, 14 - 38, 7 - 38.5, 3 - 39, and the remaining two - 40. And if we take the arithmetic mean, then this value in general for the hospital will be more than 38 degrees! But in almost half of the patients completely And here it will be more correct to use the weighted average value, and the "weight" of each value will be the number of people. In this case, the calculation result will be 37.25 degrees. The difference is obvious.
In the case of weighted average calculations, the "weight" can be taken as the number of shipments, the number of people working on a particular day, in general, anything that can be measured and affect the final result.
Varieties
The weighted average corresponds to the arithmetic mean discussed at the beginning of the article. However, the first value, as already mentioned, also takes into account the weight of each number used in the calculations. In addition, there are also geometric and harmonic weighted mean values.
There is another interesting variation used in the series of numbers. This is a weighted moving average. It is on its basis that trends are calculated. In addition to the values \u200b\u200bthemselves and their weight, periodicity is also used there. And when calculating the average value at some point in time, the values \u200b\u200bfor the previous time intervals are also taken into account.
Calculating all of these values \u200b\u200bis not that difficult, but in practice, only the usual weighted average is usually used.
Calculation methods
In an age of massive computing, there is no need to manually calculate the weighted average. However, it will be useful to know the calculation formula so that you can check and, if necessary, correct the results obtained.
The easiest way to consider the calculation is with a specific example.
It is necessary to find out what is the average wage at this enterprise, taking into account the number of workers receiving this or that earnings.
So, the calculation of the weighted average is done using the following formula:
x \u003d (a 1 * w 1 + a 2 * w 2 + ... + a n * w n) / (w 1 + w 2 + ... + w n)
For example, the calculation will be like this:
x \u003d (32 * 20 + 33 * 35 + 34 * 14 + 40 * 6) / (20 + 35 + 14 + 6) \u003d (640 + 1155 + 476 + 240) / 75 \u003d 33.48
Obviously, there is no particular difficulty in manually calculating the weighted average. The formula for calculating this value in one of the most popular applications with formulas - Excel - looks like the SUMPRODUCT (series of numbers; series of weights) / SUM (series of weights) function.
The most common form of statistical indicators used in socio-economic research is the average value, which is a generalized quantitative characteristic of an attribute of a statistical population. The average values \u200b\u200bare, as it were, "representatives" of the entire series of observations. The average can be determined in many cases through the initial ratio of the average (ISR) or its logical formula:. So, for example, to calculate the average wage of employees of an enterprise, it is necessary to divide the total wage fund by the number of employees: The numerator of the initial ratio of the average is its determining indicator. For average wages, such a defining indicator is the payroll. For each indicator used in socio-economic analysis, only one true baseline ratio can be compiled to calculate the average. It should also be added that in order to more accurately estimate the standard deviation for small samples (with the number of elements less than 30), in the denominator of the expression under the root it is necessary to use not n, a n-1.
The concept and types of average values
Average value is a generalizing indicator of a statistical population, which compensates for individual differences in the values \u200b\u200bof statistical quantities, allowing you to compare different populations with each other. Exists 2 class averages: power and structural. Structural averages include fashion and median but most often used power averagesof various types.Power averages
Power means can be simple and weighted.
The simple average is calculated in the presence of two or more ungrouped statistical quantities, arranged in an arbitrary order according to the following general power-law average formula (for different values \u200b\u200bof k (m)):
The weighted average is calculated from the grouped statistics using the following general formula:
Where x - the average value of the studied phenomenon; x i - i -th variant of the averaged feature;
f i - weight of the i -th option.
Where X - the values \u200b\u200bof individual statistical quantities or the middle of the grouping intervals;
m is an exponent, on the value of which the following types of power mean values \u200b\u200bdepend:
at m \u003d -1 average harmonic;
for m \u003d 0, the geometric mean;
for m \u003d 1, the arithmetic mean;
for m \u003d 2 root mean square;
for m \u003d 3, the average is cubic.
Using general formulas of simple and weighted averages for different exponents m, we obtain particular formulas of each type, which will be further considered in detail.
Arithmetic mean
Arithmetic mean - the initial moment of the first order, the mathematical expectation of the values \u200b\u200bof a random variable with a large number of tests;
The arithmetic mean is the most commonly used average, which is obtained by substituting m \u003d 1 in the general formula. Arithmetic mean simple looks like this:
or 
Where X - values \u200b\u200bof quantities for which it is necessary to calculate the average value; N is the total number of X values \u200b\u200b(the number of units in the studied population).
For example, a student passed 4 exams and received the following grades: 3, 4, 4 and 5. Calculate the average score using the simple arithmetic mean formula: (3 + 4 + 4 + 5) / 4 \u003d 16/4 \u003d 4. Arithmetic mean weighted looks like this:
Where f is the number of quantities with the same X value (frequency). \u003e For example, a student passed 4 exams and received the following grades: 3, 4, 4 and 5. Calculate the average score using the arithmetic weighted average formula: (3 * 1 + 4 * 2 + 5 * 1) / 4 \u003d 16/4 \u003d 4 ... If the X values \u200b\u200bare specified as intervals, then the midpoints of the X intervals are used for calculations, which are defined as the half-sum of the upper and lower boundaries of the interval. And if the X interval does not have a lower or upper boundary (open interval), then the range (the difference between the upper and lower boundaries) of the adjacent X interval is used to find it. For example, at the enterprise there are 10 employees with up to 3 years of work experience, 20 with 3 to 5 years of experience, 5 employees with more than 5 years of experience. Then we calculate the average length of service of employees using the formula of the arithmetic weighted average, taking as X the middle of the intervals of experience (2, 4 and 6 years): (2 * 10 + 4 * 20 + 6 * 5) / (10 + 20 + 5) \u003d 3.71 years.
AVERAGE function
This function calculates the average (arithmetic) of its arguments.
AVERAGE (number1, number2, ...)
Number1, number2, ... are between 1 and 30 arguments for which the average is calculated.
Arguments must be numbers or names, arrays or references containing numbers. If the argument, which is an array or a reference, contains texts, booleans, or empty cells, then those values \u200b\u200bare ignored; however, cells that contain null values \u200b\u200bare counted.
AVERAGE function
Computes the arithmetic mean of the values \u200b\u200bspecified in the argument list. In addition to numbers, text and logical values \u200b\u200bsuch as TRUE and FALSE can be used in the calculation.
AVERAGE (value1, value2, ...)
Value1, value2, ... is from 1 to 30 cells, cell ranges, or values \u200b\u200bfor which to average.
Arguments must be numbers, names, arrays, or references. Arrays and links containing text are interpreted as 0 (zero). Empty text ("") is interpreted as 0 (zero). Arguments containing TRUE are interpreted as 1, Arguments containing FALSE are interpreted as 0 (zero).
The arithmetic mean is used most often, but there are cases when it is necessary to use other types of averages. We will consider such cases further.
Average harmonic
Harmonic mean to determine the average sum of reciprocals;
Average harmonic it is used when the original data does not contain frequencies f for individual values \u200b\u200bof X, but are presented as their product Xf. Denoting Xf \u003d w, we express f \u003d w / X, and substituting these designations into the formula for the arithmetic weighted average, we obtain the formula for the harmonic weighted average:
Thus, the weighted average harmonic is used when the frequencies f are unknown, but w \u003d Xf is known. In cases where all w \u003d 1, that is, the individual values \u200b\u200bof X occur 1 time, the formula for the harmonic simple is applied: 
or 
For example, a car drove from point A to point B at a speed of 90 km / h, and back - at a speed of 110 km / h. To determine the average speed, we apply the simple harmonic average formula, since in the example the distance w 1 \u003d w 2 is given (the distance from point A to point B is the same as from B to A), which is equal to the product of speed (X) and time ( f). Average speed \u003d (1 + 1) / (1/90 + 1/110) \u003d 99 km / h.
SRGARM function
Returns the harmonic mean of a data set. Harmonic mean is the reciprocal of the arithmetic mean of reciprocals.
SRGARM (number1; number2; ...)
Number1, number2, ... are between 1 and 30 arguments for which the average is calculated. You can use an array or an array reference instead of arguments, separated by semicolons.
The harmonic mean is always less than the geometric mean, which is always less than the arithmetic mean.
Geometric mean
Geometric mean for assessing the average growth rate of random variables, finding the value of a feature equidistant from the minimum and maximum values;
Geometric mean used in determining average relative changes. The geometric mean gives the most accurate averaging result if the task is to find a value of X that would be equidistant from both the maximum and minimum values \u200b\u200bof X. For example, between 2005 and 2008 inflation index in Russia it was: in 2005 - 1.109; in 2006 - 1,090; in 2007 - 1,119; in 2008 - 1.133. Since the inflation index is a relative change (dynamics index), the average value should be calculated using the geometric mean: (1.109 * 1.090 * 1.119 * 1.133) ^ (1/4) \u003d 1.1126, that is, for the period from 2005 to 2008 prices grew by an average of 11.26% annually. An erroneous calculation using the arithmetic mean would give an incorrect result of 11.28%.SRGEOM function
Returns the geometric mean of an array or interval of positive numbers. For example, the SRGEOM function can be used to calculate average growth rates if you have specified a variable rate compound income.
SRGEOM (number1; number2; ...)
Number1, number2, ... are between 1 and 30 arguments for which the geometric mean is calculated. You can use an array or an array reference instead of arguments, separated by semicolons.
Root mean square
The root mean square is the initial moment of the second order.
Root mean square It is used in cases where the initial values \u200b\u200bof X can be both positive and negative, for example, when calculating average deviations.
The main application of the quadratic mean is to measure the variation in X values. Average cubic
Average cubic - the initial moment of the third order.
Average cubic it is used extremely rarely, for example, when calculating the poverty indices of the population for developing countries (INN-1) and for developed (INN-2), proposed and calculated by the UN.
In statistics, various types of averages are used, which are divided into two large classes:
Power means (harmonic mean, geometric mean, arithmetic mean, mean square, cubic mean);
Structural means (fashion, median).
To calculate power averagesyou must use all available characteristic values. Fashionand medianare determined only by the distribution structure, therefore they are called structural, positional averages. The median and mode are often used as an average characteristic in those populations where the calculation of the power mean is impossible or impractical.
The most common type of average is the arithmetic mean. Under arithmetic meanthe meaning of a feature is understood that each unit of the population would have if the total of all values \u200b\u200bof the feature were distributed evenly among all units of the population. The calculation of this value is reduced to the summation of all values \u200b\u200bof the varying attribute and dividing the resulting sum by the total number of units in the population. For example, five workers fulfilled an order for the manufacture of parts, while the first made 5 parts, the second - 7, the third - 4, the fourth - 10, the fifth - 12. Since in the initial data, the value of each option was encountered only once, to determine
to determine the average output of one worker, you should apply the simple arithmetic mean formula:
that is, in our example, the average output of one worker is equal to
Along with the simple arithmetic mean, study weighted arithmetic mean.For example, let's calculate the average age of students in a group of 20, whose ages range from 18 to 22, where xi - variants of the averaged feature, fi - frequency, which shows how many times i-thvalue in aggregate (Table 5.1).
Table 5.1
Average age of students
Applying the formula for the arithmetic weighted average, we get:
There is a certain rule for choosing the weighted arithmetic mean: if there is a series of data on two indicators, for one of which it is necessary to calculate
the average value, and at the same time the numerical values \u200b\u200bof the denominator of its logical formula are known, and the values \u200b\u200bof the numerator are unknown, but can be found as the product of these indicators, then the average value should be calculated using the formula of the arithmetic weighted average.
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 be only another type of average - average harmonic.At present, the computational properties of the arithmetic mean have lost their relevance in the calculation of generalizing statistical indicators in connection with the widespread introduction of electronic computing technology. The average harmonic value, which can also be simple and weighted, has acquired great practical importance. If the numerical values \u200b\u200bof the numerator of a logical formula are known, and the values \u200b\u200bof the denominator are unknown, but can be found as a quotient division of one indicator by another, then the average value is calculated using the harmonic weighted average formula.
For example, let it be known that the car traveled the first 210 km at 70 km / h and the remaining 150 km at 75 km / h. It is impossible to determine the average speed of a car throughout the entire journey of 360 km using the arithmetic mean formula. Since the options are speeds in individual sections xj\u003d 70 km / h and X2\u003d 75 km / h, and the weights (fi) are the corresponding segments of the path, then the products of the options by the weights will have neither physical nor economic meaning. In this case, the quotients from dividing the sections of the path into the corresponding speeds (options xi), that is, the time spent on the passage of individual sections of the path (fi / xi). If the segments of the path are denoted by fi, then the entire path is expressed as? Fi, and the time spent on the entire path - how? fi / xi , Then the average speed can be found as the quotient of dividing the entire path by the total time spent:
In our example, we get:
If, when using the average harmonic weights of all options (f) are equal, then instead of the weighted one, you can use simple (unweighted) harmonic mean:
where xi are individual options; n - the number of variants of the averaged feature. In the example with speed, the simple harmonic average could be applied if the path segments traveled at different speeds were equal.
Any average value should be calculated so that when it replaces each variant of the averaged feature, the value of some final, generalizing indicator, which is associated with the averaged indicator, does not change. So, when replacing the actual speeds on individual segments of the path with their average value (average speed), the total distance should not change.
The form (formula) of the average value is determined by the nature (mechanism) of the relationship of this final indicator with the average, therefore the final indicator, the value of which should not change when replacing the options with their average value, is called defining indicator.To derive the average formula, you need to compose and solve an equation using the relationship of the averaged indicator with the determining one. This equation is constructed by replacing the variants of the averaged feature (indicator) with their average value.
In addition to the arithmetic mean and harmonic mean, statistics use other types (forms) of the average. They are all special cases. power-law average.If we calculate all kinds of power-law averages for the same data, then the values
they will be the same, here the rule applies majo-ranksmedium. With an increase in the exponent of averages, the mean value itself also increases. The formulas most often used in practical research for calculating various types of power mean values \u200b\u200bare presented in Table. 5.2.
Table 5.2
Types of power averages
Geometric mean applies when available ngrowth factors, while the individual values \u200b\u200bof the feature are, as a rule, the relative values \u200b\u200bof the dynamics, built in the form of chain quantities, as a relation to the previous level of each level in the series of dynamics. The average thus characterizes the average growth rate. Average geometric simplecalculated by the formula
Formula geometric weighted meanlooks like this:
The formulas given are identical, but one is applied at the current rates or growth rates, and the second - at the absolute values \u200b\u200bof the series levels.
Root mean squareis used when calculating with the values \u200b\u200bof square functions, is used to measure the degree of variability of individual values \u200b\u200bof a feature around the arithmetic mean in distribution series and is calculated by the formula
Weighted mean squarecalculated using a different formula:
Average cubicis used when calculating with the values \u200b\u200bof cubic functions and is calculated by the formula
weighted average cubic:
All the above average values \u200b\u200bcan be presented in the form of a general formula:
where is the average value; - individual value; n - the number of units of the studied population; k Is an exponent that determines the type of average.
When using the same initial data, the more kin the general formula for the power-law average, the larger the average. From this it follows that there is a regular relationship between the magnitudes of the power averages:
The average values \u200b\u200bdescribed above give a generalized idea of \u200b\u200bthe studied aggregate, and from this point of view, their theoretical, applied and cognitive value is indisputable. But it happens that the value of the average does not coincide with any of the really existing options, therefore, in addition to the averages considered in the statistical analysis, it is advisable to use the values \u200b\u200bof specific options, which occupy a quite definite position in an ordered (ranked) series of values \u200b\u200bof a feature. Among these values, the most common are structural,or descriptive, medium - mode (Mo) and median (Me).
Fashion - the value of a feature that is most often found in a given population. With regard to the variation series, the mode is the most frequent value of the ranked series, i.e., the variant with the highest frequency. Fashion can be used to determine which stores are more frequently visited, the most common price for a product. It shows the size of a feature characteristic of a significant part of the population, and is determined by the formula
where x0 is the lower boundary of the interval; h - the size of the interval; fm - interval frequency; fm_1 - frequency of the previous interval; fm +1 - frequency of the next interval.
Mediancalled the variant located in the center of the ranked row. The median divides the row into two equal parts in such a way that the same number of population units are located on either side of it. At the same time, in one half of the units of the population, the value of the varying attribute is less than the median, in the other, it is greater. The median is used when studying an element whose value is greater than or equal to or simultaneously less than or equal to half of the elements of the distribution series. The median gives a general idea of \u200b\u200bwhere the attribute values \u200b\u200bare 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 border of the values \u200b\u200bof the varying attribute, which half of the population units have. The problem of finding the median for a discrete variation series is easy to solve. If we assign ordinal numbers to all units of the series, then the ordinal number of the median variant is defined as (n +1) / 2 with an odd number of members n. If the number of members of the series is an even number, then the median will be the average of the two options having ordinal numbers n/ 2 and n/ 2 + 1.
When determining the median in the interval variation series, the interval in which it is located (median interval) is first determined. This interval is characterized by the fact that its accumulated sum of frequencies is equal to or exceeds the half-sum of all frequencies of the series. The median of the interval variation series is calculated using the formula
where X0 - the lower boundary of the interval; h - the size of the interval; fm - interval frequency; f- the number of members of the series;
M -1 - the sum of the accumulated members of the series preceding this one.
Along with the median, for a more complete characterization of the structure of the studied population, other values \u200b\u200bof the options are used, which occupy a quite definite position in the ranked series. These include quartilesand deciles.Quartiles divide the series by the sum of frequencies into 4 equal parts, and deciles into 10 equal parts. There are three quartiles and nine deciles.
The median and mode, in contrast to the arithmetic mean, do not extinguish individual differences in the values \u200b\u200bof the varying attribute and therefore are additional and very important characteristics of the statistical population. In practice, they are often used instead of or alongside the average. It is especially advisable to calculate the median and mode in those cases when the studied population contains a certain number of units with a very large or very small value of the variable characteristic. These, not very characteristic for the aggregate values \u200b\u200bof the options, affecting the value of the arithmetic mean, do not affect the values \u200b\u200bof the median and mode, which makes the latter very valuable indicators for economic and statistical analysis.
Average values \u200b\u200bare widespread in statistics. Average values \u200b\u200bcharacterize the qualitative indicators of commercial activity: distribution costs, profit, profitability, etc.
Average is one of the common generalization techniques. A correct understanding of the essence of the average determines its special significance in the conditions of a market economy, when the average, through the single and random, makes it possible to identify the general and necessary, to reveal the tendency of the laws of economic development.
average value - these are generalizing indicators in which the action of general conditions, patterns of the phenomenon under study are expressed.
Statistical averages are calculated on the basis of mass data of a correctly statistically organized mass observation (continuous and selective). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). For example, if you calculate the average wage in cooperatives and state-owned enterprises, and extend the result to the entire population, then the average is fictitious, since it is calculated over a heterogeneous population, and such an average loses all meaning.
With the help of the average, there is, as it were, smoothing out the differences in the value of the attribute, which arise for one reason or another in individual units of observation.
For example, the average output of a salesperson depends on many reasons: qualifications, length of service, age, form of service, health, etc.
Average output reflects the general property of the entire population.
The average value is a reflection of the values \u200b\u200bof the trait under study, therefore, it is measured in the same dimension as this trait.
Each average value characterizes the studied population by any one attribute. In order to get a complete and comprehensive picture of the studied population in terms of a number of essential features, in general, it is necessary to have a system of average values \u200b\u200bthat can describe the phenomenon from different angles.
There are various averages:
arithmetic mean;
geometric mean;
average harmonic;
root mean square;
average chronological.
Let's consider some types of averages that are most often used in statistics.
Arithmetic mean
The simple arithmetic mean (unweighted) is equal to the sum of the individual values \u200b\u200bof the attribute, divided by the number of these values.
Individual values \u200b\u200bof a feature are called variants and are denoted by x (); the number of units in the population is denoted by n, the average value of the feature is denoted by 
... Therefore, the simple arithmetic mean is:
According to the data of the discrete distribution series, it can be seen that the same values \u200b\u200bof the attribute (variants) are repeated several times. So, option x occurs in aggregate 2 times, and option x - 16 times, etc.
The number of identical values \u200b\u200bof a feature in the distribution series is called the frequency or weight and is denoted by the symbol n.
Let's calculate the average wage of one worker 
in rubles:
The wage bill for each group of workers is equal to the product of the options by the frequency, and the sum of these products gives the total wage bill of all workers.
Accordingly, the calculations can be presented in general form:
The resulting formula is called the weighted arithmetic mean.
Statistical material as a result of processing can be presented not only in the form of discrete distribution series, but also in the form of interval variation series with closed or open intervals.
The calculation of the average for the grouped data is made according to the formula for the arithmetic weighted average:
In the practice of economic statistics, sometimes it is necessary to calculate the average by group means or by means of individual parts of the population (private means). In such cases, group or partial averages are taken as options (x), on the basis of which the total average is calculated as the usual weighted arithmetic mean.
Basic properties of the arithmetic mean .
The arithmetic mean has a number of properties:
1. From a decrease or increase in the frequencies of each value of the attribute x in n times, the value of the arithmetic mean will not change.
If all frequencies are divided or multiplied by any number, then the value of the average will not change.
2. The common factor of individual values \u200b\u200bof the attribute can be taken out of the mean sign:
3. The average of the sum (difference) of two or more values \u200b\u200bis equal to the sum (difference) of their average:
4. If x \u003d c, where c is a constant, then 
.
5. The sum of the deviations of the values \u200b\u200bof the attribute X from the arithmetic mean x is equal to zero:
Average harmonic.
Along with the arithmetic mean, statistics use the harmonic mean, the reciprocal of the arithmetic mean of the reciprocal values \u200b\u200bof the attribute. Like the arithmetic mean, it can be simple and weighted.
The characteristics of the variation series, along with the mean, are the mode and the median.
Fashion - This is the value of a feature (option), which is most often repeated in the studied population. For discrete distribution series, the mode will be the value of the variant with the highest frequency.
For interval series of distribution with equal intervals, the mode is determined by the formula:
where 
- the initial value of the interval containing the mode;
- the value of the modal interval;
- the frequency of the modal interval;
- frequency of the interval preceding the modal;
is the frequency of the interval following the modal.
Median - this is a variant located in the middle of the variation series. If the distribution series is discrete and has an odd number of members, then the median will be the variant located in the middle of the ordered row (an ordered row is the arrangement of the units of the population in ascending or descending order).
