In order to find the average value in Excel (no matter whether it is a numeric, text, percentage or other value), there are many functions. And each of them has its own characteristics and advantages. Indeed, in this task certain conditions may be set.
For example, the average values of a series of numbers in Excel are calculated using statistical functions. You can also manually enter your own formula. Let's consider various options.
How to find the arithmetic mean of numbers?
To find the arithmetic mean, you need to add up all the numbers in the set and divide the sum by the quantity. For example, a student’s grades in computer science: 3, 4, 3, 5, 5. What is included in the quarter: 4. We found the arithmetic mean using the formula: =(3+4+3+5+5)/5.
How to quickly do this using Excel functions? Let's take for example a series of random numbers in a string:
Or: make the active cell and simply enter the formula manually: =AVERAGE(A1:A8).
Now let's see what else the AVERAGE function can do.
Let's find the arithmetic mean of the first two and last three numbers. Formula: =AVERAGE(A1:B1,F1:H1). Result:
Condition average
The condition for finding the arithmetic mean can be a numerical criterion or a text one. We will use the function: =AVERAGEIF().
Find the arithmetic mean of numbers that are greater than or equal to 10.
Function: =AVERAGEIF(A1:A8,">=10")
The result of using the AVERAGEIF function under the condition ">=10": The third argument – “Averaging range” – is omitted. First of all, it is not required. Secondly, the range analyzed by the program contains ONLY numeric values. The cells specified in the first argument will be searched according to the condition specified in the second argument.
Attention! The search criterion can be specified in the cell. And make a link to it in the formula.
Let's find the average value of the numbers using the text criterion. For example, the average sales of the product “tables”.
The function will look like this: =AVERAGEIF($A$2:$A$12,A7,$B$2:$B$12). Range – a column with product names. The search criterion is a link to a cell with the word “tables” (you can insert the word “tables” instead of link A7). Averaging range – those cells from which data will be taken to calculate the average value.
As a result of calculating the function, we obtain the following value:
Attention! For a text criterion (condition), the averaging range must be specified.
How to calculate the weighted average price in Excel?
How did we find out the weighted average price?
Formula: =SUMPRODUCT(C2:C12,B2:B12)/SUM(C2:C12).
Using the SUMPRODUCT formula, we find out the total revenue after selling the entire quantity of goods. And the SUM function sums up the quantity of goods. By dividing the total revenue from the sale of goods by the total number of units of goods, we found the weighted average price. This indicator takes into account the “weight” of each price. Its share in the total mass of values.
Standard deviation: formula in Excel
Distinguish between average standard deviation By population and by sample. In the first case, this is the root of general variance. In the second, from the sample variance.
To calculate this statistical indicator, a dispersion formula is compiled. The root is extracted from it. But in Excel there is a ready-made function for finding the standard deviation.
The standard deviation is tied to the scale of the source data. This is not enough for a figurative representation of the variation of the analyzed range. To obtain the relative level of data scatter, the coefficient of variation is calculated:
standard deviation / arithmetic mean
The formula in Excel looks like this:
STDEV (range of values) / AVERAGE (range of values).
The coefficient of variation is calculated as a percentage. Therefore, we set the percentage format in the cell.
The characteristics of units of statistical aggregates are different in their meaning, for example, the wages of workers in the same profession of an enterprise are not the same for the same period of time, market prices for the same products, crop yields in the district’s farms, etc. Therefore, in order to determine the value of a characteristic that is characteristic of the entire population of units being studied, average values are calculated.
average value –
this is a generalizing characteristic of a set of individual values of some quantitative characteristic.
The population studied on a quantitative basis consists of individual values; they are influenced by common reasons, and individual conditions. In the average value, deviations characteristic of individual values are canceled out. The average, being a function of a set of individual values, represents the entire aggregate with one value and reflects what is common to all its units.
The average calculated for populations consisting of qualitatively homogeneous units is called typical average. For example, you can calculate the average monthly salary of an employee of a particular professional group (miner, doctor, librarian). Of course, monthly levels wages miners, due to differences in their qualifications, length of service, time worked per month and many other factors, differ from each other and from the level of average wages. However, the average level reflects the main factors that influence the level of wages, and cancels out the differences that arise due to individual characteristics employee. The average salary reflects the typical level of remuneration for a given type of worker. Obtaining a typical average should be preceded by an analysis of how qualitatively homogeneous the given population is. If the set consists of them individual parts, it should be divided into typical groups (average temperature in the hospital).
Average values used as characteristics for heterogeneous populations are called system averages. For example, the average value of gross domestic product (GDP) per capita, the average value of consumption of various groups of goods per person and other similar values that represent the general characteristics of the state as a unified economic system.
The average must be calculated for populations consisting of sufficient large number units. Compliance with this condition is necessary for the law to come into force large numbers, as a result of which random deviations of individual values from the general trend are mutually canceled out.
Types of averages and methods for calculating them
The choice of the type of average is determined by the economic content of a certain indicator and source data. However, any average value must be calculated so that when it replaces each variant of the averaged characteristic, the final, generalizing, or, as it is commonly called, does not change. defining indicator, which is associated with the averaged indicator. For example, when replacing actual speeds on individual sections of the path with their average speed, the total distance traveled should not change vehicle at the same time; when replacing the actual wages of individual employees of an enterprise with the average wage, the wage fund should not change. Consequently, in each specific case, depending on the nature of the available data, there is only one true average value of the indicator that is adequate to the properties and essence of the socio-economic phenomenon being studied.
The most commonly used are the arithmetic mean, harmonic mean, geometric mean, quadratic mean and cubic mean.
The listed averages belong to the class sedate averages and are combined by the general formula:
,
where is the average value of the characteristic being studied;
m – average degree index;
– current value (variant) of the characteristic being averaged;
n – number of features.
Depending on the value of the exponent m, the following types of power averages are distinguished:
when m = -1 – harmonic mean;
at m = 0 – geometric mean;
for m = 1 – arithmetic mean;
for m = 2 – root mean square;
at m = 3 – average cubic.
When using the same initial data, the larger the exponent m in the above formula, the more value average size:
.
This property of power averages to increase with increasing exponent of the defining function is called the rule of majority of averages.
Each of the marked averages can take two forms: simple And weighted.
Simple medium form used when the average is calculated from primary (ungrouped) data. Weighted form– when calculating the average based on secondary (grouped) data.
Arithmetic mean
The arithmetic mean is used when the volume of the population is the sum of all individual values of a varying characteristic. It should be noted that if the type of average is not specified, the arithmetic average is assumed. Its logical formula looks like: 
Simple arithmetic mean calculated based on ungrouped data
according to the formula: 
or ,
where are the individual values of the characteristic;
j is the serial number of the observation unit, which is characterized by the value ;
N – number of observation units (volume of the population).
Example. The lecture “Summary and grouping of statistical data” examined the results of observing the work experience of a team of 10 people. Let's calculate the average work experience of the team's workers. 5, 3, 5, 4, 3, 4, 5, 4, 2, 4.
Using the simple arithmetic mean formula, we can also calculate averages in chronological series, if the time intervals for which the characteristic values are presented are equal.
Example. The volume of products sold for the first quarter amounted to 47 den. units, for the second 54, for the third 65 and for the fourth 58 den. units The average quarterly turnover is (47+54+65+58)/4 = 56 den. units
If momentary indicators are given in a chronological series, then when calculating the average they are replaced by half-sums of the values at the beginning and end of the period.
If there are more than two moments and the intervals between them are equal, then the average is calculated using the formula for the average chronological
,
where n is the number of time points
In the case when the data is grouped by characteristic values
(i.e., a discrete variational distribution series has been constructed) with arithmetic average weighted calculated using either frequencies or frequencies of observation of specific values of a characteristic, the number of which (k) is significant less number observations (N) .
,
,
where k is the number of groups of the variation series,
i – group number of the variation series.
Since , a , we obtain the formulas used for practical calculations:
And
Example. Let's calculate the average length of service of work teams in a grouped row.
a) using frequencies:
b) using frequencies:
In the case when the data is grouped by intervals
, i.e. are presented in the form of interval distribution series; when calculating the arithmetic mean, the middle of the interval is taken as the value of the attribute, based on the assumption of a uniform distribution of population units over a given interval. The calculation is carried out using the formulas:
And
where is the middle of the interval: ,
where and are the lower and upper boundaries of the intervals (provided that upper limit of this interval coincides with the lower boundary of the next interval).
Example. Let's calculate the arithmetic mean of the interval variation series constructed based on the results of a study of the annual wages of 30 workers (see lecture “Summary and grouping of statistical data”).
Table 1 – Interval variation series distribution.
Intervals, UAH |
Frequency, people |
Frequency, |
The middle of the interval |
||
600-700 |
3 |
0,10 |
(600+700):2=650 |
1950 |
65 |
UAH or 
UAH
Arithmetic means calculated on the basis of source data and interval variation series may not coincide due to the uneven distribution of attribute values within the intervals. In this case, for a more accurate calculation of the weighted arithmetic mean, one should use not the middles of the intervals, but the simple arithmetic means calculated for each group ( group averages). The average calculated from group means using a weighted calculation formula is called general average.
The arithmetic mean has a number of properties.
1. The sum of deviations from the average option is zero:
.
2. If all the values of the option increase or decrease by the amount A, then the average value increases or decreases by the same amount A: 
3. If each option is increased or decreased by B times, then the average value will also increase or decrease by the same number of times:
or 
4. The sum of the products of the option by the frequencies is equal to the product of the average value by the sum of the frequencies: 
5. If all frequencies are divided or multiplied by any number, then the arithmetic mean will not change: 
6) if in all intervals the frequencies are equal to each other, then the weighted arithmetic mean is equal to the simple arithmetic mean: 
,
where k is the number of groups of the variation series.
Using the properties of the average allows you to simplify its calculation.
Let us assume that all options (x) are first reduced by the same number A, and then reduced by a factor of B. The greatest simplification is achieved when the value of the middle of the interval with the highest frequency is chosen as A, and the value of the interval (for series with identical intervals) is selected as B. The quantity A is called the origin, so this method of calculating the average is called way b ohm reference from conditional zero or way of moments.
After such a transformation, we obtain a new variational distribution series, the variants of which are equal to . Their arithmetic mean, called moment of the first order, is expressed by the formula and, according to the second and third properties, the arithmetic mean is equal to the mean of the original version, reduced first by A, and then by B times, i.e.
For getting real average(average of the original series) you need to multiply the first-order moment by B and add A: 
The calculation of the arithmetic mean using the method of moments is illustrated by the data in Table. 2.
Table 2 – Distribution of factory shop workers by length of service
Employees' length of service, years |
Amount of workers |
Middle of the interval |
|||
0 – 5 |
12 |
2,5 |
15 |
3 |
36 |
Finding the first order moment 
. Then, knowing that A = 17.5 and B = 5, we calculate the average length of service of the workshop workers:
years
Harmonic mean
As shown above, the arithmetic mean is used to calculate the average value of a characteristic in cases where its variants x and their frequencies f are known.
If statistical information does not contain frequencies f for individual options x of the population, but is presented as their product, the formula is applied weighted harmonic mean. To calculate the average, let's denote where . Substituting these expressions into the formula for the arithmetic weighted average, we obtain the formula for the harmonic weighted average: 
,
where is the volume (weight) of the indicator attribute values in the interval numbered i (i=1,2, …, k).
Thus, the harmonic mean is used in cases where it is not the options themselves that are subject to summation, but their reciprocals: 
.
In cases where the weight of each option is equal to one, i.e. individual values of the inverse characteristic occur once, applied mean harmonic simple:
,
where are individual variants of the inverse characteristic, occurring once;
N – number option.
If there are harmonic averages for two parts of a population, then the overall average for the entire population is calculated using the formula: 
and is called weighted harmonic mean of group means.
Example. During trading on the currency exchange, three transactions were concluded in the first hour of operation. Data on the amount of hryvnia sales and the hryvnia exchange rate against the US dollar are given in table. 3 (columns 2 and 3). Determine the average exchange rate of the hryvnia against the US dollar for the first hour of trading.
Table 3 – Data on the progress of trading on the foreign exchange exchange
The average dollar exchange rate is determined by the ratio of the amount of hryvnia sold during all transactions to the amount of dollars acquired as a result of the same transactions. The final amount of the sale of the hryvnia is known from column 2 of the table, and the number of dollars purchased in each transaction is determined by dividing the amount of the sale of the hryvnia by its exchange rate (column 4). A total of $22 million was purchased during three transactions. This means that the average exchange rate of the hryvnia for one dollar was 
.
The resulting value is real, because replacing it with actual hryvnia exchange rates in transactions will not change the final amount of hryvnia sales, which serves as defining indicator: million UAH
If the arithmetic mean were used for the calculation, i.e. 
hryvnia, then at the exchange rate for the purchase of 22 million dollars. it would be necessary to spend 110.66 million UAH, which is not true.
Geometric mean
The geometric mean is used to analyze the dynamics of phenomena and allows one to determine the average growth coefficient. When calculating the geometric mean, individual values of a characteristic are relative indicators of dynamics, constructed in the form of chain values, as the ratio of each level to the previous one.
The simple geometric mean is calculated using the formula: 
,
where is the sign of the product,
N – number of averaged values.
Example. The number of registered crimes over 4 years increased by 1.57 times, including for the 1st – 1.08 times, for the 2nd – 1.1 times, for the 3rd – 1.18 and for the 4th – 1.12 times. Then the average annual growth rate of the number of crimes is: , i.e. the number of registered crimes grew annually by an average of 12%.
1,8
-0,8
0,2
1,0
1,4
1
3
4
1
1
3,24
0,64
0,04
1
1,96
3,24
1,92
0,16
1
1,96
To calculate the weighted mean square, we determine and enter into the table and . Then the average deviation of the length of products from the given norm is equal to: 
Arithmetic mean in in this case would be unsuitable, because as a result we would get zero deviation.
The use of the mean square will be discussed further in terms of variation.
It gets lost in calculating the average.
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.
note
If you need to find the geometric mean for just two numbers, then you don’t need an engineering calculator: take the second root ( Square root) from any number can be done using the most ordinary calculator.
Unlike the arithmetic mean, the geometric mean is not as strongly affected by large deviations and fluctuations between individual values in the set of indicators under study.
Sources:
- Online calculator that calculates the geometric mean
- average geometric formula
Average value is one of the characteristics of a set of numbers. Represents a number that cannot be outside the range determined by the largest and lowest values in this set of numbers. Average arithmetic value is the most commonly used type of average.
Instructions
Add up all the numbers in the set and divide them by the number of terms to get the arithmetic mean. Depending on the specific calculation conditions, it is sometimes easier to divide each of the numbers by the number of values in the set and sum the result.
Use, for example, included in the Windows OS if it is not possible to calculate the arithmetic average in your head. You can open it using the program launch dialog. To do this, press the hot keys WIN + R or click the Start button and select Run from the main menu. Then type calc in the input field and press Enter or click the OK button. The same can be done through the main menu - open it, go to the “All programs” section and in the “Standard” section and select the “Calculator” line.
Enter all the numbers in the set sequentially by pressing the Plus key after each of them (except the last one) or clicking the corresponding button in the calculator interface. You can also enter numbers either from the keyboard or by clicking the corresponding interface buttons.
Press the slash key or click this in the calculator interface after entering the last set value and type the number of numbers in the sequence. Then press the equal sign and the calculator will calculate and display the arithmetic mean.
You can use a table editor for the same purpose. Microsoft Excel. In this case, launch the editor and enter all the values of the sequence of numbers into the adjacent cells. If, after entering each number, you press Enter or the down or right arrow key, the editor itself will move the input focus to the adjacent cell.
Click the cell next to the last number entered if you don't want to just see the average. Expand the Greek sigma (Σ) drop-down menu for the Edit commands on the Home tab. Select the line " Average"and the editor will insert the necessary formula for calculating the average arithmetic value into the selected cell. Press the Enter key and the value will be calculated.
The arithmetic mean is one of the measures of central tendency, widely used in mathematics and statistical calculations. Finding the arithmetic average for several values is very simple, but each task has its own nuances, which are simply necessary to know in order to perform correct calculations.
What is an arithmetic mean
The arithmetic mean determines the average value for the entire original array of numbers. In other words, from a certain set of numbers a value common to all elements is selected, the mathematical comparison of which with all elements is approximately equal. The arithmetic average is used primarily in the preparation of financial and statistical reports or for calculating the results of similar experiments.How to find the arithmetic mean
Search for the average arithmetic number for an array of numbers, you should start 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 μ (mu) or x (x with a bar). Next, the algebraic sum should be divided by the number of numbers in the array. In the example under consideration there were five numbers, so 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 has additional conditions. In these cases, finding the arithmetic mean of numbers with different signs comes down to three steps:1. Finding the general arithmetic average using the standard method;
2. Finding the arithmetic mean of negative numbers.
3. Calculation of the arithmetic mean of positive numbers.
The responses for each action are written separated by commas.
Natural and decimal fractions
If an array of numbers is presented decimals, the solution is carried out using the method of calculating the arithmetic mean of integers, but the result is reduced according to the requirements of the problem for the accuracy of the answer.When working with natural fractions they should be reduced to a common denominator, which is multiplied by the number of numbers in the array. The numerator of the answer will be the sum of the given numerators of the original fractional elements.
- Engineering calculator.
Instructions
Keep in mind that in general, the geometric mean of numbers is found by multiplying these numbers and taking the root of the power from them, which 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 root of the power from the product.
To find the geometric mean of two numbers, use the basic rule. Find their product, then take the square root of it, since the number is two, which corresponds to the power of the root. For example, in order to find the geometric mean of the numbers 16 and 4, find their product 16 4=64. From the resulting number, extract the square root √64=8. This will be the desired value. Please note that the arithmetic mean of these two numbers is greater than and equal to 10. If the entire root is not extracted, round the result to the desired order.
To find the geometric mean of more than two 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, to find the geometric mean of the numbers 2, 4, and 64, find their product. 2 4 64=512. Since you need to find the result of the geometric mean of three numbers, take 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 of 1/3, press the "=" button. We get the result of raising 512 to the 1/3 power, which corresponds to the third root. Get 512^1/3=8. This is the geometric mean of the numbers 2.4 and 64.
Using an engineering calculator, you can find the geometric mean in another way. Find the log button on your keyboard. After that, take the logarithm for each 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. For example, 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 the number equal to the sum decimal logarithms of the numbers 2, 4 and 64. Divide the resulting number by 3, since this is the number of numbers for which the geometric mean is sought. From the result, take the antilogarithm by switching the case button and use the same log key. The result will be the number 8, this is the desired geometric mean.
Average values are widely used in statistics. Average values characterize the qualitative indicators of commercial activity: distribution costs, profit, profitability, etc.
Average - This is one of the common generalization techniques. A correct understanding of the essence of the average determines its special significance in the conditions market economy, when the average through the individual and random allows us to identify the general and necessary, to identify the trend of patterns of economic development.
average value - these are generalizing indicators in which the effects of general conditions and patterns of the phenomenon being studied are expressed.
Statistical averages are calculated on the basis of mass data from correctly statistically organized mass observation (continuous and selective). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). For example, if you calculate the average wage in cooperatives and state-owned enterprises, and extend the result to the entire population, then the average is fictitious, since it is calculated for a heterogeneous population, and such an average loses all meaning.
With the help of the average, differences in the value of the characteristic, which arise for one reason or another in individual units of observation.
For example, the average productivity of a salesperson depends on many reasons: qualifications, length of service, age, form of service, health, etc.
Average output reflects the general property of the entire population.
The average value is a reflection of the values of the characteristic being studied, therefore, it is measured in the same dimension as this characteristic.
Each average value characterizes the population under study according to any one characteristic. In order to obtain a complete and comprehensive understanding of the population under study according to a number of essential characteristics, in general it is necessary to have a system of average values that can describe the phenomenon from different angles.
There are different averages:
arithmetic mean;
geometric mean;
harmonic mean;
mean square;
average chronological.
Let's look at some types of averages that are most often used in statistics.
Arithmetic mean
The simple arithmetic mean (unweighted) is equal to the sum of the individual values of the attribute divided by the number of these values.
Individual values of a characteristic are called variants and are denoted by x(); the number of population units is denoted by n, the average value of the characteristic is denoted by 
. Therefore, the arithmetic simple mean is equal to:
According to the discrete distribution series data, it is clear that the same characteristic values (variants) are repeated several times. Thus, option x occurs 2 times in total, and option x 16 times, etc.
The number of identical values of a characteristic in the distribution series is called frequency or weight and is denoted by the symbol n.
Let's calculate the average salary of one worker 
in rub.:
The wage fund for each group of workers is equal to the product of options and frequency, and the sum of these products gives the total wage fund of all workers.
In accordance with this, the calculations can be presented in general form:
The resulting formula is called the weighted arithmetic mean.
As a result of processing, statistical material can be presented not only in the form of discrete distribution series, but also in the form of interval variation series with closed or open intervals.
The average for grouped data is calculated using the weighted arithmetic average formula:
In the practice of economic statistics, it is sometimes necessary to calculate the average using group averages or averages of individual parts of the population (partial averages). In such cases, group or private averages are taken as options (x), on the basis of which the overall average is calculated as an ordinary weighted arithmetic average.
Basic properties of the arithmetic mean .
The arithmetic mean has a number of properties:
1. The value of the arithmetic mean will not change from decreasing or increasing the frequency of each value of the characteristic x by n times.
If all frequencies are divided or multiplied by any number, the average value will not change.
2. The common multiplier of individual values of a characteristic can be taken beyond the sign of the average:
3. The average of the sum (difference) of two or more quantities is equal to the sum (difference) of their averages:
4. If x = c, where c is a constant value, then 
.
5. The sum of deviations of the values of attribute X from the arithmetic mean x is equal to zero:
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.
Characteristics of variation series, along with averages, are mode and median.
Fashion - this is the value of a characteristic (variant) that is most often repeated in the population under study. For discrete distribution series, the mode will be the value of the variant with the highest frequency.
For interval distribution series with equal intervals, the mode is determined by the formula:
Where 
- initial value of the interval containing the mode;
- the value of the modal interval;
- frequency of the modal interval;
- frequency of the interval preceding the modal one;
- frequency of the interval following the modal one.
Median - this is an option located in the middle of the variation series. If the distribution series is discrete and has an odd number of members, then the median will be the option located in the middle of the ordered series (an ordered series is the arrangement of population units in ascending or descending order).
In statistics, various types of averages are used, which are divided into two large classes:
Power means (harmonic mean, geometric mean, arithmetic mean, quadratic mean, cubic mean);
Structural means (mode, median).
To calculate power averages it is necessary to use all available characteristic values. Fashion And median are determined only by the structure of the distribution, therefore they are called structural, positional averages. Median and mode are often used as average characteristic in those populations where calculating the average power law is impossible or impractical.
The most common type of average is the arithmetic mean. Under arithmetic mean is understood as the value of a characteristic that each unit of the population would have if the total sum of all values of the characteristic were distributed evenly among all units of the population. The calculation of this value comes down to summing all the values of the varying characteristic and dividing the resulting amount by the total number of units in the population. For example, five workers fulfilled an order for the production of parts, while the first made 5 parts, the second – 7, the third – 4, the fourth – 10, the fifth – 12. Since in the source data the value of each option occurred only once, to determine
To determine the average output of one worker, one should apply the simple arithmetic average formula:
i.e. in our example, the average output of one worker is equal to
Along with the simple arithmetic mean, they study weighted arithmetic average. For example, let's calculate average age students in a group of 20 people, whose ages range from 18 to 22 years, where xi– variants of the characteristic being averaged, fi– frequency, which shows how many times it occurs i-th value in the aggregate (Table 5.1).
Table 5.1
Average age of students
Applying the weighted arithmetic mean formula, we get:
To select a weighted arithmetic mean, there is certain rule: if there is a series of data on two indicators, for one of which it is necessary to calculate
average value, and at the same time the numerical values of the denominator of its logical formula are known, and the values of the numerator are unknown, but can be found as the product of these indicators, then the average value should be calculated using the arithmetic weighted average formula.
In some cases, the nature of the initial statistical data is such that the calculation of the arithmetic average loses its meaning and the only generalizing indicator can only be another type of average value - harmonic mean. Currently, the computational properties of the arithmetic mean have lost their relevance in the calculation of general statistical indicators due to the widespread introduction of electronic computing technology. The harmonic mean value, which can also be simple and weighted, has acquired great practical importance. If the numerical values of the numerator of a logical formula are known, and the values of the denominator are unknown, but can be found as a partial division of one indicator by another, then the average value is calculated using the harmonic weighted average formula.
For example, let it be known that the car covered the first 210 km at a speed of 70 km/h, and the remaining 150 km at a speed of 75 km/h. It is impossible to determine the average speed of a car over the entire journey of 360 km using the arithmetic average formula. Since the options are speeds in individual sections xj= 70 km/h and X2= 75 km/h, and the weights (fi) are considered to be the corresponding sections of the path, then the products of the options and the weights will have neither physical nor economic meaning. In this case, the quotients acquire meaning from dividing the sections of the path into the corresponding speeds (options xi), i.e., the time spent on passing individual sections of the path (fi / xi). If the segments of the path are denoted by fi, then the entire path will be expressed as?fi, and the time spent on the entire path will be expressed as?fi. fi / xi , Then the average speed can be found as the quotient of the entire path divided by the total time spent:
In our example we get:
If, when using the harmonic mean, the weights of all options (f) are equal, then instead of the weighted one you can use simple (unweighted) harmonic mean:
where xi are individual options; n– number of variants of the characteristic being averaged. In the speed example, simple harmonic mean could be applied if the path segments traveled at different speeds were equal.
Any average value must be calculated so that when it replaces each variant of the averaged characteristic, the value of some final, general indicator that is associated with the averaged indicator does not change. Thus, when replacing actual speeds on individual sections of the route with their average value (average speed), the total distance should not change.
The form (formula) of the average value is determined by the nature (mechanism) of the relationship of this final indicator with the averaged one, therefore the final indicator, the value of which should not change when replacing options with their average value, is called defining indicator. To derive the formula for the average, you need to create and solve an equation using the relationship between the averaged indicator and the determining one. This equation is constructed by replacing the variants of the characteristic (indicator) being averaged with their average value.
In addition to the arithmetic mean and harmonic mean, other types (forms) of the mean are used in statistics. They are all special cases power average. If we calculate all types of power averages for the same data, then the values
they will turn out to be the same, the rule applies here majo-rate average. As the exponent of the average increases, the average value itself increases. The most frequently used calculation formulas in practical research various types power average values are presented in table. 5.2.
Table 5.2
Types of power means
The geometric mean is used when there is n growth coefficients, while individual values of the characteristic are, as a rule, relative values dynamics constructed in the form of chain values, as a ratio to the previous level of each level in a series of dynamics. The average thus characterizes the average growth rate. Average geometric simple calculated by the formula
Formula weighted geometric mean has the following form:
The above formulas are identical, but one is applied for current coefficients or growth rates, and the second is applied for absolute values of series levels.
Mean square used in calculations with the values of quadratic functions, used to measure the degree of fluctuation of individual values of a characteristic around the arithmetic mean in the distribution series and is calculated by the formula
Weighted mean square calculated using another formula:
Average cubic is used when calculating with values of cubic functions and is calculated by the formula
average cubic weighted:
All average values discussed above can be presented as a general formula:
where is the average value; – individual meaning; n– number of units of the population being studied; k– exponent that determines the type of average.
When using the same source data, the more k V general formula power average, the larger the average value. It follows from this that there is a natural relationship between the values of power averages:
The average values described above give a generalized idea of the population being studied, and from this point of view, their theoretical, applied and educational significance is indisputable. But it happens that the average value does not coincide with any of the actually existing options, therefore, in addition to the considered averages, statistical analysis It is advisable to use the values of specific options that occupy a well-defined position in the ordered (ranked) series of attribute values. Among these quantities, the most commonly used are structural, or descriptive, average– mode (Mo) and median (Me).
Fashion– the value of a characteristic that is most often found in a given population. In relation to a variational series, the mode is the most frequently occurring value of the ranked series, that is, the option with the highest frequency. Fashion can be used in determining the stores that are visited more often, the most common price for any product. It shows the size of a feature characteristic of a significant part of the population and is determined by the formula
where x0 is the lower limit of the interval; h– interval size; fm– interval frequency; fm_ 1 – frequency of the previous interval; fm+ 1 – frequency of the next interval.
Median the option located in the center of the ranked row is called. The median divides the series into two equal parts in such a way that there are the same number of population units on either side of it. In this case, one half of the units in the population has a value of the varying characteristic that is less than the median, while the other half has a value greater than it. The median is used when studying an element whose value is greater than or equal to, or at the same time less than or equal to, half of the elements of a distribution series. The median gives general idea about where the values of the attribute are concentrated, in other words, where their center is located.
The descriptive nature of the median is manifested in the fact that it characterizes the quantitative limit of the values of a varying characteristic that half of the units in the population possess. The problem of finding the median for a discrete variation series is easily solved. If all units of the series are given serial numbers, then the serial number of the median option is determined as (n + 1) / 2 with an odd number of members of n. If the number of members of the series is an even number, then the median will be the average value of two options that have serial numbers n/ 2 and n/ 2 + 1.
When determining the median in interval variation series, first determine the interval in which it is located (median interval). This interval is characterized by the fact that its accumulated sum of frequencies is equal to or exceeds half the sum of all frequencies of the series. The median of an interval variation series is calculated using the formula
Where X0– lower limit of the interval; h– interval size; fm– interval frequency; f– number of members of the series;
M -1 – the sum of the accumulated terms of the series preceding the given one.
Along with the median for more full characteristics the structures of the population under study also use other values of options that occupy a very specific position in the ranked series. These include quartiles And deciles. Quartiles divide the series according to the sum of frequencies into 4 equal parts, and deciles - into 10 equal parts. There are three quartiles and nine deciles.
The median and mode, unlike the arithmetic mean, do not eliminate individual differences in the values of a variable characteristic and therefore are additional and very important characteristics of the statistical population. In practice, they are often used instead of the average or along with it. It is especially advisable to calculate the median and mode in cases where the population under study contains a certain number of units with a very large or very small value of the varying characteristic. These values of the options, which are not very characteristic of the population, while influencing the value of the arithmetic mean, do not affect the values of the median and mode, which makes the latter very valuable indicators for economic and statistical analysis.
