Most of all in eq. In practice, we have to use the arithmetic mean, which can be calculated as the simple and weighted arithmetic mean.
Arithmetic average (SA)-n The most common type of average. It is used in cases where the volume of a varying characteristic for the entire population is the sum of the values of the characteristics of its individual units. Social phenomena are characterized by the additivity (totality) of the volumes of a varying characteristic; this determines the scope of application of SA and explains its prevalence as a general indicator, for example: the general salary fund is the sum of the salaries of all employees.
To calculate SA, you need to divide the sum of all feature values by their number. SA is used in 2 forms.
Let's first consider a simple arithmetic average.
1-CA simple (initial, defining form) is equal to the simple sum of the individual values of the characteristic being averaged, divided by the total number of these values (used when there are ungrouped index values of the characteristic):
The calculations made can be generalized into the following formula:
(1)
Where 
- the average value of the varying characteristic, i.e., the simple arithmetic average;
means summation, i.e. the addition of individual characteristics;
x- individual values of a varying characteristic, which are called variants;
n - number of units of the population
Example 1, it is required to find the average output of one worker (mechanic), if it is known how many parts each of 15 workers produced, i.e. given a series of ind. attribute values, pcs.: 21; 20; 20; 19; 21; 19; 18; 22; 19; 20; 21; 20; 18; 19; 20.
Simple SA is calculated using formula (1), pcs.:
Example2. Let's calculate SA based on conditional data for 20 stores included in the trading company (Table 1). Table 1
Distribution of stores of the trading company "Vesna" by sales area, sq. M
|
Store no. |
Store no. | ||
To calculate the average store area ( 
) it is necessary to add up the areas of all stores and divide the resulting result by the number of stores:
Thus, the average store area for this group of retail enterprises is 71 sq.m.
Therefore, to determine a simple SA, you need to divide the sum of all values of a given attribute by the number of units possessing this attribute.
2
Where f 1
,
f 2
,
… ,f n
–
weight (frequency of repetition of identical signs); – the sum of the products of the magnitude of features and their frequencies; – the total number of population units.
(2)
Where X- options;
f- frequency (weight).
Weighted SA is the quotient of dividing the sum of the products of options and their corresponding frequencies by the sum of all frequencies. Frequencies ( f) appearing in the SA formula are usually called scales, as a result of which the SA calculated taking into account the weights is called weighted.
We will illustrate the technique of calculating weighted SA using example 1 discussed above. To do this, we will group the initial data and place them in the table.
The average of the grouped data is determined as follows: first, the options are multiplied by the frequencies, then the products are added and the resulting sum is divided by the sum of the frequencies.
According to formula (2), the weighted SA is equal, pcs.: 
P
Distribution of Vesna stores by sales area, sq. m
Thus, the result was the same. However, this will already be a weighted arithmetic mean value.
In the previous example, we calculated the arithmetic average provided that the absolute frequencies (number of stores) are known. However, in a number of cases, absolute frequencies are absent, but relative frequencies are known, or, as they are commonly called, frequencies that show the proportion or the proportion of frequencies in the entire set.
When calculating SA weighted use frequencies allows you to simplify calculations when the frequency is expressed in large, multi-digit numbers. The calculation is made in the same way, however, since the average value turns out to be increased by 100 times, the result should be divided by 100.
Then the formula for the arithmetic weighted average will look like:
Where d– frequency, i.e. the share of each frequency in the total sum of all frequencies.
(3)In our example 2, we first determine the share of stores by group in the total number of stores of the Vesna company. So, for the first group the specific gravity corresponds to 10% 
. We get the following data Table3
In mathematics, the arithmetic mean of numbers (or simply the mean) is the sum of all the numbers in a given set divided by the number of numbers. This is the most generalized and widespread concept of average value. As you already understood, to find you need to sum up all the numbers given to you, and divide the resulting result by the number of terms.
What is the arithmetic mean?
Let's look at an example.
Example 1. Given numbers: 6, 7, 11. You need to find their average value.
Solution.
First, let's find the sum of all these numbers.
Now divide the resulting sum by the number of terms. Since we have three terms, we will therefore 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 can be clearly seen in the illustration.
The average is a bit like “evening out” a series of numbers. As you can see, the piles of pencils have become the same level.
Let's look at another example to consolidate the knowledge gained.
Example 2. Given numbers: 3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29. You need to find their arithmetic mean.
Solution.
Find the amount.
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 let's look at negative numbers. Let's remember how to summarize them. For example, you have two numbers 1 and -4. Let's find their sum.
1 + (-4) = 1 - 4 = -3
Knowing this, let's look at another example.
Example 3. Find the average value of a series of numbers: 3, -7, 5, 13, -2.
Solution.
Find the sum of numbers.
3 + (-7) + 5 + 13 + (-2) = 12
Since there are 5 terms, 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 Microsoft Office software package. Let's look at a brief instruction, the value of 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 refer to ranges and arrays).
To make it more clear, let’s try out the knowledge we have 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
- Select AVERAGE. After this, 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, you should have the answer in cell C7 - 13.7. When you click on cell C7, the function (=Average(C1:C6)) will appear in the formula bar.
This feature is very useful for accounting, invoices, or when you just need to find the average of a very long series of numbers. Therefore, it is often used in offices and large companies. This allows you to keep your records in order and makes it possible to quickly calculate something (for example, average monthly income). You can also use Excel to find the average value of a function.
The average value is a general indicator characterizing the typical level of a phenomenon. It expresses the value of a characteristic per unit of the population.
The average value is:
1) the most typical value of the attribute for the population;
2) the volume of the population attribute, distributed equally among the units of the population.
The characteristic for which the average value is calculated is called “averaged” in statistics.
The average always generalizes the quantitative variation of a trait, i.e. in average values, individual differences between units in the population due to random circumstances are eliminated. In contrast to the average, the absolute value characterizing the level of a characteristic of an individual unit of a population does not allow one to compare the values of a characteristic among units belonging to different populations. So, if you need to compare the levels of remuneration of workers at two enterprises, then you cannot compare two employees of different enterprises on this basis. The compensation of workers selected for comparison may not be typical for these enterprises. If we compare the size of wage funds at the enterprises under consideration, the number of employees is not taken into account and, therefore, it is impossible to determine where the level of wages is higher. Ultimately, only average indicators can be compared, i.e. How much does one employee earn on average at each enterprise? Thus, there is a need to calculate the average value as a generalizing characteristic of the population.
It is important to note that during the averaging process, the total value of the attribute levels or its final value (in the case of calculating average levels in a dynamics series) must remain unchanged. In other words, when calculating the average value, the volume of the characteristic under study should not be distorted, and the expressions compiled when calculating the average must necessarily make sense.
Calculating the average is one of the common generalization techniques; the average indicator denies what is common (typical) to all units of the population being studied, while at the same time it ignores the differences of individual units. In every phenomenon and its development there is a combination of chance and necessity. When calculating averages, due to the action of the law of large numbers, the randomness cancels out and balances out, so it is possible to abstract from the unimportant features of the phenomenon, from the quantitative values of the characteristic in each specific case. The ability to abstract from the randomness of individual values and fluctuations lies the scientific value of averages as generalizing characteristics of aggregates.
In order for the average to be truly representative, it must be calculated taking into account certain principles.
Let us dwell on some general principles for the use of averages.
1. The average must be determined for populations consisting of qualitatively homogeneous units.
2. The average must be calculated for a population consisting of a sufficiently large number of units.
3. The average must be calculated for a population whose units are in a normal, natural state.
4. The average should be calculated taking into account the economic content of the indicator under study.
5.2. Types of averages and methods for calculating them
Let us now consider the types of average values, features of their calculation and areas of application. Average values are divided into two large classes: power averages, structural averages.
Power means include the most well-known and frequently used types, such as geometric mean, arithmetic mean and square mean.
The mode and median are considered as structural averages.
Let's focus on power averages. Power averages, depending on the presentation of the source data, can be simple or weighted. Simple average It is calculated based on ungrouped data and has the following general form:
,
where X i is the variant (value) of the characteristic being averaged;
n – number option.
Weighted average is calculated based on grouped data and has a general appearance
,
where X i is the variant (value) of the characteristic being averaged or the middle value of the interval in which the variant is measured;
m – average degree index;
f i – frequency showing how many times the i-e value of the averaged characteristic occurs.
If you calculate all types of averages for the same initial data, then their values will turn out to be different. The rule of majority of averages applies here: as the exponent m increases, the corresponding average value also increases:
In statistical practice, arithmetic means and harmonic weighted means are used more often than other types of weighted averages.
Types of power means
|
Kind of power |
Index |
Calculation formula |
|
|
Simple |
Weighted |
||
|
Harmonic |
|||
|
Geometric |
|
|
|
|
Arithmetic |
|||
|
Quadratic |
|||
|
Cubic |
|||
The harmonic mean has a more complex structure than the arithmetic mean. The harmonic mean is used for calculations when not the units of the population - the carriers of the characteristic - are used as weights, but the product of these units by the values of the characteristic (i.e. m = Xf). The average harmonic simple should be resorted to in cases of determining, for example, the average cost of labor, time, materials per unit of production, per one part for two (three, four, etc.) enterprises, workers engaged in the manufacture of the same type of product , the same part, product.
The main requirement for the formula for calculating the average value is that all stages of the calculation have a real meaningful justification; the resulting average value should replace the individual values of the attribute for each object without disrupting the connection between the individual and summary indicators. In other words, the average value must be calculated in such a way that when each individual value of the averaged indicator is replaced by its average value, some final summary indicator, connected in one way or another with the averaged indicator, remains unchanged. This total is called defining since the nature of its relationship with individual values determines the specific formula for calculating the average value. Let us demonstrate this rule using the example of the geometric mean.
Geometric mean formula
used most often when calculating the average value based on individual relative dynamics.
The geometric mean is used if a sequence of chain relative dynamics is given, indicating, for example, an increase in production volume compared to the level of the previous year: i 1, i 2, i 3,…, i n. Obviously, the volume of production in the last year is determined by its initial level (q 0) and subsequent increase over the years:
q n =q 0 × i 1 × i 2 ×…×i n .
Taking q n as the determining indicator and replacing the individual values of the dynamics indicators with average ones, we arrive at the relation
From here
A special type of average values - structural averages - is used to study the internal structure of the distribution series of attribute values, as well as to estimate the average value (power type), if, according to the available statistical data, its calculation cannot be performed (for example, if in the considered example there were no data both the volume of production and the amount of costs by group of enterprises).
Indicators are most often used as structural averages fashion – the most frequently repeated value of the attribute – and medians – the value of a characteristic that divides the ordered sequence of its values into two equal parts. As a result, for one half of the units in the population the value of the attribute does not exceed the median level, and for the other half it is not less than it.
If the characteristic being studied has discrete values, then there are no particular difficulties in calculating the mode and median. If data on the values of attribute X are presented in the form of ordered intervals of its change (interval series), the calculation of the mode and median becomes somewhat more complicated. Since the median value divides the entire population into two equal parts, it ends up in one of the intervals of characteristic X. Using interpolation, the value of the median is found in this median interval:
,
where X Me is the lower limit of the median interval;
h Me – its value;
(Sum m)/2 – half of the total number of observations or half the volume of the indicator that is used as a weighting in the formulas for calculating the average value (in absolute or relative terms);
S Me-1 – the sum of observations (or the volume of the weighting attribute) accumulated before the beginning of the median interval;
m Me – the number of observations or the volume of the weighting characteristic in the median interval (also in absolute or relative terms).
When calculating the modal value of a characteristic based on the data of an interval series, it is necessary to pay attention to the fact that the intervals are identical, since the repeatability indicator of the values of the characteristic X depends on this. For an interval series with equal intervals, the magnitude of the mode is determined as
,
where X Mo is the lower value of the modal interval;
m Mo – number of observations or volume of the weighting characteristic in the modal interval (in absolute or relative terms);
m Mo-1 – the same for the interval preceding the modal one;
m Mo+1 – the same for the interval following the modal one;
h – the value of the interval of change of the characteristic in groups.
TASK 1
The following data is available for the group of industrial enterprises for the reporting year
|
№ enterprises |
Product volume, million rubles. |
Average number of employees, people. |
Profit, thousand rubles |
|
|
197,7 |
10,0 |
13,5 |
||
|
22,8 |
1500 |
136,2 |
||
|
465,5 |
18,4 |
1412 |
97,6 |
|
|
296,2 |
12,6 |
1200 |
44,4 |
|
|
584,1 |
22,0 |
1485 |
146,0 |
|
|
480,0 |
119,0 |
1420 |
110,4 |
|
|
57805 |
21,6 |
1390 |
138,7 |
|
|
204,7 |
30,6 |
|||
|
466,8 |
19,4 |
1375 |
111,8 |
|
|
292,2 |
113,6 |
1200 |
49,6 |
|
|
423,1 |
17,6 |
1365 |
105,8 |
|
|
192,6 |
30,7 |
|||
|
360,5 |
14,0 |
1290 |
64,8 |
|
|
280,3 |
10,2 |
33,3 |
It is required to group enterprises for the exchange of products, taking the following intervals:
from 400 to 600 million rubles.
For each group and for all together, determine the number of enterprises, volume of production, average number of employees, average output per employee. Present the grouping results in the form of a statistical table. Formulate a conclusion.
SOLUTION
We will group enterprises by product exchange, calculate the number of enterprises, volume of production, and the average number of employees using the simple average formula. The results of grouping and calculations are summarized in a table.
Groups by product volume
№
enterprises
Product volume, million rubles.
Average annual cost of fixed assets, million rubles.
Medium sleep
juicy number of employees, people.
Profit, thousand rubles
Average output per employee
1 group
up to 200 million rubles
1,8,12
197,7
204,7
192,6
10,0
9,4
8,8
900
817
13,5
30,6
30,7
28,2
2567
74,8
0,23
Average level
198,3
24,9
2nd group
from 200 to 400 million rubles.
4,10,13,14
196,2
292,2
360,5
280,3
12,6
113,6
14,0
10,2
1200
1200
1290
44,4
49,6
64,8
33,3
1129,2
150,4
4590
192,1
0,25
Average level
282,3
37,6
1530
64,0
3 group
from 400 to
600 million
2,3,5,6,7,9,11
592
465,5
584,1
480,0
578,5
466,8
423,1
22,8
18,4
22,0
119,0
21,6
19,4
17,6
1500
1412
1485
1420
1390
1375
1365
136,2
97,6
146,0
110,4
138,7
111,8
105,8
3590
240,8
9974
846,5
0,36
Average level
512,9
34,4
1421
120,9
Total in aggregate
5314,2
419,4
17131
1113,4
0,31
On average
379,6
59,9
1223,6
79,5
Conclusion. Thus, in the population under consideration, the largest number of enterprises in terms of production volume fell into the third group - seven, or half of the enterprises. The average annual cost of fixed assets is also in this group, as well as the large average number of employees - 9974 people; enterprises of the first group are the least profitable.
TASK 2
The following data is available on the company's enterprises
Number of the enterprise included in the company
I quarter
II quarter
Product output, thousand rubles.
Man-days worked by workers
Average output per worker per day, rub.
59390,13
up to 200 million rubles
from 200 to 400 million rubles.
In mathematics and statistics average arithmetic (or easy average) of a set of numbers is the sum of all the numbers in this set divided by their number. The arithmetic mean is a particularly universal and most common representation of an average.
You will need
- Knowledge of mathematics.
Instructions
1. Let a set of four numbers be given. Need to be discovered average meaning this kit. To do this, we first find the sum of all these numbers. Possible numbers are 1, 3, 8, 7. Their sum is S = 1 + 3 + 8 + 7 = 19. The set of numbers must consist of numbers of the same sign, otherwise the sense in calculating the average value is lost.
2. Average meaning set of numbers is equal to the sum of numbers S divided by the number of these numbers. That is, it turns out that average meaning equals: 19/4 = 4.75.
3. For a set of numbers it is also possible to detect not only average arithmetic, but also average geometric. The geometric mean of several regular real numbers is a number that can replace any of these numbers so that their product does not change. The geometric mean G is sought using the formula: the Nth root of the product of a set of numbers, where N is the number in the set. Let's look at the same set of numbers: 1, 3, 8, 7. Let's find them average geometric. To do this, let's calculate the product: 1*3*8*7 = 168. Now from the number 168 you need to extract the 4th root: G = (168)^1/4 = 3.61. Thus average the geometric set of numbers is 3.61.
Average The geometric average is generally used less often than the arithmetic average, however, it can be useful when calculating the average value of indicators that change over time (the salary of an individual employee, the dynamics of academic performance indicators, etc.).
You will need
- Engineering calculator
Instructions
1. In order to find the geometric mean of a series of numbers, you first need to multiply all these numbers. Let's say you are given a set of five indicators: 12, 3, 6, 9 and 4. Let's multiply all these numbers: 12x3x6x9x4=7776.
2. Now from the resulting number you need to extract the root of a power equal to the number of elements of the series. In our case, from the number 7776 it will be necessary to extract the fifth root using an engineering calculator. The number obtained after this operation - in this case the number 6 - will be the geometric mean for the initial group of numbers.
3. If you don’t have an engineering calculator at hand, then you can calculate the geometric mean of a series of numbers using the SRGEOM function in Excel or using one of the online calculators specifically designed for calculating geometric mean values.
Note!
If you need to find the geometric mean of each for 2 numbers, then you do not need an engineering calculator: you can extract the second root (square root) of any number using the most ordinary calculator.
Helpful advice
Unlike the arithmetic mean, the geometric mean is not so powerfully affected by huge deviations and fluctuations between individual values in the set of indicators under study.
Average value is one of the collations of a set of numbers. Represents a number that cannot fall outside the range defined by the largest and smallest values in that set of numbers. Average arithmetic value is a particularly commonly used type of average.
Instructions
1. Add up all the numbers in the set and divide them by the number of terms to get the arithmetic mean. Depending on certain calculation conditions, it is sometimes easier to divide each of the numbers by the number of values in the set and sum the total.
2. Use, say, the calculator included with Windows OS if calculating the arithmetic average in your head is not possible. You can open it with support from the program launch dialog. To do this, press the “hot keys” WIN + R or click the “Start” button and select the “Run” command from the main menu. After that, type calc in the input field and press Enter on your keyboard or click the “OK” button. The same can be done through the main menu - open it, go to the “All programs” section and to the “Typical” segments and select the “Calculator” line.
3. Enter all the numbers of the set step by step by pressing the Plus key on the keyboard after all of them (besides the last one) or by clicking the corresponding button in the calculator interface. You can also enter numbers either from the keyboard or by clicking the corresponding interface buttons.
4. Press the slash key or click this icon in the calculator interface after entering the last value of the set and type the number of numbers in the sequence. After that, press the equal sign and the calculator will calculate and display the arithmetic mean.
5. You can use the Microsoft Excel spreadsheet editor for the same purpose. In this case, launch the editor and enter all the values of the sequence of numbers into the adjacent cells. If, after entering the entire number, you press Enter or the down or right arrow key, the editor itself will move the input focus to the adjacent cell.
6. Select all entered values and in the lower left corner of the editor window (in the status bar) you will see the arithmetic mean value for the selected cells.
7. Click the cell next to the last number entered if you just want to see the average. Expand the drop-down list with the image of the Greek letter sigma (Σ) in the Editing command group on the Main tab. Select the line " Average" and the editor will insert the necessary formula for calculating the arithmetic mean into the selected cell. Press the Enter key and the value will be calculated.
The arithmetic mean is one of the measures of central propensity, widely used in mathematics and statistical calculations. It is very easy to find the arithmetic average for several values, but every problem has its own nuances, which you need to know in order to perform correct calculations.
What is an arithmetic mean
The arithmetic mean defines the average value for each initial array of numbers. In other words, from a certain set of numbers a value that is universal for all elements is selected, the mathematical comparison of which with all elements is approximately equal. The arithmetic average is used preferably in the preparation of financial and statistical reports or for calculating the quantitative results of similar skills.
How to find the arithmetic mean
Finding the arithmetic mean for an array of numbers should begin by determining the algebraic sum of these values. For example, if the array contains the numbers 23, 43, 10, 74 and 34, then their algebraic sum will be equal to 184. When writing, the arithmetic mean is denoted by the letter? (mu) or x (x with a line). Next, the algebraic sum should be divided by the number of numbers in the array. In the example under consideration there were five numbers, therefore the arithmetic mean will be equal to 184/5 and will be 36.8.
Features of working with negative numbers
If the array contains negative numbers, then the arithmetic mean is found using a similar algorithm. The difference only exists when calculating in the programming environment, or if the problem contains additional data. In these cases, finding the arithmetic mean of numbers with different signs comes down to three steps: 1. Finding the universal arithmetic mean using the standard method;2. Finding the arithmetic mean of negative numbers.3. Calculation of the arithmetic mean of positive numbers. The results of each action are written separated by commas.
Natural and decimal fractions
If an array of numbers is represented by decimal fractions, the solution is carried out using the method of calculating the arithmetic mean of integers, but the reduction of the total is made according to the requirements of the problem for the accuracy of the result. When working with natural fractions, they should be reduced to a common denominator, the one that is multiplied by the number of numbers in the array. The numerator of the result will be the sum of the given numerators of the initial fractional elements.
The geometric mean of numbers depends not only on the absolute value of the numbers themselves, but also on their number. It is impossible to confuse the geometric mean and the arithmetic mean of numbers, since they are found using different methodologies. In this case, the geometric mean is invariably less than or equal to the arithmetic mean.
You will need
- Engineering calculator.
Instructions
1. Consider that in the general case the geometric mean of numbers is found by multiplying these numbers and taking from them the root of the power that corresponds to the number of numbers. For example, if you need to find the geometric mean of five numbers, then you will need to extract the fifth root from the product.
2. To find the geometric mean of 2 numbers, use the basic rule. Find their product, then take the square root of the number two, which corresponds to the degree of the root. Let's say, in order to find the geometric mean of the numbers 16 and 4, find their product 16 4 = 64. From the resulting number, take the square root?64=8. This will be the desired value. Please note that the arithmetic mean of these 2 numbers is larger and equal to 10. If the root is not extracted in its entirety, round the total to the required order.
3. To find the geometric mean of more than 2 numbers, also use the basic rule. To do this, find the product of all numbers for which you need to find the geometric mean. From the resulting product, extract the root of the power equal to the number of numbers. For example, in order to find the geometric mean of the numbers 2, 4 and 64, find their product. 2 4 64=512. Because it is necessary to find the result of the geometric mean of 3 numbers, extract the third root from the product. It is difficult to do this verbally, so use an engineering calculator. For this purpose it has a button “x^y”. Dial the number 512, press the “x^y” button, then dial the number 3 and press the “1/x” button to find the value 1/3, press the “=” button. We get the result of raising 512 to the power of 1/3, which corresponds to the third root. Get 512^1/3=8. This is the geometric mean of the numbers 2.4 and 64.
4. With the support of an engineering calculator, you can find the geometric mean using another method. Find the log button on your keyboard. After this, take the logarithm for all of the numbers, find their sum and divide it by the number of numbers. Take the antilogarithm from the resulting number. This will be the geometric mean of the numbers. Let's say, in order to find the geometric mean of the same numbers 2, 4 and 64, perform a set of operations on the calculator. Dial the number 2, then press the log button, press the “+” button, dial the number 4 and press log and “+” again, dial 64, press log and “=”. The result will be a number equal to the sum of the decimal logarithms of the numbers 2, 4 and 64. Divide the resulting number by 3, since this is the number of numbers by which the geometric mean is sought. From the total, take the antilogarithm by switching the register button and use the same log key. The result will be the number 8, this is the desired geometric mean.
Note!
The average value cannot be larger than the largest number in the set and smaller than the smallest.
Helpful advice
In mathematical statistics, the average value of a quantity is called the mathematical expectation.
Discipline: Statistics
Option No. 2
Average values used in statistics
Introduction………………………………………………………………………………….3
Theoretical task
Average value in statistics, its essence and conditions of application.
1.1. The essence of average size and conditions of use………….4
1.2. Types of averages………………………………………………………8
Practical task
Task 1,2,3…………………………………………………………………………………14
Conclusion………………………………………………………………………………….21
List of references………………………………………………………...23
Introduction
This test consists of two parts – theoretical and practical. In the theoretical part, such an important statistical category as the average value will be examined in detail in order to identify its essence and conditions of application, as well as highlight the types of averages and methods for their calculation.
Statistics, as we know, studies massive socio-economic phenomena. Each of these phenomena may have a different quantitative expression of the same characteristic. For example, wages of workers of the same profession or market prices for the same product, etc. Average values characterize the qualitative indicators of commercial activity: distribution costs, profit, profitability, etc.
To study any population according to varying (quantitatively changing) characteristics, statistics uses average values.
Medium sized entity
The average value is a generalizing quantitative characteristic of a set of similar phenomena based on one varying characteristic. In economic practice, a wide range of indicators are used, calculated as average values.
The most important property of the average value is that it represents the value of a certain characteristic in the entire population with one number, despite its quantitative differences in individual units of the population, and expresses what is common to all units of the population under study. Thus, through the characteristics of a unit of a population, it characterizes the entire population as a whole.
Average values are related to the law of large numbers. The essence of this connection is that during averaging, random deviations of individual values, due to the action of the law of large numbers, cancel each other out and the main development trend, necessity, and pattern are revealed in the average. Average values allow you to compare indicators related to populations with different numbers of units.
In modern conditions of development of market relations in the economy, averages serve as a tool for studying the objective patterns of socio-economic phenomena. However, in economic analysis one cannot limit oneself only to average indicators, since general favorable averages may hide large serious shortcomings in the activities of individual economic entities, and the sprouts of a new, progressive one. For example, the distribution of the population by income makes it possible to identify the formation of new social groups. Therefore, along with average statistical data, it is necessary to take into account the characteristics of individual units of the population.
The average value is the resultant of all factors influencing the phenomenon under study. That is, when calculating average values, the influence of random (perturbation, individual) factors cancels out and, thus, it is possible to determine the pattern inherent in the phenomenon under study. Adolphe Quetelet emphasized that the significance of the method of averages is the possibility of transition from the individual to the general, from the random to the regular, and the existence of averages is a category of objective reality.
Statistics studies mass phenomena and processes. Each of these phenomena has both common to the entire set and special, individual properties. The difference between individual phenomena is called variation. Another property of mass phenomena is their inherent similarity of characteristics of individual phenomena. So, the interaction of elements of a set leads to a limitation of the variation of at least part of their properties. This trend exists objectively. It is in its objectivity that lies the reason for the widest use of average values in practice and in theory.
The average value in statistics is a general indicator that characterizes the typical level of a phenomenon in specific conditions of place and time, reflecting the value of a varying characteristic per unit of a qualitatively homogeneous population.
In economic practice, a wide range of indicators are used, calculated as average values.
Using the method of averages, statistics solves many problems.
The main significance of averages lies in their generalizing function, that is, the replacement of many different individual values of a characteristic with an average value that characterizes the entire set of phenomena.
If the average value generalizes qualitatively homogeneous values of a characteristic, then it is a typical characteristic of the characteristic in a given population.
However, it is incorrect to reduce the role of average values only to the characterization of typical values of characteristics in populations homogeneous for a given characteristic. In practice, much more often modern statistics use average values that generalize clearly homogeneous phenomena.
The average national income per capita, the average grain yield throughout the country, the average consumption of various food products - these are the characteristics of the state as a single economic system, these are the so-called system averages.
System averages can characterize both spatial or object systems that exist simultaneously (state, industry, region, planet Earth, etc.) and dynamic systems extended over time (year, decade, season, etc.).
The most important property of the average value is that it reflects what is common to all units of the population under study. The attribute values of individual units of the population fluctuate in one direction or another under the influence of many factors, among which there may be both basic and random. For example, the stock price of a corporation as a whole is determined by its financial position. At the same time, on certain days and on certain exchanges, these shares, due to prevailing circumstances, may be sold at a higher or lower rate. The essence of the average lies in the fact that it cancels out the deviations of the characteristic values of individual units of the population caused by the action of random factors, and takes into account the changes caused by the action of the main factors. This allows the average to reflect the typical level of the trait and abstract from the individual characteristics inherent in individual units.
Calculating the average is one of the most common generalization techniques; the average indicator reflects what is common (typical) for all units of the population being studied, while at the same time it ignores the differences of individual units. In every phenomenon and its development there is a combination of chance and necessity.
The average is a summary characteristic of the laws of the process in the conditions in which it occurs.
Each average characterizes the population under study according to any one characteristic, but to characterize any population, describe its typical features and qualitative features, a system of average indicators is needed. Therefore, in the practice of domestic statistics, to study socio-economic phenomena, as a rule, a system of average indicators is calculated. So, for example, the average wage indicator is assessed together with indicators of average output, capital-labor ratio and energy-labor ratio, the degree of mechanization and automation of work, etc.
The average should be calculated taking into account the economic content of the indicator under study. Therefore, for a specific indicator used in socio-economic analysis, only one true value of the average can be calculated based on the scientific method of calculation.
The average value is one of the most important generalizing statistical indicators, characterizing a set of similar phenomena according to some quantitatively varying characteristic. Averages in statistics are general indicators, numbers expressing the typical characteristic dimensions of social phenomena according to one quantitatively varying characteristic.
Types of averages
The types of average values differ primarily in what property, what parameter of the initial varying mass of individual values of the attribute must be kept unchanged.
Arithmetic mean
The arithmetic mean is the average value of a characteristic, during the calculation of which the total volume of the characteristic in the aggregate remains unchanged. Otherwise, we can say that the arithmetic mean is the average term. When calculating it, the total volume of the attribute is mentally distributed equally among all units of the population.
The arithmetic mean is used if the values of the characteristic being averaged (x) and the number of population units with a certain characteristic value (f) are known.
The arithmetic average can be simple or weighted.
Simple arithmetic mean
Simple is used if each value of attribute x occurs once, i.e. for each x the value of the attribute is f=1, or if the source data is not ordered and it is unknown how many units have certain attribute values.
The formula for the arithmetic mean is simple:
where is the average value; x – the value of the averaged characteristic (variant), – the number of units of the population being studied.
Arithmetic average weighted
Unlike a simple average, a weighted arithmetic average is used if each value of attribute x occurs several times, i.e. for each value of the feature f≠1. This average is widely used in calculating the average based on a discrete distribution series:
where is the number of groups, x is the value of the characteristic being averaged, f is the weight of the characteristic value (frequency, if f is the number of units in the population; frequency, if f is the proportion of units with option x in the total volume of the population).
Harmonic mean
Along with the arithmetic mean, statistics uses the harmonic mean, the inverse of the arithmetic mean of the inverse values of the attribute. Like the arithmetic mean, it can be simple and weighted. It is used when the necessary weights (f i) in the initial data are not specified directly, but are included as a factor in one of the available indicators (i.e., when the numerator of the initial ratio of the average is known, but its denominator is unknown).
Harmonic mean weighted
The product xf gives the volume of the averaged characteristic x for a set of units and is denoted w. If the source data contains values of the characteristic x being averaged and the volume of the characteristic being averaged w, then the harmonic weighted method is used to calculate the average:
where x is the value of the averaged characteristic x (variant); w – weight of variants x, volume of the averaged characteristic.
Harmonic mean unweighted (simple)
This medium form, used much less frequently, has the following form:
where x is the value of the characteristic being averaged; n – number of x values.
Those. this is the reciprocal of the simple arithmetic mean of the reciprocal values of the attribute.
In practice, the harmonic simple mean is rarely used in cases where the values of w for population units are equal.
Mean square and mean cubic
In a number of cases in economic practice, there is a need to calculate the average size of a characteristic, expressed in square or cubic units of measurement. Then the mean square is used (for example, to calculate the average size of a side and square sections, the average diameters of pipes, trunks, etc.) and the average cubic (for example, when determining the average length of a side and cubes).
If, when replacing individual values of a characteristic with an average value, it is necessary to keep the sum of the squares of the original values unchanged, then the average will be a quadratic average value, simple or weighted.
Simple mean square
Simple is used if each value of the attribute x occurs once, in general it has the form:
where is the square of the values of the characteristic being averaged; - the number of units in the population.
Weighted mean square
The weighted mean square is applied if each value of the averaged characteristic x occurs f times:
,
where f is the weight of options x.
Cubic average simple and weighted
The average cubic prime is the cube root of the quotient of dividing the sum of the cubes of individual attribute values by their number:
where are the values of the attribute, n is their number.
Average cubic weighted:
,
where f is the weight of the options x.
The square and cubic means have limited use in statistical practice. The mean square statistic is widely used, but not from the options themselves x , and from their deviations from the average when calculating variation indices.
The average can be calculated not for all, but for some part of the units in the population. An example of such an average could be the progressive average as one of the partial averages, calculated not for everyone, but only for the “best” (for example, for indicators above or below individual averages).
Geometric mean
If the values of the characteristic being averaged are significantly different from each other or are specified by coefficients (growth rates, price indices), then the geometric mean is used for calculation.
The geometric mean is calculated by extracting the root of the degree and from the products of individual values - variants of the characteristic X:
where n is the number of options; P - product sign.
The geometric mean is most widely used to determine the average rate of change in dynamics series, as well as in distribution series.
Average values are general indicators in which the effect of general conditions and the pattern of the phenomenon being studied are expressed. Statistical averages are calculated on the basis of mass data from correctly statistically organized mass observation (continuous or sample). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). The use of averages should proceed from a dialectical understanding of the categories of general and individual, mass and individual.
The combination of general means with group means makes it possible to limit qualitatively homogeneous populations. By dividing the mass of objects that make up this or that complex phenomenon into internally homogeneous, but qualitatively different groups, characterizing each of the groups with its average, it is possible to reveal the reserves of the process of an emerging new quality. For example, the distribution of the population by income allows us to identify the formation of new social groups. In the analytical part, we looked at a particular example of using the average value. To summarize, we can say that the scope and use of averages in statistics is quite wide.
Practical task
Task No. 1
Determine the average purchase rate and average sale rate of one and $ US
Average purchase rate
Average selling rate
Task No. 2
The dynamics of the volume of own public catering products in the Chelyabinsk region for 1996-2004 are presented in the table in comparable prices (million rubles)
Close rows A and B. To analyze the series of dynamics of production of finished products, calculate:
1. Absolute growth, chain and base growth and growth rates
2. Average annual production of finished products
3. Average annual growth rate and increase in the company’s products
4. Perform analytical alignment of the dynamics series and calculate the forecast for 2005
5. Graphically depict a series of dynamics
6. Draw a conclusion based on the dynamics results
1) yi B = yi-y1 yi C = yi-y1
y2 B = 2.175 – 2.04 y2 C = 2.175 – 2.04 = 0.135
y3B = 2.505 – 2.04 y3 C = 2.505 – 2.175 = 0.33
y4 B = 2.73 – 2.04 y4 C = 2.73 – 2.505 = 0.225
y5 B = 1.5 – 2.04 y5 C = 1.5 – 2.73 = 1.23
y6 B = 3.34 – 2.04 y6 C = 3.34 – 1.5 = 1.84
y7 B = 3.6 3 – 2.04 y7 C = 3.6 3 – 3.34 = 0.29
y8 B = 3.96 – 2.04 y8 C = 3.96 – 3.63 = 0.33
y9 B = 4.41–2.04 y9 C = 4.41 – 3.96 = 0.45
Tr B2 
Tr Ts2 
Tr B3 
Tr Ts3 
Tr B4 
Tr Ts4 
Tr B5 
Tr Ts5
Tr B6 
Tr Ts6 
Tr B7 
Tr Ts7
Tr B8 
Tr Ts8
Tr B9 
Tr Ts9
Tr B = (TprB *100%) – 100%
Tr B2 = (1.066*100%) – 100% = 6.6%
Tr Ts3 = (1.151*100%) – 100% = 15.1%
2)y 
million rubles – average product productivity
2,921 + 0,294*(-4) = 2,921-1,176 = 1,745
2,921 + 0,294*(-3) = 2,921-0,882 = 2,039
(yt-y) = (1.745-2.04) = 0.087
(yt-yt) = (1.745-2.921) = 1.382
(y-yt) = (2.04-2.921) = 0.776
Tp 
By 
y2005=2.921+1.496*4=2.921+5.984=8.905
8,905+2,306*1,496=12,354
8,905-2,306*1,496=5,456
5,456 2005 12,354
Task No. 3
Statistical data on wholesale supplies of food and non-food items and the retail trade network of the region in 2003 and 2004 are presented in the corresponding graphs.
According to Tables 1 and 2, it is required
1. Find the general index of the wholesale supply of food products in actual prices;
2. Find the general index of the actual volume of food supply;
3. Compare general indices and draw the appropriate conclusion;
4. Find the general index of supply of non-food products in actual prices;
5. Find the general index of the physical volume of supply of non-food products;
6. Compare the obtained indices and draw conclusions on non-food products;
7. Find the consolidated general supply indexes of the entire commodity mass in actual prices;
8. Find the consolidated general index of physical volume (for the entire commodity mass of goods);
9. Compare the resulting summary indices and draw the appropriate conclusion.
| Base period |
Reporting period (2004) |
Supplies of the reporting period at prices of the base period |
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| 1,291-0,681=0,61= - 39 Conclusion In conclusion, let's summarize. Average values are general indicators in which the effect of general conditions and the pattern of the phenomenon being studied are expressed. Statistical averages are calculated on the basis of mass data from correctly statistically organized mass observation (continuous or sample). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). The use of averages should proceed from a dialectical understanding of the categories of general and individual, mass and individual. The average reflects what is common in each individual, individual object; therefore, the average becomes of great importance for identifying patterns inherent in mass social phenomena and invisible in individual phenomena. The deviation of the individual from the general is a manifestation of the development process. In some isolated cases, elements of the new, advanced may be laid down. In this case, it is specific factors, taken against the background of average values, that characterize the development process. Therefore, the average reflects the characteristic, typical, real level of the phenomena being studied. The characteristics of these levels and their changes in time and space are one of the main problems of averages. Thus, through the averages, for example, characteristic of enterprises at a certain stage of economic development is manifested; changes in the well-being of the population are reflected in average wages, family income in general and for individual social groups, and the level of consumption of products, goods and services. The average indicator is a typical value (ordinary, normal, prevailing as a whole), but it is such because it is formed in the normal, natural conditions of the existence of a specific mass phenomenon, considered as a whole. The average reflects the objective property of the phenomenon. In reality, often only deviant phenomena exist, and the average as a phenomenon may not exist, although the concept of typicality of a phenomenon is borrowed from reality. The average value is a reflection of the value of the characteristic being studied and, therefore, is measured in the same dimension as this characteristic. However, there are various ways to approximate the level of population distribution for comparing summary characteristics that are not directly comparable to each other, for example, the average population size in relation to the territory (average population density). Depending on which factor needs to be eliminated, the content of the average will also be determined. The combination of general means with group means makes it possible to limit qualitatively homogeneous populations. By dividing the mass of objects that make up this or that complex phenomenon into internally homogeneous, but qualitatively different groups, characterizing each of the groups with its average, it is possible to reveal the reserves of the process of an emerging new quality. For example, the distribution of the population by income allows us to identify the formation of new social groups. In the analytical part, we looked at a particular example of using the average value. To summarize, we can say that the scope and use of averages in statistics is quite wide. Bibliography 1. Gusarov, V.M. Theory of statistics by quality [Text]: textbook. allowance / V.M. Gusarov manual for universities. - M., 1998 2. Edronova, N.N. General theory of statistics [Text]: textbook / Ed. N.N. Edronova - M.: Finance and Statistics 2001 - 648 p. 3. Eliseeva I.I., Yuzbashev M.M. General theory of statistics [Text]: Textbook / Ed. Corresponding member RAS I.I. Eliseeva. – 4th ed., revised. and additional - M.: Finance and Statistics, 1999. - 480 pp.: ill. 4. Efimova M.R., Petrova E.V., Rumyantsev V.N. General theory of statistics: [Text]: Textbook. - M.: INFRA-M, 1996. - 416 p. 5. Ryauzova, N.N. General theory of statistics [Text]: textbook / Ed. N.N. Ryauzova - M.: Finance and Statistics, 1984. Gusarov V.M. Theory of statistics: Textbook. A manual for universities. - M., 1998.-P.60. Eliseeva I.I., Yuzbashev M.M. General theory of statistics. - M., 1999.-P.76. Gusarov V.M. Theory of statistics: Textbook. A manual for universities. -M., 1998.-P.61. | |||||||