The average value of a feature is a statistic formula. Moscow State University of Printing Arts
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 as common 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 gross domestic product (GDP) per capita, the average consumption of various groups of goods per person and other similar values that represent 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 quantities 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 individual sections of the path, their average speed should not change the total distance traveled vehicle at 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, there are the following types power averages:
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 input data, the larger the exponent m in the above formula, the more value medium size:
.
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.
simple form 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 mean value is not specified, the arithmetic mean 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.
The arithmetic mean simple formula 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 number 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 upper bound of this interval coincides with the lower bound 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 the same 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. .
To receive 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). Define average rate 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 rate. When calculating the geometric mean, the individual values of the attribute 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.
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 a 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 °).
Types of average values and methods for their calculation
At the stage of statistical processing, a variety of research tasks can be set, for the solution of which it is necessary to choose the appropriate average. In doing so, it is necessary to follow next rule: the values that represent the numerator and denominator of the mean must be logically related to each other.
- power averages;
- structural averages.
Let us introduce the following notation:
The values for which the average is calculated;
Average, where the line above indicates that the averaging of individual values takes place;
Frequency (repeatability of individual trait values).
Various averages are derived from general formula power mean:
(5.1)
for k = 1 - arithmetic mean; k = -1 - harmonic mean; k = 0 - geometric mean; k = -2 - root mean square.
Averages are either simple or weighted. weighted averages are called quantities that take into account that some variants of the values of the attribute may have different numbers, and therefore each variant has to be multiplied by this number. In other words, "weights" are the number of units in the population in different groups, i.e. each option is "weighted" by its frequency. The frequency f is called statistical weight or weighing average.
Arithmetic mean- the most common type of medium. It is used when the calculation is carried out on ungrouped statistical data, where you want to get the average summand. The arithmetic mean is such an average value of a feature, upon receipt of which the total volume of the feature in the population remains unchanged.
The arithmetic mean formula ( simple) has the form
where n is the population size.
For example, the average salary of employees of an enterprise is calculated as the arithmetic average:
The determining indicators here are the wages of each employee and the number of employees of the enterprise. When calculating the average, the total amount of wages remained the same, but distributed, as it were, equally among all workers. For example, it is necessary to calculate the average salary of employees of a small company where 8 people are employed:
When calculating averages, individual values of the attribute that is averaged can be repeated, so the average is calculated using grouped data. In this case we are talking about using arithmetic mean weighted, which looks like
(5.3)
So, we need to calculate the average share price of a joint-stock company at the stock exchange. It is known that transactions were carried out within 5 days (5 transactions), the number of shares sold at the sales rate was distributed as follows:
1 - 800 ac. - 1010 rubles
2 - 650 ac. - 990 rub.
3 - 700 ak. - 1015 rubles.
4 - 550 ac. - 900 rub.
5 - 850 ak. - 1150 rubles.
The initial ratio for determining the average share price is the ratio of the total amount of transactions (TCA) to the number of shares sold (KPA):
OSS = 1010 800+990 650+1015 700+900 550+1150 850= 3 634 500;
CPA = 800+650+700+550+850=3550.
In this case, the average share price was equal to
It is necessary to know the properties of the arithmetic mean, which is very important both for its use and for its calculation. There are three main properties that most of all led to the widespread use of the arithmetic mean in statistical and economic calculations.
Property one (zero): the sum of positive deviations of individual values of a trait from its mean value is equal to the sum of negative deviations. This is a very important property, since it shows that any deviations (both with + and with -) due to random causes will be mutually canceled.
Proof:
Property two (minimum): the sum of the squared deviations of the individual values of the trait from the arithmetic mean is less than from any other number (a), i.e. is the minimum number.
Proof.
Compose the sum of the squared deviations from the variable a:
(5.4)
To find the extremum of this function, it is necessary to equate its derivative with respect to a to zero:
From here we get:
(5.5)
Therefore, the extremum of the sum of squared deviations is reached at . This extremum is the minimum, since the function cannot have a maximum.
Property three: the arithmetic mean of a constant is equal to this constant: at a = const.
In addition to these three most important properties of the arithmetic mean, there are so-called design properties, which are gradually losing their significance due to the use of electronic computers:
- if the individual value of the attribute of each unit is multiplied or divided by a constant number, then the arithmetic mean will increase or decrease by the same amount;
- the arithmetic mean will not change if the weight (frequency) of each feature value is divided by a constant number;
- if the individual values of the attribute of each unit are reduced or increased by the same amount, then the arithmetic mean will decrease or increase by the same amount.
Average harmonic. This average is called the reciprocal arithmetic average, since this value is used when k = -1.
Simple harmonic mean is used when the weights of the characteristic values are the same. Its formula can be derived from the base formula by substituting k = -1:
For example, we need to calculate the average speed of two cars that have traveled the same path, but at different speeds: the first at 100 km/h, the second at 90 km/h. Using the harmonic mean method, we calculate the average speed:
In statistical practice, harmonic weighted is more often used, the formula of which has the form
This formula is used in cases where the weights (or volumes of phenomena) for each attribute are not equal. In the original ratio, the numerator is known to calculate the average, but the denominator is unknown.
For example, 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 units sold (we are talking about different goods), but we know the sums of sales of these different goods. Let's say we need to know average price goods sold:
We get
Geometric mean. Most often, the geometric mean finds its application in determining the average growth rate (average growth rates), when the individual values of the trait are presented as relative values. It is also used if it is necessary to find the average between the minimum and maximum values of a characteristic (for example, between 100 and 1000000). There are formulas for simple and weighted geometric mean.
For a simple geometric mean
For the weighted geometric mean
RMS. The main scope of its application is the measurement of the variation of a trait in the population (calculation of the average standard deviation).
Simple root mean square formula
Weighted Root Mean Square Formula
(5.11)
As a result, it can be said that right choice the type of average value in each particular case depends on the successful solution of the problems of statistical research. The choice of the average assumes the following sequence:
a) the establishment of a generalizing indicator of the population;
b) determination of a mathematical ratio of values for a given generalizing indicator;
c) replacement of individual values by average values;
d) calculation of the average using the corresponding equation.
Mean values and variation
average value- this is a generalizing indicator that characterizes a qualitatively homogeneous population according to a certain quantitative attribute. For example, average age 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 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 "weighted" 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 are the individual values of the variable attribute (options), and N is 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, and then calculate the average value using the weighted arithmetic mean formula, in which x" i is substituted for 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 we are given an interval series with open first and the last intervals, then 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 attribute:
where 1/ x i are the reciprocal values of the variants, 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 variants, and w i are the volumetric values of the variants ( 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 are the individual values of the variable trait (options), and
N is the 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's 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 is the lower limit of the median interval; i is the value of the median interval; S Me-1 is the cumulative frequency of the interval that precedes the median; f Me is the 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 is the lower limit of the modal interval; i is the value of the modal interval; f Mo is the frequency of the modal interval; f Mo-1 is the frequency of the interval preceding the modal; f Mo+1 is the 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 trait 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:
Range of variation (range of fluctuations) - important indicator fluctuations of the sign, but it makes it possible to see only extreme deviations, which limits the scope of its application. 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 variation different signs 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 option in this problem is the term of imprisonment, and the frequency is the number of individual options. 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.
simple arithmetic mean
Average values
Average values are widely used in statistics.
average value is a generalizing indicator in which the expression of action is found general conditions, patterns of development of the phenomenon under study.
Statistical averages are calculated on the basis of mass data of a correctly statistically organized observation (continuous and sample). 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 we calculate the average salary in joint-stock companies and state-owned enterprises, and extend the result to the entire population, then the average is fictitious, since it is calculated on a heterogeneous population, and such an average loses all meaning.
With the help of the average, there is, as it were, a smoothing of differences in the magnitude of the feature that arise for one reason or another in individual units of observation.
For example, the average output of an individual seller depends on many factors: qualifications, length of service, age, form of service, health, and so on. Average output reflects general characteristics the whole aggregate.
The average value is measured in the same units as the feature itself.
Each average value characterizes the studied population according to any one attribute. In order to get a complete and comprehensive picture of the population under study in terms of a number of essential features, it is necessary to have a system of average values that can describe the phenomenon from different angles.
Exist different kinds medium:
arithmetic mean;
average harmonic;
geometric mean;
root mean square;
average cubic.
The averages of all the types listed above, in turn, are divided into simple (unweighted) and weighted.
Consider the types of averages that are used in statistics.
The simple arithmetic mean (unweighted) is equal to the sum of the individual values of the characteristic, divided by the number of these values.
Separate values of a feature are called variants and are denoted by х i ( 
); the number of population units is denoted by n, the average value of the feature - by 
. Therefore, the simple arithmetic mean is:
or 
Example 1 Table 1
Data on the production of products A by workers per shift
In this example, the variable attribute is the release of products per shift.
The numerical values of the attribute (16, 17, etc.) are called variants. Let us determine the average output of products by the workers of this group:
PCS.
A simple arithmetic mean is used in cases where there are individual values of a characteristic, i.e. the data is not grouped. If the data is presented in the form of distribution series or groupings, then the average is calculated differently.
Arithmetic weighted average
The arithmetic weighted average is equal to the sum of the products of each individual value of the attribute (variant) by the corresponding frequency, divided by the sum of all frequencies.
The number of identical feature values in the distribution series is called frequency or weight and is denoted by f i .
In accordance with this, the arithmetic weighted average looks like this:
or 
It can be seen from the formula that the average depends not only on the values of the attribute, but also on their frequencies, i.e. on the composition of the population, on its structure.
Example 2 table 2
Worker wage data
According to the data of the discrete distribution series, it can be seen that the same values of the attribute (options) are repeated several times. So, variant x 1 occurs in the aggregate 2 times, and variant x 2 - 6 times, etc.
Calculate the average wage per worker:
The wage fund for each group of workers is equal to the product of options and frequency ( 
), and the sum of these products gives the total wage fund of all workers ( 
).
If the calculation were performed using the simple arithmetic average formula, the average earnings would be 3,000 rubles. (). Comparing the obtained result with the initial data, it is obvious that the average wage should be significantly higher (more than half of the workers receive wages above 3,000 rubles). Therefore, the calculation of the simple arithmetic mean in such cases will be erroneous.
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.
Consider the calculation of the arithmetic mean for such series.
The average is:
MeanMean- numerical characteristic of a set of numbers or functions; - some number enclosed between the smallest and largest of their values.
|
Basic information
The starting point for the formation of the theory of averages was the study of proportions by the school of Pythagoras. At the same time, no strict distinction was made between the concepts of average and proportion. A significant impetus to the development of the theory of proportions from an arithmetic point of view was given by Greek mathematicians - Nicomachus of Geras (late I - early II century AD) and Pappus of Alexandria (III century AD). The first stage in the development of the concept of average is the stage when the average began to be considered the central member of a continuous proportion. But the concept of the mean as the central value of the progression does not make it possible to derive the concept of the mean with respect to a sequence of n terms, regardless of the order in which they follow each other. For this purpose it is necessary to resort to a formal generalization of averages. The next stage is the transition from continuous proportions to progressions - arithmetic, geometric and harmonic.
In the history of statistics, for the first time, the widespread use of averages is associated with the name of the English scientist W. Petty. W. Petty was one of the first who tried to give the average value a statistical meaning, linking it with economic categories. But Petty did not produce a description of the concept of the average value, its allocation. A. Quetelet is considered to be the founder of the theory of average values. He was one of the first to consistently develop the theory of averages, trying to bring a mathematical basis for it. A. Quetelet singled out two types of averages - actual averages and arithmetic averages. Properly averages represent a thing, a number, really existing. Actually averages or average statistical ones should be derived from phenomena of the same quality, identical in their internal meaning. Arithmetic averages are numbers that give the closest possible idea of many numbers, different, albeit homogeneous.
Each type of average can be either a simple average or a weighted average. The correctness of the choice of the average form follows from the material nature of the object of study. Simple average formulas are used if the individual values of the averaged feature do not repeat. When in practical research individual values of the trait under study occur several times in the units of the population under study, then the frequency of repetitions of the individual trait values is present in the calculation formulas of power averages. In this case, they are called weighted average formulas.
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The topic of arithmetic and geometric mean is included in the mathematics program for grades 6-7. Since the paragraph is quite easy to understand, it is quickly passed, and the conclusion is school year students forget it. But knowledge in basic statistics is needed for passing the exam, as well as for international SAT exams. Yes and for Everyday life developed analytical thinking never hurts.
How to calculate the arithmetic and geometric mean of numbers
Suppose there is a series of numbers: 11, 4, and 3. The arithmetic mean is the sum of all numbers divided by the number of given numbers. That is, in the case of numbers 11, 4, 3, the answer will be 6. How is 6 obtained?
Solution: (11 + 4 + 3) / 3 = 6
The denominator must contain a number equal to the number of numbers whose average is to be found. The sum is divisible by 3, since there are three terms.
Now we need to deal with the geometric mean. Let's say there is a series of numbers: 4, 2 and 8.
The geometric mean is the product of all given numbers, which is under a root with a degree equal to the number of given numbers. That is, in the case of numbers 4, 2 and 8, the answer is 4. Here's how it happened:
Solution: ∛(4 × 2 × 8) = 4
In both options, whole answers were obtained, since special numbers were taken as an example. This is not always the case. In most cases, the answer has to be rounded or left at the root. For example, for the numbers 11, 7, and 20, the arithmetic mean is ≈ 12.67, and the geometric mean is ∛1540. And for the numbers 6 and 5, the answers, respectively, will be 5.5 and √30.
Can it happen that the arithmetic mean becomes equal to the geometric mean?
Of course it can. But only in two cases. If there is a series of numbers consisting only of either ones or zeros. It is also noteworthy that the answer does not depend on their number.
Proof with units: (1 + 1 + 1) / 3 = 3 / 3 = 1 (arithmetic mean).
∛(1 × 1 × 1) = ∛1 = 1 (geometric mean).
Proof with zeros: (0 + 0) / 2=0 (arithmetic mean).
√(0 × 0) = 0 (geometric mean).
There is no other option and there cannot be.
Average values are widely used in statistics. Average values characterize the qualitative indicators of commercial activity: distribution costs, profit, profitability, etc.
Medium This is one of the most common generalizations. A correct understanding of the essence of the average determines its special significance in a market economy, when the average, through a single and random one, makes it possible to identify the general and necessary, to identify the trend of patterns of economic development.
average value - these are generalizing indicators in which they find expression of the action of general conditions, patterns of the phenomenon under study.
Statistical averages are calculated on the basis of mass data of 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 we calculate the average wages in cooperatives and state-owned enterprises, and extend the result to the entire population, then the average is fictitious, since it is calculated for a heterogeneous population, and such an average loses all meaning.
With the help of the average, there is, as it were, a smoothing of differences in the magnitude of the feature that arise for one reason or another in individual units of observation.
For example, the average output of a salesperson depends on many factors: qualifications, length of service, age, form of service, health, and so on.
The average output reflects the general property of the entire population.
The average value is a reflection of the values of the studied trait, therefore, it is measured in the same dimension as this trait.
Each average value characterizes the studied population according to any one attribute. In order to get a complete and comprehensive picture of the population under study in terms of a number of essential features, it is generally necessary to have a system of average values that can describe the phenomenon from different angles.
There are various averages:
arithmetic mean;
geometric mean;
average harmonic;
root mean square;
chronological average.
Consider some types of averages that are most commonly used in statistics.
Arithmetic mean
The simple arithmetic mean (unweighted) is equal to the sum of the individual values of the characteristic, divided by the number of these values.
The individual values of the attribute are called variants and are denoted by x (); the number of population units is denoted by n, the average value of the feature - by 
. Therefore, the simple arithmetic mean is:
According to the data of the discrete distribution series, it can be seen that the same values of the attribute (options) are repeated several times. So, variant x occurs in the aggregate 2 times, and variant x - 16 times, etc.
The number of identical values of a feature in the distribution series is called the frequency or weight and is denoted by the symbol n.
Calculate the average wage per worker 
in rubles:
The wage bill for each group of workers is equal to the product of options and frequency, and the sum of these products gives the total wage bill of all workers.
In accordance with this, the calculations can be presented in a 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 grouped data is carried out according to the weighted arithmetic mean formula:
In the practice of economic statistics, sometimes it is necessary to calculate the average by group averages or by averages of individual parts of the population (partial averages). 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 arithmetic weighted average.
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 by n times, the value of the arithmetic mean will not change.
If all frequencies are divided or multiplied by some number, then the value of the average will not change.
2. The total multiplier of the individual values of the attribute can be taken out of the sign of the average:
3. The average sum (difference) of two or more quantities is equal to the sum (difference) of their averages:
4. If x \u003d c, where c is a constant value, then 
.
5. The sum of the deviations of the values of the feature 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 of the attribute. Like the arithmetic mean, it can be simple and weighted.
Along with the averages, the characteristics of the variation series are the mode and the median.
Fashion - this is the value of the trait (variant), the most frequently repeated in the studied population. For discrete distribution series, the mode will be the value of the variant with the highest frequency.
For interval distribution series with equal intervals, the mode is determined by the formula:
where 
- initial value of the interval containing the mode;
- the value of the modal interval;
- modal interval frequency;
- frequency of the interval preceding the modal;
- frequency of the interval following the modal.
Median is the variant located in the middle of the variation row. 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 series (an ordered series is the arrangement of population units in ascending or descending order).
Every person in modern world, planning to take out a loan or stocking vegetables for the winter, periodically confronts such a concept as "average". Let's find out: what it is, what types and classes of it exist, and why it is used in statistics and other disciplines.
Average value - what is it?
A similar name (SV) is a generalized characteristic of a set of homogeneous phenomena, determined by any one quantitative variable attribute.
However, people far from such abstruse definitions understand this concept as an average amount of something. For example, before taking a loan, a bank employee will definitely ask a potential client to provide data on the average income for the year, that is, the total amount of money a person earns. It is calculated by summing the earnings for the entire year and dividing by the number of months. Thus, the bank will be able to determine whether its client will be able to repay the debt on time.
Why is it being used?
As a rule, average values are widely used in order to give a final characterization of certain social phenomena, which are massive. They can also be used for smaller calculations, as in the case of a loan, in the example above.
However, most often averages are still used for global purposes. An example of one of them is the calculation of the amount of electricity consumed by citizens during one calendar month. Based on the data obtained, maximum norms are subsequently set for the categories of the population that enjoy benefits from the state.
Also, with the help of average values, a warranty period for the service of certain household appliances, cars, buildings, etc. Based on the data collected in this way, modern standards of work and rest were once developed.
In fact, any phenomenon modern life, which is of a mass nature, in one way or another is necessarily connected with the concept under consideration.
Applications
This phenomenon is widely used in almost all exact sciences, especially those of an experimental nature.
Finding the average is of great importance in medicine, engineering, cooking, economics, politics, and so on.
Based on the data obtained from such generalizations, develop medical preparations, curricula, set minimum living wages and wages, build study schedules, produce furniture, clothing and footwear, hygiene items, and much more.
In mathematics, this term is called "average value" and is used to implement decisions various examples and tasks. The simplest of these are addition and subtraction with ordinary fractions. After all, as is known, in order to solve similar examples Both fractions must be reduced to a common denominator.
Also, in the queen of the exact sciences, the term “average value of a random variable” is often used, which is close in meaning. To most, it is more familiar as "expectation", more often considered in probability theory. It is worth noting that a similar phenomenon also applies when performing statistical calculations.
Average value in statistics
However, most often the concept under study is used in statistics. As you know, this science itself specializes in the calculation and analysis quantitative characteristics mass social events. Therefore, the average value in statistics is used as a specialized method for achieving its main objectives - the collection and analysis of information.
The essence of this statistical method is to replace the individual unique values of the trait under consideration with a certain balanced average value.
An example is the famous food joke. So, at a certain factory on Tuesdays for lunch, his bosses usually eat meat casserole, and ordinary workers eat stewed cabbage. Based on these data, we can conclude that, on average, the plant's staff dines on cabbage rolls on Tuesdays.
Although this example is slightly exaggerated, it illustrates the main drawback of the average value search method - the leveling of the individual characteristics of objects or personalities.
The averages are used not only to analyze the collected information, but also to plan and predict further actions.
It is also used to evaluate the results achieved (for example, the implementation of the plan for growing and harvesting wheat for the spring-summer season).
How to calculate
Although, depending on the type of SW, there are different formulas her calculations, general theory statistics, as a rule, only one method of calculating the average value of a feature is used. To do this, you must first add together the values of all phenomena, and then divide the resulting sum by their number.
When making such calculations, it is worth remembering that the average value always has the same dimension (or units) as a separate unit of the population.
Conditions for correct calculation
The formula discussed above is very simple and universal, so it is almost impossible to make a mistake in it. However, it is always worth considering two aspects, otherwise the data obtained will not reflect the real situation.
CB classes
Having found answers to the main questions: "The average value - what is it?", "Where is it used?" and "How can I calculate it?", it is worth knowing what classes and types of CB exist.
First of all, this phenomenon is divided into 2 classes. These are structural and power averages.
Types of power SW
Each of the above classes, in turn, is divided into types. The power class has four of them.
- Medium arithmetic value- This is the most common type of SV. It is an average term, in determining which the total volume of the considered attribute in the data set is equally distributed among all units of this set.
This type is divided into subspecies: simple and weighted arithmetic SV.
- The mean harmonic value is an indicator that is the reciprocal of the simple arithmetic mean, calculated from the reciprocal values of the characteristic in question.
It is used in cases where the individual values of the feature and the product are known, but the frequency data are not.
- The geometric mean is most often used in the analysis of the growth rates of economic phenomena. It makes it possible to keep the product of the individual values of a given quantity unchanged, rather than the sum.
It also happens to be simple and balanced.
- The root mean square value is used in the calculation of individual indicators of indicators, such as the coefficient of variation, which characterizes the rhythm of output, etc.
Also, with its help, the average diameters of pipes, wheels, the average sides of a square and similar figures are calculated.
Like all other types of average SW, the root mean square is simple and weighted.
Types of structural quantities
In addition to average SWs, structural types are often used in statistics. They are better suited for calculating the relative characteristics of the values of a variable attribute and internal structure distribution lines.
There are two such types.
