Calculation of the arithmetic mean formula. How to calculate the average in Excel
In the process of various calculations and working with data, it is often necessary to calculate their average value. It is calculated by adding the numbers and dividing the total by their number. Let's find out how to calculate the average of a set of numbers using Microsoft Excel in various ways.
The easiest and most famous way to find the arithmetic mean of a set of numbers is to use a special button on the Microsoft Excel ribbon. Select a range of numbers located in a column or row of a document. While in the “Home” tab, click on the “AutoSum” button, which is located on the ribbon in the “Editing” tool block. From the drop-down list, select “Average”.
After this, using the “AVERAGE” function, the calculation is made. The arithmetic mean of a given set of numbers is displayed in the cell under the selected column, or to the right of the selected row.
This method is good for its simplicity and convenience. But it also has significant drawbacks. Using this method, you can calculate the average value of only those numbers that are arranged in a row in one column or in one row. But you cannot work with an array of cells, or with scattered cells on a sheet, using this method.
For example, if you select two columns and calculate the arithmetic mean using the method described above, then the answer will be given for each column separately, and not for the entire array of cells.
Calculation using the Function Wizard
For cases when you need to calculate the arithmetic average of an array of cells, or scattered cells, you can use the Function Wizard. It uses the same “AVERAGE” function, known to us from the first calculation method, but does it in a slightly different way.
Click on the cell where we want the result of calculating the average value to be displayed. Click on the “Insert Function” button, which is located to the left of the formula bar. Or, type the combination Shift+F3 on the keyboard.
The Function Wizard starts. In the list of functions presented, look for “AVERAGE”. Select it and click on the “OK” button.
The arguments window for this function opens. The function arguments are entered into the “Number” fields. These can be either ordinary numbers or the addresses of the cells where these numbers are located. If you are uncomfortable entering cell addresses manually, you should click on the button located to the right of the data entry field.
After this, the function arguments window will be minimized, and you will be able to select the group of cells on the sheet that you take for the calculation. Then, again click on the button to the left of the data entry field to return to the function arguments window.
If you want to calculate the arithmetic mean between numbers located in separate groups of cells, then do the same actions mentioned above in the “Number 2” field. And so on until all the necessary groups of cells are selected.
After that, click on the “OK” button.
The result of calculating the arithmetic mean will be highlighted in the cell that you selected before launching the Function Wizard.
Formula bar
There is a third way to launch the AVERAGE function. To do this, go to the “Formulas” tab. Select the cell in which the result will be displayed. After that, in the “Function Library” tool group on the ribbon, click on the “Other Functions” button. A list appears in which you need to sequentially go through the items “Statistical” and “AVERAGE”.
Then, exactly the same window of function arguments is launched as when using the Function Wizard, the work of which we described in detail above.
Further actions are exactly the same.
Manual function entry
But, do not forget that you can always enter the “AVERAGE” function manually if you wish. It will have the following pattern: “=AVERAGE(cell_range_address(number); cell_range_address(number)).
Of course, this method is not as convenient as the previous ones, and requires the user to keep certain formulas in his head, but it is more flexible.
Calculation of average value by condition
In addition to the usual calculation of the average value, it is possible to calculate the average value by condition. In this case, only those numbers from the selected range that meet a certain condition will be taken into account. For example, if these numbers are greater or less than a specific value.
For these purposes, the “AVERAGEIF” function is used. Like the AVERAGE function, you can launch it through the Function Wizard, from the formula bar, or by manually entering it into a cell. After the function arguments window has opened, you need to enter its parameters. In the “Range” field, enter the range of cells whose values will participate in determining the average arithmetic number. We do this in the same way as with the “AVERAGE” function.
But in the “Condition” field we must indicate a specific value, numbers greater or less than which will participate in the calculation. This can be done using comparison signs. For example, we took the expression “>=15000”. That is, for the calculation, only cells in the range containing numbers greater than or equal to 15000 will be taken. If necessary, instead of a specific number, you can specify the address of the cell in which the corresponding number is located.
The “Averaging range” field is optional. Entering data into it is only required when using cells with text content.
When all the data has been entered, click on the “OK” button.
After this, the result of calculating the arithmetic average for the selected range is displayed in a pre-selected cell, with the exception of cells whose data does not meet the conditions.
As you can see, in Microsoft Excel there are a number of tools with which you can calculate the average value of a selected series of numbers. Moreover, there is a function that automatically selects numbers from the range that do not meet a user-preset criterion. This makes calculations in Microsoft Excel even more user-friendly.
Suppose you need to find the average number of days to complete tasks by different employees. Or you want to calculate a time interval of 10 years Average temperature on a certain day. Calculating the average of a series of numbers in several ways.
The mean is a function of the measure of central tendency, which is the center of a series of numbers in a statistical distribution. Three majority general criteria central tendencies stand out.
Average The arithmetic mean is calculated by adding a series of numbers and then dividing the number of those numbers. For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by 6.5;
Median The average number of a series of numbers. Half the numbers have values that are greater than the Median, and half the numbers have values that are less than the Median. For example, the median of 2, 3, 3, 5, 7 and 10 is 4.
Mode The most common number in a group of numbers. For example, mode 2, 3, 3, 5, 7 and 10 - 3.
These three measures of central tendency, the symmetrical distribution of a series of numbers, are the same. In an asymmetrical distribution of a number of numbers, they can be different.
Calculate the average of cells that are contiguous in the same row or column
Follow these steps:
Calculating the average of random cells
To perform this task, use the function AVERAGE. Copy the table below onto a blank sheet of paper.
Calculation of weighted average
SUMPRODUCT And amounts. Example vThis calculates average price units of measure paid across three purchases, where each purchase is for a different number of units of measure at different prices per unit.
Copy the table below onto a blank sheet of paper.
Calculating the average of numbers, excluding zero values
To perform this task, use the functions AVERAGE And If. Copy the table below and keep in mind that in this example, to make it easier to understand, copy it onto a blank sheet of paper.
Now let's talk about how to calculate average.
In its classical form, the general theory of statistics offers us one version of the rules for choosing an average value.
First, you need to create the correct logical formula for calculating the average value (AFV). For each average value there is always only one logical formula for calculating it, so it is difficult to make a mistake here. But we must always remember that in the numerator (this is what is on top of the fraction) the sum of all phenomena, and in the denominator (this is what is at the bottom of the fraction) total quantity elements.
After the logical formula has been compiled, you can use the rules (for ease of understanding, we will simplify and shorten them):
1. If the source data (determined by frequency) contains the denominator of a logical formula, then the calculation is carried out using the weighted arithmetic mean formula.
2. If the numerator of a logical formula is presented in the source data, then the calculation is carried out using the weighted harmonic average formula.
3. If the problem presents both the numerator and the denominator of a logical formula (this rarely happens), then we carry out the calculation using this formula or the simple arithmetic average formula.
This is the classic idea of choosing the right formula for calculating the average. Next, we present the sequence of actions when solving problems for calculating the average value.
Algorithm for solving problems on calculating the average value
A. Determine the method for calculating the average value - simple or weighted . If the data is presented in a table, then we use a weighted method, if the data is presented by a simple enumeration, then we use a simple calculation method.
B. Determine or arrange symbols – x – option, f – frequency . The option is for which phenomenon you want to find the average value. The remaining data in the table will be the frequency.
B. We determine the form for calculating the average value - arithmetic or harmonic . The determination is carried out using the frequency column. The arithmetic form is used if the frequencies are specified by an explicit quantity (conditionally, you can substitute the word pieces, the number of elements “pieces”). The harmonic form is used if frequencies are specified not by an explicit quantity, but by a complex indicator (the product of the averaged quantity and frequency).
The most difficult thing is to guess where and what quantity is given, especially for a student inexperienced in such matters. In such a situation, you can use one of the following methods. For some tasks (economic), a statement developed over years of practice is suitable (point B.1). In other situations, you will have to use point B.2.
B.1 If the frequency is given in monetary units (in rubles), then the harmonic average is used for calculation, this statement is always true, if the identified frequency is given in money, in other situations this rule does not apply.
B.2 Use the rules for choosing the average value indicated above in this article. If the frequency is given by the denominator of the logical formula for calculating the average value, then we calculate using the arithmetic mean form; if the frequency is given by the numerator of the logical formula for calculating the average value, then we calculate using the harmonic mean form.
Let's look at examples of using this algorithm.
A. Since the data is presented in a line, we use a simple calculation method.
B.V. We only have data on the amount of pensions, and they will be our option - x. The data is presented as a simple number (12 people), for calculation we use the simple arithmetic average.
The average pension for a pensioner is 9208.3 rubles.
B. Since we need to find medium size payments per child, then the options are in the first column, put the designation x there, the second column automatically becomes the frequency f.
B. The frequency (number of children) is given by an explicit quantity (you can substitute the word pieces of children, from the point of view of the Russian language this is an incorrect phrase, but, in fact, it is very convenient to check), which means that the weighted arithmetic mean is used for the calculation.
The same problem can be solved not by a formulaic method, but by a tabular method, that is, entering all the data of intermediate calculations into a table.
As a result, all that needs to be done now is to separate the two totals in the correct order.
The average payment per child per month was 1,910 rubles.
A. Since the data is presented in the table, we use a weighted form for calculation.
B. Frequency (production cost) is given by an implicit quantity (frequency is given in rubles point of algorithm B1), which means that the weighted harmonic average is used for the calculation. In general, in essence, the cost of production is a complex indicator, which is obtained by multiplying the cost of a unit of a product by the number of such products, this is the essence of the harmonic average.
In order for this problem to be solved using the arithmetic mean formula, it is necessary that instead of the production cost there should be a number of products with the corresponding cost.
Please note that the sum in the denominator obtained after calculations is 410 (120+80+210) this is the total number of products produced.
The average cost per unit of product was 314.4 rubles.
A. Since the data is presented in the table, we use a weighted form for calculation.
B. Since we need to find the average cost per unit of product, the options are in the first column, we put the designation x there, the second column automatically becomes the frequency f.
B. Frequency ( total number absences) is given by an implicit number (this is the product of two indicators of the number of absences and the number of students with such a number of absences), which means that the weighted harmonic average is used for the calculation. We will use point of algorithm B2.
In order for this problem to be solved using the arithmetic mean formula, it is necessary that instead of the total number of absences there should be the number of students.
We create a logical formula for calculating the average number of absences per student.
Frequency by task condition Total number of omissions. In the logical formula, this indicator is in the numerator, which means we use the harmonic mean formula.
Please note that the sum in the denominator, resulting after calculations 31 (18+8+5), is the total number of students.
The average number of absences per student is 13.8 days.
Average value - this is a general indicator that characterizes a qualitatively homogeneous population according to a certain quantitative characteristic. For example, middle age persons convicted of theft.
In judicial statistics, average values are used to characterize:
Average time for consideration of cases of this category;
Average claim size;
Average number of defendants per case;
Average damage;
Average workload of judges, etc.
The average is always a named value and has the same dimension as the characteristic of an individual unit of the population. Each average value characterizes the population being studied according to any one varying characteristic, therefore, behind each average value lies a series of distribution of units of this population according to the characteristic being studied. The choice of the type of average is determined by the content of the indicator and the initial data for calculating the average value.
All types of averages used in statistical research are divided 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 . Simple form The average value is used to obtain the average value of the characteristic being studied, when the calculation is carried out using ungrouped statistical data, or when each option in the aggregate occurs only once. Weighted averages are values that take into account that variants of attribute values may have different numbers, and therefore each variant has to be multiplied by the corresponding frequency. In other words, each option is “weighted” by its frequency. Frequency is called statistical weight.
Simple arithmetic mean- the most common type of average. It is equal to the sum of the individual values of the attribute divided by the total number of these values:
Where x 1 ,x 2 , … ,x N are the individual values of the varying characteristic (variants), and N is the number of units in the population.
Arithmetic average weighted used in cases where data is presented in the form of distribution series or groupings. It is calculated as the sum of the products of options and their corresponding frequencies, divided by the sum of the frequencies of all options:
Where x i- meaning i th variants of the characteristic; f i- frequency i th options.
Thus, each variant value is weighted by its frequency, which is why frequencies are sometimes called statistical weights.
Comment. When we're talking about about the arithmetic mean without indicating its type, the arithmetic mean is simple.
Table 12.
Solution. To calculate, we use the weighted arithmetic average formula:
Thus, on average there are two defendants per criminal case.
If the calculation of the average value is carried out using data grouped in the form of interval distribution series, then you first need to determine the middle values of each interval x"i, and then calculate the average value using the arithmetic weighted average formula, into which x"i is substituted instead of xi.
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.
Solution. In order to determine the average age of criminals based on an interval variation series, it is necessary to first find the middle 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 values of the first and last intervals are equal to 10.
Now we find the average age of criminals using the weighted arithmetic average formula:
Thus, the average age of criminals convicted of theft is approximately 27 years.
Mean harmonic simple represents the reciprocal of the arithmetic mean of the reciprocal values of the attribute:
where 1/ x i are the inverse values of the options, and N is the number of units in the population.
Example. To determine the average annual workload on judges of a district court when considering criminal cases, a study of the workload of 5 judges of this court was conducted. 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 on one criminal case and the average annual workload on judges of a given district court when considering criminal cases.
Solution. To determine the average time spent on one criminal case, we use the harmonic average formula:
To simplify the calculations, in the example we take the number of days in a year to be 365, including weekends (this does not affect the calculation methodology, 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 a given district court when considering criminal cases will be: 365 (days) : 5.56 ≈ 65.6 (cases).
If we were to use the simple arithmetic average 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 on judges turned out to be less.
Let's check the validity of this approach. To do this, we will 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 (cases), 365(days) : 5.6 ≈ 65.2 (cases), 365(days) : 6.3 ≈ 58 (cases),
365(days) : 4.9 ≈ 74.5 (cases), 365(days) : 5.4 ≈ 68 (cases).
Now let’s calculate the average annual workload for judges of a given district court when considering criminal cases:
Those. the average annual load is the same as when using the harmonic average.
Thus, the use of the arithmetic average in this case is unlawful.
In cases where the variants of a feature and their volumetric values (the product of variants and frequency) are known, but the frequencies themselves are unknown, the weighted harmonic average formula is used:
,
Where x i are the values of the attribute options, and w i are the volumetric values of the options ( w i = x i f i).
Example. Data on the price of a unit of the same type of product produced by various institutions of the penal system, and on the volume of its sales are given in Table 14.
Table 14
Find the average selling price of the product.
Solution. When calculating the average price, we must use the ratio of the sales amount to the number of units sold. We do not know the number of units sold, but we know the amount of sales of goods. Therefore, to find the average price of goods sold, we will use the weighted harmonic average formula. We get
If you use the arithmetic average 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 attribute variants:
,
Where x 1 ,x 2 , … ,x N- individual values of the varying characteristic (variants), and
N- the number of units in the population.
This type of average is used to calculate the average growth rates of time series.
Mean square used to calculate average square deviation, which is an indicator of variation, and will be discussed below.
To determine the structure of the population, special average indicators 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 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 units of a statistical population can be ordered in ascending or descending order of variants of the characteristic being studied.
Median (Me)- this is the value that corresponds to the option located in the middle of the ranked series. Thus, the median is that version 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 units in the population).
If the series consists of an odd number of terms, then the median is equal to the option with number N Me. If the series consists of an even number of terms, 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 the row is given 1, 5, 7, 9, 11, 14, 15, 16, i.e. series with an even number of terms (N = 8), then N Me = (8 + 1) / 2 = 4.5. This means that 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. The frequencies of the option, starting from the first, are summed until the median number is exceeded. The value of the last summed option will be the median.
Example. Find the median number of accused per criminal case using the data in Table 12.
Solution. In this case, the volume of the variation series is N = 154, therefore, N Me = (154 + 1) / 2 = 77.5. Having summed 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 an interval variation series, the distribution first indicates the interval in which the median will be located. They call him median . This is the first interval whose accumulated 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- lower limit of the median interval; i is the value of the median interval; S Me-1- accumulated frequency of the interval that precedes the median; f Me- frequency of the median interval.
Example. Find the median age of offenders convicted of theft based on the statistics presented in Table 13.
Solution. Statistical data is presented by an interval variation series, which means that we first determine the median interval. The volume of the population is N = 162, therefore, the median interval is the interval 18-28, because this is the first interval whose accumulated frequency (15 + 90 = 105) exceeds half the volume (162: 2 = 81) of the interval variation series. Now we determine the numerical value of the median using the above formula:
Thus, half of those convicted of theft are under 25 years of age.
Fashion (Mo) They call the value of a characteristic that is most often found in units of the population. Fashion is used to identify the value of a characteristic that is most widespread. For a discrete series, the mode will be the option with the highest frequency. For example, for the discrete series presented in Table 3 Mo= 1, since this value corresponds to the highest frequency - 75. To determine the mode of the interval series, first determine modal interval (the 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 using the formula:
Where x Mo- lower limit of the modal interval; i is the value of the modal interval; f Mo- frequency of the modal interval; f Mo-1- frequency of the interval preceding the modal one; f Mo+1- frequency of the interval following the modal one.
Example. Find the age of the criminals convicted of theft, data on which are presented in Table 13.
Solution. The highest frequency corresponds to the interval 18-28, therefore, the mode should be in this interval. Its value is determined by the above formula:
Thus, the largest number of criminals convicted of theft are 24 years old.
The average value provides a general characteristic of the entirety of the phenomenon being studied. However, two populations that have the same average values may differ significantly from each other in the degree of fluctuation (variation) in the value of the characteristic being studied. For example, in one court the following terms of imprisonment were imposed: 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 populations 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 spread is quite large, the average term of imprisonment does not reflect the entire population. Thus, if the individual values of a characteristic differ little from each other, then the arithmetic mean will be a fairly indicative characteristic of the properties of a given population. Otherwise, the arithmetic mean will be an unreliable characteristic of this population and its use in practice will be ineffective. Therefore, it is necessary to take into account the variation in the values of the characteristic being studied.
Variation- these are differences in the values of any characteristic among different units of a given population at 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 a characteristic are formed under the combined influence of various factors (conditions), which are combined differently 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) scope of variation;
2) average linear deviation;
3) dispersion;
4) standard deviation;
5) coefficient of variation.
Let's briefly look at each of them.
Range of 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 a characteristic for units of a given population:
Range of variation (range of fluctuations) - important indicator the variability of the sign, but it makes it possible to see only extreme deviations, which limits the scope of its application. To more accurately characterize the variation of a trait based on its variability, other indicators are used.
Average linear deviation represents the arithmetic mean of the absolute values of deviations of individual values of a characteristic from the average 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 dispersion of the values of the characteristic being studied relative to its average value. Dispersion is defined as the average of the deviations squared.
Simple variance for ungrouped data:
.
Variance weighted for the variation series:
Comment. In practice, it is better to use the following formulas to calculate variance:
For 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 measures of scattering discussed above (range of variation, dispersion, standard deviation) are absolute indicators, by which it is not always possible to judge the degree of variability of a characteristic. In some problems it is necessary to use relative scattering indices, one of which is coefficient of variation.
Coefficient of variation- the ratio of the standard deviation to the arithmetic mean, expressed as a percentage:
The coefficient of variation is used not only for a comparative assessment of variation different signs or the same characteristic 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. The following data are available on the terms of imprisonment of 50 convicts delivered to serve a sentence imposed by the court in a correctional institution of the penal 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 series of distributions by terms of imprisonment.
2. Find the mean, variance and standard deviation.
3. Calculate the coefficient of variation and make a conclusion about the homogeneity or heterogeneity of the population being studied.
Solution. To construct a discrete distribution series, it is necessary to determine options 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:
Let's find the mean and variance. Since statistical data is represented by a discrete variation series, we will use the formulas for the weighted arithmetic mean and dispersion to calculate them. We get:
= 
= 4,1;
= 
5,21.
Now we calculate the standard deviation:
Finding the coefficient of variation:
Consequently, the statistical population is quantitatively heterogeneous.
Every person in modern world When planning to take out a loan or stocking up on vegetables for the winter, you periodically come across such a concept as “average value”. Let's find out: what it is, what types and classes 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 characteristic.
However, people who are far from such abstruse definitions understand this concept as an average amount of something. For example, before taking out a loan, a bank employee will definitely ask a potential client to provide data on average income for the year, that is, the total amount of money a person earns. It is calculated by summing up 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 used?
As a rule, average values are widely used to give a summary characterization of certain social phenomena, which are of a mass nature. They can also be used for smaller scale calculations, as in the case of a loan in the example above.
However, most often average values 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 standards are subsequently established for categories of the population enjoying benefits from the state.
Also, using average values, the warranty service life of certain household appliances, cars, buildings, etc. Based on data collected in this way, modern standards of work and rest were once developed.
Virtually any phenomenon modern life, which is of a mass nature, is in one way or another necessarily connected with the concept under consideration.
Areas of application
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, etc.
Based on the data obtained from such generalizations, they develop medicinal preparations, training programs, set minimum living wages and salaries, build educational schedules, produce furniture, clothing and shoes, hygiene products and much more.
In mathematics, this term is called the “average value” and is used to make decisions various examples and tasks. The simplest ones are addition and subtraction with ordinary fractions. After all, as is known, to solve similar examples It is necessary to reduce both fractions to a common denominator.
Also in the queen of exact sciences the term “average value of a random variable”, which is similar in meaning, is often used. It is more familiar to most as “mathematical 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, the concept being studied is most often used in statistics. As you know, this science itself specializes in calculation and analysis quantitative characteristics mass social phenomena. Therefore, the average value in statistics is used as a specialized method for achieving its main objectives - collecting and analyzing information.
The essence of this statistical method is to replace the individual unique values of the characteristic under consideration with a certain balanced average value.
An example is the famous food joke. So, at a certain factory on Tuesdays for lunch, its bosses usually eat meat casserole, and ordinary workers eat stewed cabbage. Based on these data, we can conclude that, on average, the plant staff dine on cabbage rolls on Tuesdays.
Although this example is slightly exaggerated, it illustrates the main drawback of the method of finding the average value - leveling individual characteristics objects or persons.
In average values they are used not only for analyzing the collected information, but also for planning and predicting 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 correctly
Although, depending on the type of SV, there are different formulas its calculations, in general theory statistics, as a rule, only one method is used to calculate the average value of a characteristic. To do this, you first need to 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 the individual 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 with it. However, it is always worth considering two aspects, otherwise the data obtained will not reflect the real situation.
SV classes
Having found answers to the basic questions: “What is the average value?”, “Where is it used?” and “How can you calculate it?”, it is worth finding out what classes and types of SVs exist.
First of all, this phenomenon is divided into 2 classes. These are structural and power averages.
Types of power SVs
Each of the above classes, in turn, is divided into types. The sedate class has four.
- Average arithmetic quantity- This is the most common type of SV. It is the average term, in determining which the total volume of the characteristic under consideration in a set of data is equally distributed among all units of this set.
This type is divided into subtypes: simple and weighted arithmetic SV.
- The harmonic mean is an indicator that is the inverse of the simple arithmetic mean, calculated from the reciprocal values of the characteristic under consideration.
It is used in cases where the individual values of the attribute and the product are known, but the frequency data are not.
- The geometric average is most often used when analyzing the growth rates of economic phenomena. It makes it possible to preserve unchanged the product of the individual values of a given quantity, and not the sum.
It can also be simple and balanced.
- The mean square value is used when calculating individual indicators, such as the coefficient of variation, characterizing the rhythm of product output, etc.
It is also used to calculate the average diameters of pipes, wheels, average sides of a square and similar figures.
Like all other types of averages, the root mean square can be simple and weighted.
Types of structural quantities
In addition to average SVs, structural types are often used in statistics. They are better suited for calculating the relative characteristics of the values of a varying characteristic and internal structure distribution rows.
There are two such types.
