average value- this is a generalizing indicator that characterizes a qualitatively homogeneous population according to a certain quantitative attribute. For example, the average age of persons convicted of theft.
In judicial statistics, averages are used to characterize:
Average terms of consideration of cases of this category;
Medium size claim;
The average number of defendants per case;
Average amount of damage;
Average workload of judges, etc.
The average value is always named and has the same dimension as the attribute of a separate unit of the population. Each average value characterizes the studied population according to any one varying attribute, therefore, behind any average, there is a series of distribution of units of this population according to the studied attribute. The choice of the type of average is determined by the content of the indicator and the initial data for calculating the average.
All types of averages used in statistical studies fall into two categories:
1) power averages;
2) structural averages.
The first category of averages includes: arithmetic mean, harmonic mean, geometric mean and root mean square . The second category is fashion and median. Moreover, each of the listed types of power averages can have two forms: simple and weighted . The simple form of the mean is used to obtain the mean of the trait under study when the calculation is based on ungrouped statistics, or when each variant occurs only once in the population. Weighted averages are called values that take into account that the options for the values of a feature can have different numbers, and therefore each option has to be multiplied by the corresponding frequency. In other words, each option is "weighed" by its frequency. The frequency is called the statistical weight.
simple arithmetic mean- the most common type of medium. It is equal to the sum of individual characteristic values divided by the total number of these values:
where x 1 ,x 2 , … ,x N- individual values of the variable attribute (options), and N - the number of population units.
Arithmetic weighted average used when the data is presented in the form of distribution series or groupings. It is calculated as the sum of the products of the options and their corresponding frequencies, divided by the sum of the frequencies of all options:
where x i- meaning i-th variants of the feature; fi- frequency i th options.
Thus, each variant value is weighted by its frequency, which is why the frequencies are sometimes called statistical weights.
Comment. When it comes to the arithmetic mean without specifying its type, the simple arithmetic mean is meant.
Table 12
Decision. For the calculation, we use the formula of the arithmetic weighted average:
Thus, on average, there are two defendants per criminal case.
If the calculation of the average value is carried out according to data grouped in the form of interval distribution series, then first you need to determine the median values of each interval x "i, then calculate the average value using the weighted arithmetic mean formula, in which x" i is substituted instead of x i.
Example. Data on the age of criminals convicted of theft are presented in the table:
Table 13
Determine the average age of criminals convicted of theft.
Decision. In order to determine the average age of criminals based on the interval variation series, you must first find the median values of the intervals. Since an interval series with open first and last intervals is given, the values of these intervals are taken equal to the values of adjacent closed intervals. In our case, the value of the first and last intervals are 10.
Now we find the average age of criminals using the weighted arithmetic mean formula:
Thus, the average age of offenders convicted of theft is approximately 27 years.
Average harmonic simple is the reciprocal of the arithmetic mean of the reciprocal values of the feature:
where 1/ x i are the reciprocals of the options, and N is the number of population units.
Example. In order to determine the average annual workload for judges of a district court when considering criminal cases, a survey was conducted on the workload of 5 judges of this court. The average time spent on one criminal case for each of the surveyed judges turned out to be equal (in days): 6, 0, 5, 6, 6, 3, 4, 9, 5, 4. Find the average costs for one criminal case and the average annual workload on the judges of this district court when considering criminal cases.
Decision. To determine the average time spent on one criminal case, we use the harmonic simple formula:
To simplify the calculations in the example, let's take the number of days in a year equal to 365, including weekends (this does not affect the calculation method, and when calculating a similar indicator in practice, it is necessary to substitute the number of working days in a particular year instead of 365 days). Then the average annual workload for judges of this district court when considering criminal cases will be: 365 (days): 5.56 ≈ 65.6 (cases).
If we used the simple arithmetic mean formula to determine the average time spent on one criminal case, we would get:
365 (days): 5.64 ≈ 64.7 (cases), i.e. the average workload for judges was less.
Let's check the validity of this approach. To do this, we use data on the time spent on one criminal case for each judge and calculate the number of criminal cases considered by each of them per year.
We get accordingly:
365(days) : 6 ≈ 61 (case), 365(days) : 5.6 ≈ 65.2 (case), 365(days) : 6.3 ≈ 58 (case),
365(days) : 4.9 ≈ 74.5 (cases), 365(days) : 5.4 ≈ 68 (cases).
Now we calculate the average annual workload for judges of this district court when considering criminal cases:
Those. the average annual load is the same as when using the harmonic mean.
Thus, the use of the arithmetic mean in this case is illegal.
In cases where the variants of a feature are known, their volumetric values (the product of the variants by the frequency), but the frequencies themselves are unknown, the harmonic weighted average formula is applied:
,
where x i are the values of the trait options, and w i are the volumetric values of the options ( w i = x i f i).
Example. Data on the price of a unit of the same type of goods produced by various institutions of the penitentiary system, and on the volume of its implementation are given in table 14.
Table 14
Find the average selling price of the product.
Decision. When calculating the average price, we must use the ratio of the amount sold to the number of units sold. We do not know the number of sold units, but we know the amount of sales of goods. Therefore, to find the average price of goods sold, we use the harmonic weighted average formula. We get
If you use the arithmetic mean formula here, you can get an average price that will be unrealistic:
Geometric mean is calculated by extracting the root of degree N from the product of all values of the feature options:
,
where x 1 ,x 2 , … ,x N- individual values of the variable trait (options), and
N- number of population units.
This type of average is used to calculate the average growth rates of time series.
root mean square is used to calculate the standard deviation, which is an indicator of variation, and will be discussed below.
To determine the structure of the population, special averages are used, which include median and fashion , or the so-called structural averages. If the arithmetic mean is calculated based on the use of all variants of the attribute values, then the median and mode characterize the value of the variant that occupies a certain average position in the ranked (ordered) series. The ordering of units of the statistical population can be carried out in ascending or descending order of the variants of the trait under study.
Median (Me) is the value that corresponds to the variant in the middle of the ranked series. Thus, the median is that variant of the ranked series, on both sides of which in this series there should be an equal number of population units.
To find the median, you first need to determine its serial number in the ranked series using the formula:
where N is the volume of the series (the number of population units).
If the series consists of an odd number of members, then the median is equal to the variant with the number N Me . If the series consists of an even number of members, then the median is defined as the arithmetic mean of two adjacent options located in the middle.
Example. Given a ranked series 1, 2, 3, 3, 6, 7, 9, 9, 10. The volume of the series is N = 9, which means N Me = (9 + 1) / 2 = 5. Therefore, Me = 6, i.e. . fifth option. If a row is given 1, 5, 7, 9, 11, 14, 15, 16, i.e. series with an even number of members (N = 8), then N Me = (8 + 1) / 2 = 4.5. So the median is equal to half the sum of the fourth and fifth options, i.e. Me = (9 + 11) / 2 = 10.
In a discrete variation series, the median is determined by the accumulated frequencies. Variant frequencies, starting with the first one, are summed until the median number is exceeded. The value of the last summed options will be the median.
Example. Find the median number of defendants per criminal case using the data in Table 12.
Decision. In this case, the volume of the variation series is N = 154, therefore, N Me = (154 + 1) / 2 = 77.5. Summing up the frequencies of the first and second options, we get: 75 + 43 = 118, i.e. we have surpassed the median number. So Me = 2.
In the interval variation series of the distribution, first indicate the interval in which the median will be located. He is called median . This is the first interval whose cumulative frequency exceeds half the volume of the interval variation series. Then the numerical value of the median is determined by the formula:
where x Me- the lower limit of the median interval; i - the value of the median interval; S Me-1- the accumulated frequency of the interval that precedes the median; f Me- frequency of the median interval.
Example. Find the median age of offenders convicted of theft, based on the statistics presented in Table 13.
Decision. Statistical data is represented by an interval variation series, which means that we first determine the median interval. The volume of the population N = 162, therefore, the median interval is the interval 18-28, because this is the first interval, the accumulated frequency of which (15 + 90 = 105) exceeds half the volume (162: 2 = 81) of the interval variation series. Now the numerical value of the median is determined by the above formula:
Thus, half of those convicted of theft are under 25 years old.
Fashion (Mo) name the value of the attribute, which is most often found in units of the population. Fashion is used to identify the value of the trait that has the greatest distribution. For a discrete series, the mode will be the variant with the highest frequency. For example, for a discrete series presented in Table 3 Mo= 1, since this value of the options corresponds to the highest frequency - 75. To determine the mode of the interval series, first determine modal interval (interval having the highest frequency). Then, within this interval, the value of the feature is found, which can be a mode.
Its value is found by the formula:
where x Mo- the lower limit of the modal interval; i - the value of the modal interval; f Mo- modal interval frequency; f Mo-1- frequency of the interval preceding the modal; f Mo+1- frequency of the interval following the modal.
Example. Find the age mode of criminals convicted of theft, data on which are presented in table 13.
Decision. The highest frequency corresponds to the interval 18-28, therefore, the mode must be in this interval. Its value is determined by the above formula:
Thus, the largest number of criminals convicted of theft is 24 years old.
The average value gives a generalizing characteristic of the totality of the phenomenon under study. However, two populations with the same mean values may differ significantly from each other in terms of the degree of fluctuation (variation) in the value of the studied trait. For example, in one court the following terms of imprisonment were assigned: 3, 3, 3, 4, 5, 5, 5, 12, 12, 15 years, and in another - 5, 5, 6, 6, 7, 7, 7 , 8, 8, 8 years old. In both cases, the arithmetic mean is 6.7 years. However, these aggregates differ significantly from each other in the spread of individual values of the assigned term of imprisonment relative to the average value.
And for the first court, where this variation is quite large, the average term of imprisonment does not reflect the whole population well. Thus, if the individual values of the attribute differ little from each other, then the arithmetic mean will be a fairly indicative characteristic of the properties of this population. Otherwise, the arithmetic mean will be an unreliable characteristic of this population and its application in practice is ineffective. Therefore, it is necessary to take into account the variation in the values of the studied trait.
Variation- these are differences in the values of a characteristic in different units of a given population in the same period or point in time. The term "variation" is of Latin origin - variatio, which means difference, change, fluctuation. It arises as a result of the fact that the individual values of the attribute are formed under the combined influence of various factors (conditions), which are combined in different ways in each individual case. To measure the variation of a trait, various absolute and relative indicators are used.
The main indicators of variation include the following:
1) range of variation;
2) average linear deviation;
3) dispersion;
4) standard deviation;
5) coefficient of variation.
Let's briefly dwell on each of them.
Span variation R is the most accessible absolute indicator in terms of ease of calculation, which is defined as the difference between the largest and smallest values of the attribute for the units of this population:
The range of variation (range of fluctuations) is an important indicator of the variability of a trait, but it makes it possible to see only extreme deviations, which limits its scope. For a more accurate characterization of the variation of a trait based on its fluctuation, other indicators are used.
Average linear deviation represents the arithmetic mean of the absolute values of the deviations of the individual values of the trait from the mean and is determined by the formulas:
1) for ungrouped data
2) for variation series
However, the most widely used measure of variation is dispersion . It characterizes the measure of the spread of the values of the studied trait relative to its average value. The variance is defined as the average of the deviations squared.
simple variance for ungrouped data:
.
Weighted variance for the variation series:
Comment. In practice, it is better to use the following formulas to calculate the variance:
For a simple variance
.
For weighted variance
Standard deviation is the square root of the variance:
The standard deviation is a measure of the reliability of the mean. The smaller the standard deviation, the more homogeneous the population and the better the arithmetic mean reflects the entire population.
The dispersion measures considered above (range of variation, variance, standard deviation) are absolute indicators, by which it is not always possible to judge the degree of fluctuation of a trait. In some problems, it is necessary to use relative scattering indices, one of which is the coefficient of variation.
The coefficient of variation- expressed as a percentage of the ratio of the standard deviation to the arithmetic mean:
The coefficient of variation is used not only for a comparative assessment of the variation of different traits or the same trait in different populations, but also to characterize the homogeneity of the population. The statistical population is considered quantitatively homogeneous if the coefficient of variation does not exceed 33% (for distributions close to the normal distribution).
Example. There is the following data on the terms of imprisonment of 50 convicts delivered to serve the sentence imposed by the court in a correctional institution of the penitentiary system: 5, 4, 2, 1, 6, 3, 4, 3, 2, 2, 5, 6, 4, 3 , 10, 5, 4, 1, 2, 3, 3, 4, 1, 6, 5, 3, 4, 3, 5, 12, 4, 3, 2, 4, 6, 4, 4, 3, 1 , 5, 4, 3, 12, 6, 7, 3, 4, 5, 5, 3.
1. Construct a distribution series by terms of imprisonment.
2. Find the mean, variance and standard deviation.
3. Calculate the coefficient of variation and draw a conclusion about the homogeneity or heterogeneity of the studied population.
Decision. To construct a discrete distribution series, it is necessary to determine the variants and frequencies. The variant in this problem is the term of imprisonment, and the frequency is the number of individual variant. Having calculated the frequencies, we obtain the following discrete distribution series:
Find the mean and variance. Since the statistical data are represented by a discrete variational series, we will use the formulas of the arithmetic weighted average and variance to calculate them. We get:
= 
= 4,1;
= 
5,21.
Now we calculate the standard deviation:
We find the coefficient of variation:
Consequently, the statistical population is quantitatively heterogeneous.
Arithmetic mean - a statistical indicator that shows the average value of a given data array. Such an indicator is calculated as a fraction, the numerator of which is the sum of all array values, and the denominator is their number. The arithmetic mean is an important coefficient that is used in household calculations.
The meaning of the coefficient
The arithmetic mean is an elementary indicator for comparing data and calculating an acceptable value. For example, a can of beer from a particular manufacturer is sold in different stores. But in one store it costs 67 rubles, in another - 70 rubles, in the third - 65 rubles, and in the last - 62 rubles. There is a rather large range of prices, so the buyer will be interested in the average cost of a can, so that when buying a product he can compare his costs. On average, a can of beer in the city has a price:
Average price = (67 + 70 + 65 + 62) / 4 = 66 rubles.
Knowing the average price, it is easy to determine where it is profitable to buy goods, and where you will have to overpay.
The arithmetic mean is constantly used in statistical calculations in cases where a homogeneous data set is analyzed. In the example above, this is the price of a can of beer of the same brand. However, we cannot compare the price of beer from different manufacturers or the prices of beer and lemonade, since in this case the spread of values will be greater, the average price will be blurred and unreliable, and the very meaning of the calculations will be distorted to the caricature "average temperature in the hospital." To calculate heterogeneous data arrays, the arithmetic weighted average is used, when each value receives its own weighting factor.
Calculating the arithmetic mean
The formula for calculations is extremely simple:
P = (a1 + a2 + … an) / n,
where an is the value of the quantity, n is the total number of values.
What can this indicator be used for? The first and obvious use of it is in statistics. Almost every statistical study uses the arithmetic mean. This can be the average age of marriage in Russia, the average mark in a subject for a student, or the average spending on groceries per day. As mentioned above, without taking into account the weights, the calculation of averages can give strange or absurd values.
For example, the President of the Russian Federation made a statement that, according to statistics, the average salary of a Russian is 27,000 rubles. For most people in Russia, this level of salary seemed absurd. It is not surprising if the calculation takes into account the income of oligarchs, heads of industrial enterprises, large bankers, on the one hand, and the salaries of teachers, cleaners and sellers, on the other. Even average salaries in one specialty, for example, an accountant, will have serious differences in Moscow, Kostroma and Yekaterinburg.
How to calculate averages for heterogeneous data
In payroll situations, it is important to consider the weight of each value. This means that the salaries of oligarchs and bankers would be given a weight of, for example, 0.00001, and the salaries of salespeople would be 0.12. These are numbers from the ceiling, but they roughly illustrate the prevalence of oligarchs and salesmen in Russian society.
Thus, in order to calculate the average of averages or the average value in a heterogeneous data array, it is required to use the arithmetic weighted average. Otherwise, you will receive an average salary in Russia at the level of 27,000 rubles. If you want to know your average mark in mathematics or the average number of goals scored by a selected hockey player, then the arithmetic mean calculator will suit you.
Our program is a simple and convenient calculator for calculating the arithmetic mean. You only need to enter parameter values to perform calculations.
Let's look at a couple of examples
Average Grade Calculation
Many teachers use the arithmetic mean method to determine an annual grade in a subject. Let's imagine that a child gets the following quarter grades in math: 3, 3, 5, 4. What annual grade will the teacher give him? Let's use a calculator and calculate the arithmetic mean. First, select the appropriate number of fields and enter the grade values in the cells that appear:
(3 + 3 + 5 + 4) / 4 = 3,75
The teacher will round the value in favor of the student, and the student will receive a solid four for the year.
Calculation of eaten sweets
Let's illustrate some absurdity of the arithmetic mean. Imagine that Masha and Vova had 10 sweets. Masha ate 8 candies, and Vova only 2. How many candies did each child eat on average? Using a calculator, it is easy to calculate that on average, children ate 5 sweets each, which is completely untrue and common sense. This example shows that the arithmetic mean is important for meaningful datasets.
Conclusion
The calculation of the arithmetic mean is widely used in many scientific fields. This indicator is popular not only in statistical calculations, but also in physics, mechanics, economics, medicine or finance. Use our calculators as an assistant for solving arithmetic mean problems.
Signs of units of statistical aggregates are different in their meaning, for example, the wages of workers of one profession of an enterprise are not the same for the same period of time, market prices for the same products are different, crop yields in the farms of the region, etc. Therefore, in order to determine the value of a feature characteristic of the entire population of units under study, average values are calculated.
average value –
it is a generalizing characteristic of the set of individual values of some quantitative trait.
The population studied by a quantitative attribute consists of individual values; they are influenced by both general causes and individual conditions. In the average value, the deviations characteristic of the individual values are canceled out. The average, being a function of a set of individual values, represents the entire set with one value and reflects the common thing that is inherent in all its units.
The average calculated for populations consisting of qualitatively homogeneous units is called typical average. For example, you can calculate the average monthly salary of an employee of one or another professional group (miner, doctor, librarian). Of course, the levels of monthly wages of miners, due to the difference in their qualifications, length of service, hours worked per month and many other factors, differ from each other, and from the level of average wages. However, the average level reflects the main factors that affect the level of wages, and mutually offset the differences that arise due to the individual characteristics of the employee. The average wage reflects the typical level of wages for this type of worker. Obtaining a typical average should be preceded by an analysis of how this population is qualitatively homogeneous. If the population consists of separate parts, it should be divided into typical groups (average temperature in the hospital).
Average values used as characteristics for heterogeneous populations are called system averages. For example, the average value of the gross domestic product (GDP) per capita, the average consumption of various groups of goods per person and other similar values representing the general characteristics of the state as a single economic system.
The average should be calculated for populations consisting of a sufficiently large number of units. Compliance with this condition is necessary in order for the law of large numbers to come into force, as a result of which random deviations of individual values from the general trend cancel each other out.
Types of averages and methods for calculating them
The choice of the type of average is determined by the economic content of a certain indicator and the initial data. However, any average value should be calculated so that when it replaces each variant of the averaged feature, the final, generalizing, or, as it is commonly called, defining indicator, which is related to the average. For example, when replacing the actual speeds on separate sections of the path, their average speed should not change the total distance traveled by the vehicle in the same time; when replacing the actual wages of individual employees of the enterprise with the average wage, the wage fund should not change. Consequently, in each specific case, depending on the nature of the available data, there is only one true average value of the indicator that is adequate to the properties and essence of the socio-economic phenomenon under study.
The most commonly used are the arithmetic mean, harmonic mean, geometric mean, mean square, and mean cubic.
The listed averages belong to the class power average and are combined by the general formula:
,
where is the average value of the studied trait;
m is the exponent of the mean;
– current value (variant) of the averaged feature;
n is the number of features.
Depending on the value of the exponent m, the following types of power averages are distinguished:
at m = -1 – mean harmonic ;
at m = 0 – geometric mean ;
at m = 1 – arithmetic mean;
at m = 2 – root mean square ;
at m = 3 - average cubic.
When using the same initial data, the larger the exponent m in the above formula, the larger the value of the average value:
.
This property of power-law means to increase with an increase in the exponent of the defining function is called the rule of majorance of means.
Each of the marked averages can take two forms: simple and weighted.
The simple form of the middle applies when the average is calculated on primary (ungrouped) data. weighted form– when calculating the average for secondary (grouped) data.
Arithmetic mean
The arithmetic mean is used when the volume of the population is the sum of all individual values of the varying attribute. It should be noted that if the type of average is not indicated, the arithmetic average is assumed. Its logical formula is: 
simple arithmetic mean calculated by ungrouped data
according to the formula: 
or ,
where are the individual values of the feature;
j is the serial number of the unit of observation, which is characterized by the value ;
N is the number of observation units (set size).
Example. In the lecture “Summary and grouping of statistical data”, the results of observing the work experience of a team of 10 people were considered. Calculate the average work experience of the workers of the brigade. 5, 3, 5, 4, 3, 4, 5, 4, 2, 4.
According to the formula of the arithmetic mean simple, one also calculates chronological averages, if the time intervals for which the characteristic values are presented are equal.
Example. The volume of products sold for the first quarter amounted to 47 den. units, for the second 54, for the third 65 and for the fourth 58 den. units The average quarterly turnover is (47+54+65+58)/4 = 56 den. units
If momentary indicators are given in the chronological series, then when calculating the average, they are replaced by half-sums of values at the beginning and end of the period.
If there are more than two moments and the intervals between them are equal, then the average is calculated using the formula for the average chronological
,
where n is the number of time points
When the data is grouped by attribute values
(i.e., a discrete variational distribution series is constructed) with weighted arithmetic mean is calculated using either frequencies , or frequencies of observation of specific values of the feature , the number of which (k) is significantly less than the number of observations (N) .
,
,
where k is the number of groups of the variation series,
i is the number of the group of the variation series.
Since , and , we obtain the formulas used for practical calculations:
and
Example. Let's calculate the average length of service of the working teams for the grouped series.
a) using frequencies:
b) using frequencies:
When the data is grouped by intervals
, i.e. are presented in the form of interval distribution series; when calculating the arithmetic mean, the middle of the interval is taken as the value of the feature, based on the assumption of a uniform distribution of population units in this interval. The calculation is carried out according to the formulas:
and
where is the middle of the interval: ,
where and are the lower and upper boundaries of the intervals (provided that the upper boundary of this interval coincides with the lower boundary of the next interval).
Example. Let us calculate the arithmetic mean of the interval variation series constructed from the results of a study of the annual wages of 30 workers (see the lecture "Summary and grouping of statistical data").
Table 1 - Interval variation series of distribution.
Intervals, UAH |
Frequency, pers. |
frequency, |
The middle of the interval |
||
600-700 |
3 |
0,10 |
(600+700):2=650 |
1950 |
65 |
UAH or 
UAH
The arithmetic means calculated on the basis of the initial data and interval variation series may not coincide due to the uneven distribution of the attribute values within the intervals. In this case, for a more accurate calculation of the arithmetic weighted average, one should use not the middle of the intervals, but the arithmetic simple averages calculated for each group ( group averages). The average calculated from group means using a weighted calculation formula is called general average.
The arithmetic mean has a number of properties.
1. The sum of deviations of the variant from the mean is zero:
.
2. If all values of the option increase or decrease by the value A, then the average value increases or decreases by the same value A: 
3. If each option is increased or decreased by B times, then the average value will also increase or decrease by the same number of times:
or 
4. The sum of the products of the variant by the frequencies is equal to the product of the average value by the sum of the frequencies: 
5. If all frequencies are divided or multiplied by any number, then the arithmetic mean will not change: 
6) if in all intervals the frequencies are equal to each other, then the arithmetic weighted average is equal to the simple arithmetic average: 
,
where k is the number of groups in the variation series.
Using the properties of the average allows you to simplify its calculation.
Suppose that all options (x) are first reduced by the same number A, and then reduced by a factor of B. The greatest simplification is achieved when the value of the middle of the interval with the highest frequency is chosen as A, and the value of the interval as B (for rows with equal intervals). The quantity A is called the origin, so this method of calculating the average is called way b ohm reference from conditional zero or way of moments.
After such a transformation, we obtain a new variational distribution series, the variants of which are equal to . Their arithmetic mean, called moment of the first order, is expressed by the formula and according to the second and third properties, the arithmetic mean is equal to the mean of the original version, reduced first by A, and then by B times, i.e. .
For getting real average(middle of the original row) you need to multiply the moment of the first order by B and add A: 
The calculation of the arithmetic mean by the method of moments is illustrated by the data in Table. 2.
Table 2 - Distribution of employees of the enterprise shop by length of service
Work experience, years |
Amount of workers |
Interval midpoint |
|||
0 – 5 |
12 |
2,5 |
15 |
3 |
36 |
Finding the moment of the first order 
. Then, knowing that A = 17.5, and B = 5, we calculate the average work experience of the shop workers:
years
Average harmonic
As shown above, the arithmetic mean is used to calculate the average value of a feature in cases where its variants x and their frequencies f are known.
If the statistical information does not contain frequencies f for individual options x of the population, but is presented as their product , the formula is applied average harmonic weighted. To calculate the average, denote , whence . Substituting these expressions into the weighted arithmetic mean formula, we obtain the weighted harmonic mean formula: 
,
where is the volume (weight) of the indicator attribute values in the interval with number i (i=1,2, …, k).
Thus, the harmonic mean is used in cases where it is not the options themselves that are subject to summation, but their reciprocals: 
.
In cases where the weight of each option is equal to one, i.e. individual values of the inverse feature occur once, apply simple harmonic mean:
,
where are individual variants of the inverse trait that occur once;
N is the number of options.
If there are harmonic averages for two parts of the population with a number of and, then the total average for the entire population is calculated by the formula: 
and called weighted harmonic mean of the group means.
Example. Three deals were made during the first hour of trading on the currency exchange. Data on the amount of hryvnia sales and the hryvnia exchange rate against the US dollar are given in Table. 3 (columns 2 and 3). Determine the average exchange rate of the hryvnia against the US dollar for the first hour of trading.
Table 3 - Data on the course of trading on the currency exchange
The average dollar exchange rate is determined by the ratio of the amount of hryvnias sold in the course of all transactions to the amount of dollars acquired as a result of the same transactions. The total amount of the hryvnia sale is known from column 2 of the table, and the amount of dollars purchased in each transaction is determined by dividing the hryvnia sale amount by its exchange rate (column 4). A total of $22 million was purchased during three transactions. This means that the average hryvnia exchange rate for one dollar was 
.
The resulting value is real, because his substitution of the actual hryvnia exchange rates in transactions will not change the total amount of sales of the hryvnia, which acts as defining indicator: mln. UAH
If the arithmetic mean was used for the calculation, i.e. 
hryvnia, then at the exchange rate for the purchase of 22 million dollars. UAH 110.66 million would have to be spent, which is not true.
Geometric mean
The geometric mean is used to analyze the dynamics of phenomena and allows you to determine the average growth factor. When calculating the geometric mean, the individual values of the trait are relative indicators of dynamics, built in the form of chain values, as the ratio of each level to the previous one.
The geometric simple mean is calculated by the formula: 
,
where is the sign of the product,
N is the number of averaged values.
Example. The number of registered crimes over 4 years increased by 1.57 times, including for the 1st - by 1.08 times, for the 2nd - by 1.1 times, for the 3rd - by 1.18 and for the 4th - 1.12 times. Then the average annual growth rate of the number of crimes is: , i.e. The number of registered crimes has grown by an average of 12% annually.
1,8
-0,8
0,2
1,0
1,4
1
3
4
1
1
3,24
0,64
0,04
1
1,96
3,24
1,92
0,16
1
1,96
To calculate the mean square weighted, we determine and enter in the table and. Then the average value of deviations of the length of products from a given norm is equal to: 
The arithmetic mean in this case would be unsuitable, because as a result, we would get zero deviation.
The use of the root mean square will be discussed later in the exponents of variation.
The average value is the most valuable from an analytical point of view and a universal form of expression of statistical indicators. The most common average - the arithmetic average - has a number of mathematical properties that can be used in its calculation. At the same time, when calculating a specific average, it is always advisable to rely on its logical formula, which is the ratio of the volume of the attribute to the volume of the population. For each mean, there is only one true reference ratio, which, depending on the data available, may require different forms of means. However, in all cases where the nature of the averaged value implies the presence of weights, it is impossible to use their unweighted formulas instead of the weighted average formulas.
The average value is the most characteristic value of the attribute for the population and the size of the attribute of the population distributed in equal shares between the units of the population.
The characteristic for which the average value is calculated is called averaged .
The average value is an indicator calculated by comparing absolute or relative values. The average value is
The average value reflects the influence of all factors influencing the phenomenon under study, and is the resultant for them. In other words, repaying individual deviations and eliminating the influence of cases, the average value, reflecting the general measure of the results of this action, acts as a general pattern of the phenomenon under study.
Conditions for the use of averages:
Ø homogeneity of the studied population. If some elements of the population subject to the influence of a random factor have significantly different values of the studied trait from the rest, then these elements will affect the size of the average for this population. In this case, the average will not express the most typical value of the feature for the population. If the phenomenon under study is heterogeneous, it is required to break it down into groups containing homogeneous elements. In this case, group averages are calculated - group averages, expressing the most characteristic value of the phenomenon in each group, and then the overall average value for all elements is calculated, characterizing the phenomenon as a whole. It is calculated as the average of the group means, weighted by the number of population elements included in each group;
Ø a sufficient number of units in the aggregate;
Ø the maximum and minimum values of the trait in the studied population.
Average value (indicator)- this is a generalized quantitative characteristic of a trait in a systematic population under specific conditions of place and time.
In statistics, the following forms (types) of averages are used, called power and structural:
Ø arithmetic mean(simple and weighted);
simple
In mathematics, the arithmetic mean of numbers (or simply the average) is the sum of all the numbers in a given set divided by their number. This is the most generalized and widespread concept of the average value. As you already understood, in order to find the average value, you need to sum up all the numbers given to you, and divide the result by the number of terms.
What is the arithmetic mean?
Let's look at an example.
Example 1. Numbers are given: 6, 7, 11. You need to find their average value.
Decision.
First, let's find the sum of all given numbers.
Now we divide the resulting sum by the number of terms. Since we have three terms, respectively, we will divide by three.
Therefore, the average of the numbers 6, 7 and 11 is 8. Why 8? Yes, because the sum of 6, 7 and 11 will be the same as three eights. This is clearly seen in the illustration.
The average value is somewhat reminiscent of the "alignment" of a series of numbers. As you can see, the piles of pencils have become one level.
Consider another example to consolidate the knowledge gained.
Example 2 Numbers are given: 3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29. You need to find their arithmetic mean.
Decision.
We find the sum.
3 + 7 + 5 + 13 + 20 + 23 + 39 + 23 + 40 + 23 + 14 + 12 + 56 + 23 + 29 = 330
Divide by the number of terms (in this case, 15).
Therefore, the average value of this series of numbers is 22.
Now consider negative numbers. Let's remember how to sum them up. For example, you have two numbers 1 and -4. Let's find their sum.
1 + (-4) = 1 – 4 = -3
Knowing this, consider another example.
Example 3 Find the average value of a series of numbers: 3, -7, 5, 13, -2.
Decision.
Finding the sum of numbers.
3 + (-7) + 5 + 13 + (-2) = 12
Since there are 5 terms, we divide the resulting sum by 5.
Therefore, the arithmetic mean of the numbers 3, -7, 5, 13, -2 is 2.4.
In our time of technological progress, it is much more convenient to use computer programs to find the average value. Microsoft Office Excel is one of them. Finding the average in Excel is quick and easy. Moreover, this program is included in the software package from Microsoft Office. Consider a brief instruction on how to find the arithmetic mean using this program.
In order to calculate the average value of a series of numbers, you must use the AVERAGE function. The syntax for this function is:
=Average(argument1, argument2, ... argument255)
where argument1, argument2, ... argument255 are either numbers or cell references (cells mean ranges and arrays).
To make it clearer, let's test the knowledge gained.
- Enter the numbers 11, 12, 13, 14, 15, 16 in cells C1 - C6.
- Select cell C7 by clicking on it. In this cell, we will display the average value.
- Click on the "Formulas" tab.
- Select More Functions > Statistical to open the drop down list.
- Select AVERAGE. After that, a dialog box should open.
- Select and drag cells C1-C6 there to set the range in the dialog box.
- Confirm your actions with the "OK" button.
- If you did everything correctly, in cell C7 you should have the answer - 13.7. When you click on cell C7, the function (=Average(C1:C6)) will be displayed in the formula bar.
It is very useful to use this function for accounting, invoices, or when you just need to find the average of a very long range of numbers. Therefore, it is often used in offices and large companies. This allows you to keep the records in order and makes it possible to quickly calculate something (for example, the average income per month). You can also use Excel to find the mean of a function.
Average
This term has other meanings, see the average meaning.Average(in mathematics and statistics) sets of numbers - the sum of all numbers divided by their number. It is one of the most common measures of central tendency.
It was proposed (along with the geometric mean and harmonic mean) by the Pythagoreans.
Special cases of the arithmetic mean are the mean (of the general population) and the sample mean (of samples).
Introduction
Denote the set of data X = (x 1 , x 2 , …, x n), then the sample mean is usually denoted by a horizontal bar over the variable (x ¯ (\displaystyle (\bar (x))) , pronounced " x with a dash").
The Greek letter μ is used to denote the arithmetic mean of the entire population. For a random variable for which a mean value is defined, μ is probability mean or the mathematical expectation of a random variable. If the set X is a collection of random numbers with a probability mean μ, then for any sample x i from this collection μ = E( x i) is the expectation of this sample.
In practice, the difference between μ and x ¯ (\displaystyle (\bar (x))) is that μ is a typical variable because you can see the sample rather than the entire population. Therefore, if the sample is represented randomly (in terms of probability theory), then x ¯ (\displaystyle (\bar (x))) (but not μ) can be treated as a random variable having a probability distribution on the sample (probability distribution of the mean).
Both of these quantities are calculated in the same way:
X ¯ = 1 n ∑ i = 1 n x i = 1 n (x 1 + ⋯ + x n) . (\displaystyle (\bar (x))=(\frac (1)(n))\sum _(i=1)^(n)x_(i)=(\frac (1)(n))(x_ (1)+\cdots +x_(n)).)
If X is a random variable, then the mathematical expectation X can be considered as the arithmetic mean of the values in repeated measurements of the quantity X. This is a manifestation of the law of large numbers. Therefore, the sample mean is used to estimate the unknown mathematical expectation.
In elementary algebra, it is proved that the mean n+ 1 numbers above average n numbers if and only if the new number is greater than the old average, less if and only if the new number is less than the average, and does not change if and only if the new number is equal to the average. The more n, the smaller the difference between the new and old averages.
Note that there are several other "means" available, including power-law mean, Kolmogorov mean, harmonic mean, arithmetic-geometric mean, and various weighted means (e.g., arithmetic-weighted mean, geometric-weighted mean, harmonic-weighted mean).
Examples
- For three numbers, you need to add them and divide by 3:
- For four numbers, you need to add them and divide by 4:
Or easier 5+5=10, 10:2. Because we added 2 numbers, which means that how many numbers we add, we divide by that much.
Continuous random variable
For a continuously distributed value f (x) (\displaystyle f(x)) the arithmetic mean on the interval [ a ; b ] (\displaystyle ) is defined via a definite integral:
F (x) ¯ [ a ; b ] = 1 b − a ∫ a b f (x) d x (\displaystyle (\overline (f(x)))_()=(\frac (1)(b-a))\int _(a)^(b) f(x)dx)
Some problems of using the average
Lack of robustness
Main article: Robustness in statisticsAlthough the arithmetic mean is often used as means or central trends, this concept does not apply to robust statistics, which means that the arithmetic mean is heavily influenced by "large deviations". It is noteworthy that for distributions with a large skewness, the arithmetic mean may not correspond to the concept of “average”, and the values of the mean from robust statistics (for example, the median) may better describe the central trend.
The classic example is the calculation of the average income. The arithmetic mean can be misinterpreted as a median, which can lead to the conclusion that there are more people with more income than there really are. "Mean" income is interpreted in such a way that most people's incomes are close to this number. This "average" (in the sense of the arithmetic mean) income is higher than the income of most people, since a high income with a large deviation from the average makes the arithmetic mean strongly skewed (in contrast, the median income "resists" such a skew). However, this "average" income says nothing about the number of people near the median income (and says nothing about the number of people near the modal income). However, if the concepts of "average" and "majority" are taken lightly, then one can incorrectly conclude that most people have incomes higher than they actually are. For example, a report on the "average" net income in Medina, Washington, calculated as the arithmetic average of all annual net incomes of residents, will give a surprisingly high number due to Bill Gates. Consider the sample (1, 2, 2, 2, 3, 9). The arithmetic mean is 3.17, but five of the six values are below this mean.
Compound interest
Main article: ROIIf numbers multiply, but not fold, you need to use the geometric mean, not the arithmetic mean. Most often, this incident happens when calculating the return on investment in finance.
For example, if stocks fell 10% in the first year and rose 30% in the second year, then it is incorrect to calculate the "average" increase over these two years as the arithmetic mean (−10% + 30%) / 2 = 10%; the correct average in this case is given by the compound annual growth rate, from which the annual growth is only about 8.16653826392% ≈ 8.2%.
The reason for this is that percentages have a new starting point each time: 30% is 30% from a number less than the price at the beginning of the first year: if the stock started at $30 and fell 10%, it is worth $27 at the start of the second year. If the stock is up 30%, it is worth $35.1 at the end of the second year. The arithmetic average of this growth is 10%, but since the stock has only grown by $5.1 in 2 years, an average increase of 8.2% gives a final result of $35.1:
[$30 (1 - 0.1) (1 + 0.3) = $30 (1 + 0.082) (1 + 0.082) = $35.1]. If we use the arithmetic mean of 10% in the same way, we will not get the actual value: [$30 (1 + 0.1) (1 + 0.1) = $36.3].
Compound interest at the end of year 2: 90% * 130% = 117% , i.e. a total increase of 17%, and the average annual compound interest is 117% ≈ 108.2% (\displaystyle (\sqrt (117\%))\approx 108.2\%) , that is, an average annual increase of 8.2%.
Directions
Main article: Destination statisticsWhen calculating the arithmetic mean of some variable that changes cyclically (for example, phase or angle), special care should be taken. For example, the average of 1° and 359° would be 1 ∘ + 359 ∘ 2 = (\displaystyle (\frac (1^(\circ )+359^(\circ ))(2))=) 180°. This number is incorrect for two reasons.
- First, angular measures are only defined for the range from 0° to 360° (or from 0 to 2π when measured in radians). Thus, the same pair of numbers could be written as (1° and −1°) or as (1° and 719°). The averages of each pair will be different: 1 ∘ + (− 1 ∘) 2 = 0 ∘ (\displaystyle (\frac (1^(\circ )+(-1^(\circ )))(2))=0 ^(\circ )) , 1 ∘ + 719 ∘ 2 = 360 ∘ (\displaystyle (\frac (1^(\circ )+719^(\circ ))(2))=360^(\circ )) .
- Second, in this case, a value of 0° (equivalent to 360°) would be the geometrically best mean, since the numbers deviate less from 0° than from any other value (value 0° has the smallest variance). Compare:
- the number 1° deviates from 0° by only 1°;
- the number 1° deviates from the calculated average of 180° by 179°.
The average value for a cyclic variable, calculated according to the above formula, will be artificially shifted relative to the real average to the middle of the numerical range. Because of this, the average is calculated in a different way, namely, the number with the smallest variance (center point) is chosen as the average value. Also, instead of subtracting, modulo distance (i.e., circumferential distance) is used. For example, the modular distance between 1° and 359° is 2°, not 358° (on a circle between 359° and 360°==0° - one degree, between 0° and 1° - also 1°, in total - 2 °).
Weighted average - what is it and how to calculate it?
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 also faced with the calculation of other averages. What can they be and how do they differ from each other?
Averages: Meaning and Differences
Not always accurate indicators give an understanding of the situation. In order to assess this or that situation, it is sometimes necessary to analyze a huge number of figures. And then averages come to the rescue. They allow you to assess the situation in general.
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 terms is divisible by n. That is, if you need to calculate the arithmetic mean in the sequence of values 27, 22, 34 and 37, then you need to solve the expression (27 + 22 + 34 + 37) / 4, since 4 values \u200b\u200bare used in the calculations. In this case, the desired value will be equal to 30.
Often, as part of the school course, the geometric mean is also studied. The calculation of this value is based on extracting the root of the nth degree from the product of n terms. If we take the same numbers: 27, 22, 34 and 37, then the result of the calculations will be 29.4.
The harmonic mean in a general education school is usually not the subject of study. However, 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 and the sum 1/a 1 +1/a 2 +...+1/a n . If we again take the same series of numbers for calculation, then the harmonic will be 29.6.
Weighted Average: Features
However, all of the above values may not be used everywhere. For example, in statistics, when calculating some average values, the "weight" of each number used in the calculation plays an important role. The results are more revealing and correct because they take into account more information. This group of values is collectively referred to as the "weighted average". They are not passed at school, so it is worth dwelling on them in more detail.
First of all, it is worth explaining what is meant by the "weight" of a particular value. The easiest way to explain this is with a specific example. The body temperature of each patient is measured twice a day in the hospital. Of the 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 over 38 degrees! But almost half of the patients have a completely normal temperature. And here it would be more correct to use the weighted average, and the "weight" of each value will be the number of people. In this case, the result of the calculation will be 37.25 degrees. The difference is obvious.
In the case of weighted average calculations, the "weight" can be taken as the number of shipments, the number of people working on a given day, in general, anything that can be measured and affect the final result.
Varieties
The weighted average corresponds to the arithmetic average discussed at the beginning of the article. However, the first value, as already mentioned, also takes into account the weight of each number used in the calculations. In addition, there are also weighted geometric and harmonic values.
There is another interesting variety used in series of numbers. This is a weighted moving average. It is on its basis that trends are calculated. In addition to the values themselves and their weight, periodicity is also used there. And when calculating the average value at some point in time, values for previous time periods are also taken into account.
Calculating all these values is not that difficult, but in practice, only the usual weighted average is usually used.
Calculation methods
In the age of computerization, there is no need to manually calculate the weighted average. However, it would be useful to know the calculation formula so that you can check and, if necessary, correct the results obtained.
It will be easiest to consider the calculation on 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 a particular salary.
So, the calculation of the weighted average is carried out using the following formula:
x = (a 1 *w 1 +a 2 *w 2 +...+a n *w n)/(w 1 +w 2 +...+w n)
For example, the calculation would be:
x = (32*20+33*35+34*14+40*6)/(20+35+14+6) = (640+1155+476+240)/75 = 33.48
Obviously, there is no particular difficulty in manually calculating the weighted average. The formula for calculating this value in one of the most popular applications with formulas - Excel - looks like the SUMPRODUCT (series of numbers; series of weights) / SUM (series of weights) function.
How to find average value in excel?
how to find arithmetic mean in excel?
Vladimir09854
As easy as pie. In order to find the average value in excel, you only need 3 cells. In the first we write one number, in the second - another. And in the third cell, we will score a formula that will give us the average value between these two numbers from the first and second cells. If cell No. 1 is called A1, cell No. 2 is called B1, then in the cell with the formula you need to write like this:
This formula calculates the arithmetic mean of two numbers.
For the beauty of our calculations, we can highlight the cells with lines, in the form of a plate.
There is also a function in Excel itself to determine the average value, but I use the old-fashioned method and enter the formula I need. Thus, I am sure that Excel will calculate exactly as I need, and will not come up with some kind of rounding of its own.
M3sergey
This is very easy if the data is already entered into the cells. If you are just interested in a number, just select the desired range / ranges, and the value of the sum of these numbers, their arithmetic mean and their number will appear in the status bar at the bottom right.
You can select an empty cell, click on the triangle (drop-down list) "Autosum" and select "Average" there, after which you will agree with the proposed range for calculation, or choose your own.
Finally, you can use the formulas directly - click "Insert Function" next to the formula bar and cell address. The AVERAGE function is in the "Statistical" category, and takes as arguments both numbers and cell references, etc. There you can also choose more complex options, for example, AVERAGEIF - calculation of the average by condition.
Find average in excel is a fairly simple task. Here you need to understand whether you want to use this average value in some formulas or not.
If you need to get only the value, then it is enough to select the required range of numbers, after which excel will automatically calculate the average value - it will be displayed in the status bar, the heading "Average".
In the case when you want to use the result in formulas, you can do this:
1) Sum the cells using the SUM function and divide it all by the number of numbers.
2) A more correct option is to use a special function called AVERAGE. The arguments to this function can be numbers given sequentially, or a range of numbers.
Vladimir Tikhonov
circle the values that will be involved in the calculation, click the "Formulas" tab, there you will see "AutoSum" on the left and next to it a triangle pointing down. click on this triangle and choose "Average". Voila, done) at the bottom of the column you will see the average value :)
Ekaterina Mutalapova
Let's start at the beginning and in order. What does average mean?
The mean value is the value that is the arithmetic mean, i.e. is calculated by adding a set of numbers and then dividing the total sum of numbers by their number. For example, for the numbers 2, 3, 6, 7, 2 it will be 4 (the sum of the numbers 20 is divided by their number 5)
In an Excel spreadsheet, for me personally, the easiest way was to use the formula =AVERAGE. To calculate the average value, you need to enter data into the table, write the function =AVERAGE() under the data column, and in brackets indicate the range of numbers in the cells, highlighting the column with the data. After that, press ENTER, or simply left-click on any cell. The result will be displayed in the cell below the column. On the face of it, the description is incomprehensible, but in fact it is a matter of minutes.
Adventurer 2000
The Excel program is multi-faceted, so there are several options that will allow you to find the average:
First option. You simply sum all the cells and divide by their number;
Second option. Use a special command, write in the required cell the formula "=AVERAGE (and here specify the range of cells)";
Third option. If you select the required range, then note that on the page below, the average value in these cells is also displayed.
Thus, there are a lot of ways to find the average value, you just need to choose the best one for you and use it constantly.
In Excel, using the AVERAGE function, you can calculate the simple arithmetic mean. To do this, you need to enter a number of values. Press equals and select in the Statistical category, among which select the AVERAGE function
Also, using statistical formulas, you can calculate the arithmetic weighted average, which is considered more accurate. To calculate it, we need the values of the indicator and the frequency.
How to find the average in Excel?
The situation is this. There is the following table:
The columns shaded in red contain the numerical values of the grades for the subjects. In the "Average" column, you need to calculate their average value.
The problem is this: there are 60-70 objects in total and some of them are on another sheet.
I looked in another document, the average has already been calculated, and in the cell there is a formula like
="sheet name"!|E12
but this was done by some programmer who got fired.
Tell me, please, who understands this.
Hector
In the line of functions, you insert "AVERAGE" from the proposed functions and choose from where they need to be calculated (B6: N6) for Ivanov, for example. I don’t know for sure about neighboring sheets, but for sure this is contained in the standard Windows help
Tell me how to calculate the average value in Word
Please tell me how to calculate the average value in Word. Namely, the average value of the ratings, and not the number of people who received ratings. 
Yulia pavlova
Word can do a lot with macros. Press ALT+F11 and write a macro program..
In addition, Insert-Object... will allow you to use other programs, even Excel, to create a sheet with a table inside a Word document.
But in this case, you need to write down your numbers in the table column, and put the average in the bottom cell of the same column, right?
To do this, insert a field into the bottom cell.
Insert-Field...-Formula
Field content
[=AVERAGE(ABOVE)]
returns the average of the sum of the cells above.
If the field is selected and the right mouse button is pressed, then it can be Updated if the numbers have changed,
view the code or field value, change the code directly in the field.
If something goes wrong, delete the entire field in the cell and re-create it.
AVERAGE means average, ABOVE - about, that is, a row of cells above.
I did not know all this myself, but I easily found it in HELP, of course, thinking a little.
