In mathematics and statistics average arithmetic (or easily average) of a set of numbers is the sum of all the numbers in that set divided by their number. The arithmetic mean is a particularly general and most common representation of the average.

You will need

• Knowledge in mathematics.

Instruction

1. Let a set of four numbers be given. Need to discover average meaning this kit. To do this, we first find the sum of all these numbers. These numbers are possible 1, 3, 8, 7. Their sum is equal to 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 the 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 average geometric. The geometric mean of several regular real numbers is a number that is allowed to replace any of these numbers so that their product does not change. The geometric mean G is sought by the formula: the root of the Nth degree of the product of a set of numbers, where N is the number of 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, we calculate the product: 1 * 3 * 8 * 7 = 168. Now from the number 168 you need to extract the root of the 4th degree: G = (168) ^ 1/4 = 3.61. Thus average the geometric set of numbers is 3.61.

Average the geometric mean is less commonly used in the aggregate than the arithmetic mean, however, it can be useful in calculating the average value of indicators that change over time (the salary of an individual employee, the dynamics of academic performance, etc.).

You will need

• Engineering Calculator

Instruction

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 it is necessary to extract the root of the degree 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 with support for the CPGEOM function in Excel or using one of the online calculators that are deliberately prepared 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 2nd degree root (square root) from any number using the most ordinary calculator.

Helpful advice
In contrast to the arithmetic mean, the geometric mean is not so powerfully influenced by huge deviations and fluctuations between individual values ​​in the studied set of indicators.

Average value is one of the collations of a set of numbers. Represents a number that cannot be outside the range defined by the largest and smallest values ​​in this set of numbers. Average an arithmetic value is a particularly commonly used variety of averages.

Instruction

1. Add 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 any of the numbers by the number of values ​​​​of the set and sum up the total.

2. Use, say, the calculator included with Windows OS, if calculating the arithmetic mean in your head is not possible. It can be opened with the support of the program launch dialog. To do this, press the "burning keys" WIN + R or click the "Start" button and select the "Run" command from the main menu. After that, type in the input field calc and press Enter on the 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 in the set in steps 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. Entering numbers is also allowed both from the keyboard and by clicking the corresponding interface buttons.

4. Press the slash key or click this icon 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.

5. It is allowed to use the spreadsheet editor Microsoft Excel for the same purpose. In this case, start the editor and enter all the values ​​of the sequence of numbers into 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 the entered values ​​and in the lower left corner of the editor window (in the status bar) you will see the arithmetic mean for the selected cells.

7. Click the cell next to the last number you entered if you'd rather just see the arithmetic mean. Expand the drop-down list with the image of the Greek letter sigma (Σ) in the "Editing" group of commands on the "Basic" tab. Select the line " Average” and the editor will insert the necessary formula for calculating the arithmetic mean in 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. Finding the arithmetic mean for several values ​​​​is very easy, but every task has its own nuances, which you need to know in order to perform correct calculations.

What is the arithmetic mean

The arithmetic mean determines 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 mean is preferably used when compiling financial and statistical reports or for calculating the quantitative results of similar skills performed.

How to find the arithmetic mean

The search for the arithmetic mean for an array of numbers should begin with 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 184. When writing, the arithmetic mean is denoted by the letter? (mu) or x (x with a dash). Next, the algebraic sum should be divided by the number of numbers in the array. In this example, there were five numbers, so the arithmetic mean will be 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. There is a difference only when calculating in the programming environment, or if there is additional data in the task. In these cases, finding the arithmetic mean of numbers with different signs comes down to three steps: 1. Finding the general arithmetic mean in the standard way; 2. Finding the arithmetic mean of negative numbers.3. Calculation of the arithmetic mean of positive numbers. The results of any of the actions are written separated by commas.

Natural and decimal fractions

If the array of numbers is represented by decimal fractions, the solution occurs according to the method of calculating the arithmetic mean of integers, but the total is reduced 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 reduced 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, because they are found according to different methodologies. The geometric mean is invariably less than or equal to the arithmetic mean.

You will need

• Engineering calculator.

Instruction

1. Consider that in the general case the geometric mean of numbers is found by multiplying these numbers and extracting from them the root of the degree that corresponds to the number of numbers. Say, if you need to find the geometric mean of five numbers, then from the product it will be necessary to extract the root of the fifth degree.

2. To find the geometric mean of 2 numbers, use the basic rule. Find their product, then extract the square root from it, from the fact that the number is 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, extract the square root? 64 = 8. This will be the desired value. Please note that the arithmetic mean of these 2 numbers is larger and equals 10. If the root is not taken completely, round the total to the desired order.

3. In order to find the geometric mean of more than 2 numbers, also use the basic rule. To do this, find the product of all the numbers for which you need to find the geometric mean. From the resulting product, extract the root of the degree equal to the number of numbers. Let's say, in order to find the geometric mean of the numbers 2, 4 and 64, find their product. 2 4 64=512. From the fact that it is necessary to find the total of the geometric mean of 3 numbers, that extract the root of the third degree from the product. It is difficult to do this verbally, so use an engineering calculator. To do this, 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 root of the third degree. 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, it is possible to detect the geometric mean using a different method. Find the log button on the keyboard. After that, take the logarithm of all of the numbers, find their sum, and divide it by the number of numbers. From the resulting number, take the antilogarithm. 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, make 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, from the fact that this is the number of numbers by which the geometric mean is sought. From the total, take the antilogarithm by toggling 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.

Method of averages

3.1 Essence and meaning of averages in statistics. Types of averages

Average value in statistics, a generalized characteristic of qualitatively homogeneous phenomena and processes according to some varying attribute is called, which shows the level of the attribute, related to the unit of the population. average value abstract, because characterizes the value of the attribute for some impersonal unit of the population.Essence of average magnitude lies in the fact that the general and necessary, i.e., the tendency and regularity in the development of mass phenomena, are revealed through the individual and the accidental. Features that generalize in average values ​​are inherent in all units of the population. Due to this, the average value is of great importance for identifying patterns inherent in mass phenomena and not noticeable in individual units of the population.

General principles for the use of averages:

a reasonable choice of the population unit for which the average value is calculated is necessary;

when determining the average value, it is necessary to proceed from the qualitative content of the averaged trait, take into account the relationship of the studied traits, as well as the data available for calculation;

average values ​​should be calculated according to qualitatively homogeneous aggregates, which are obtained by the grouping method, which involves the calculation of a system of generalizing indicators;

overall averages should be supported by group averages.

Depending on the nature of the primary data, the scope and method of calculation in statistics, the following are distinguished: main types of averages:

1) power averages(arithmetic mean, harmonic, geometric, root mean square and cubic);

2) structural (non-parametric) averages(mode and median).

In statistics, the correct characterization of the population under study on the basis of varying characteristics in each individual case is given only by a well-defined type of average. The question of what type of average should be applied in a particular case is resolved by a specific analysis of the population under study, as well as based on the principle of meaningfulness of the results when summing up or when weighing. These and other principles are expressed in statistics the theory of averages.

For example, the arithmetic mean and the harmonic mean are used to characterize the mean value of a variable trait in the population being studied. The geometric mean is used only when calculating the average rate of dynamics, and the mean square only when calculating the variation indicators.

Formulas for calculating average values ​​are presented in Table 3.1.

Table 3.1 - Formulas for calculating average values

Types of averages	Calculation formulas
	simple	weighted
1. Arithmetic mean
2. Average harmonic
3. Geometric mean
4. Root Mean Square

Designations:- quantities for which the average is calculated; - average, where the line above indicates that the averaging of individual values ​​takes place; - frequency (repeatability of individual trait values).

Obviously, different averages are derived from the general formula for the power mean (3.1) :

, (3.1)

for k = + 1 - arithmetic mean; k = -1 - harmonic mean; k = 0 - geometric mean; k = +2 - root mean square.

Averages are either simple or weighted. weighted averages values ​​are called that take into account that some variants of the attribute values ​​may have different numbers; in this regard, each option has to be multiplied by this number. In this case, the “weights” are the numbers of population units in different groups, i.e. each option is "weighted" by its frequency. The frequency f is called statistical weight or weighing average.

Eventually correct choice of average assumes the following sequence:

a) the establishment of a generalizing indicator of the population;

b) determination of a mathematical ratio of values ​​for a given generalizing indicator;

c) replacement of individual values ​​by average values;

d) calculation of the average using the corresponding equation.

3.2 Arithmetic mean and its properties and calculation technique. Average harmonic

Arithmetic mean- the most common type of medium size; it is calculated in those cases when the volume of the averaged attribute is formed as the sum of its values ​​for individual units of the studied statistical population.

The most important properties of the arithmetic mean:

1. The product of the average and the sum of frequencies is always equal to the sum of the products of the variant (individual values) and frequencies.

2. If any arbitrary number is subtracted (added) from each option, then the new average will decrease (increase) by the same number.

3. If each option is multiplied (divided) by some arbitrary number, then the new average will increase (decrease) by the same amount

4. If all frequencies (weights) are divided or multiplied by any number, then the arithmetic mean will not change from this.

5. The sum of deviations of individual options from the arithmetic mean is always zero.

It is possible to subtract an arbitrary constant value from all values ​​of the attribute (better is the value of the middle option or options with the highest frequency), reduce the resulting differences by a common factor (preferably by the value of the interval), and express the frequencies in particulars (in percent) and multiply the calculated average by the common factor and add an arbitrary constant value. This method of calculating the arithmetic mean is called method of calculation from conditional zero .

Geometric mean finds its application in determining the average growth rate (average growth rates), when the individual values ​​of the trait are presented as relative values. It is also used if it is necessary to find the average between the minimum and maximum values ​​of a characteristic (for example, between 100 and 1000000).

root mean square used to measure the variation of a trait in the population (calculation of the standard deviation).

In statistics it works Majority rule for means:

X harm.< Х геом. < Х арифм. < Х квадр. < Х куб.

3.3 Structural means (mode and median)

To determine the structure of the population, special averages are used, which include the median and mode, or the so-called structural averages. If the arithmetic mean is calculated based on the use of all variants of the attribute values, then the median and mode characterize the value of the variant that occupies a certain average position in the ranked variation series

Fashion- the most typical, most often encountered value of the attribute. For discrete series the mode will be the one with the highest frequency. To define fashion interval series first determine the modal interval (interval having the highest frequency). Then, within this interval, the value of the feature is found, which can be a mode.

To find a specific value of the mode of the interval series, it is necessary to use the formula (3.2)

(3.2)

where X Mo is the lower limit of the modal interval; i Mo - the value of the modal interval; f Mo is the frequency of the modal interval; f Mo-1 - the frequency of the interval preceding the modal; f Mo+1 - the frequency of the interval following the modal.

Fashion is widely used in marketing activities in the study of consumer demand, especially in determining the sizes of clothes and shoes that are in greatest demand, while regulating pricing policy.

Median - the value of the variable attribute, falling in the middle of the ranged population. For ranked series with an odd number individual values ​​(for example, 1, 2, 3, 6, 7, 9, 10) the median will be the value that is located in the center of the series, i.e. the fourth value is 6. For ranked series with an even number individual values ​​(for example, 1, 5, 7, 10, 11, 14) the median will be the arithmetic mean value, which is calculated from two adjacent values. For our case, the median is (7+10)/2= 8.5.

Thus, to find the median, it is first necessary to determine its ordinal number (its position in the ranked series) using formulas (3.3):

(if there are no frequencies)

N Me=
(if there are frequencies) (3.3)

where n is the number of units in the population.

The numerical value of the median interval series determined by the accumulated frequencies in a discrete variational series. To do this, you must first specify the interval for finding the median in the interval series of the distribution. The median is the first interval where the sum of the accumulated frequencies exceeds half of the observations of the total number of all observations.

The numerical value of the median is usually determined by the formula (3.4)

(3.4)

where x Me - the lower limit of the median interval; iMe - the value of the interval; SMe -1 - the accumulated frequency of the interval that precedes the median; fMe is the frequency of the median interval.

Within the found interval, the median is also calculated using the formula Me = xl e, where the second factor on the right side of the equation shows the location of the median within the median interval, and x is the length of this interval. The median divides the variation series in half by frequency. Define more quartiles , which divide the variation series into 4 parts of equal size in probability, and deciles dividing the series into 10 equal parts.

The most common form of statistical indicators used in socio-economic research is the average value, which is a generalized quantitative characteristic of a sign of a statistical population. Average values ​​are, as it were, "representatives" of the entire series of observations. In many cases, the average can be determined through the initial ratio of the average (ISS) or its logical formula: . So, for example, to calculate the average wages of employees of an enterprise, it is necessary to divide the total wage fund by the number of employees: The numerator of the initial ratio of the average is its defining indicator. For the average wage, such a determining indicator is the wage fund. For each indicator used in the socio-economic analysis, only one true reference ratio can be compiled to calculate the average. It should also be added that in order to more accurately estimate the standard deviation for small samples (with the number of elements less than 30), the denominator of the expression under the root should not use n, A n- 1.

The concept and types of averages

Average value- this is a generalizing indicator of the statistical population, which extinguishes individual differences in the values ​​of statistical quantities, allowing you to compare different populations with each other. Exists 2 classes average values: power and structural. Structural averages are fashion And median , but the most commonly used power averages various types.

Power averages

Power averages can be simple And weighted.

A simple average is calculated when there are two or more ungrouped statistical values, arranged in an arbitrary order according to the following general formula of the average power law (for different values ​​of k (m)):

The weighted average is calculated from the grouped statistics using the following general formula:

Where x - the average value of the phenomenon under study; x i – i-th variant of the averaged feature ;

f i is the weight of the i-th option.

Where X are the values ​​of individual statistical values ​​or the midpoints of grouping intervals;
m - exponent, on the value of which the following types of power averages depend:
at m = -1 harmonic mean;
for m = 0, the geometric mean;
for m = 1, the arithmetic mean;
at m = 2, the root mean square;
at m = 3, the average cubic.

Using the general formulas for simple and weighted averages at different exponents m, we obtain particular formulas of each type, which will be discussed in detail below.

Arithmetic mean

Arithmetic mean - the initial moment of the first order, the mathematical expectation of the values ​​of a random variable with a large number of trials;

The arithmetic mean is the most commonly used average value, which is obtained by substituting m = 1 into the general formula. Arithmetic mean simple has the following form:

or

Where X are the values ​​of the quantities for which it is necessary to calculate the average value; N is the total number of X values ​​(the number of units in the studied population).

For example, a student passed 4 exams and received the following grades: 3, 4, 4 and 5. Let's calculate the average score using the simple arithmetic mean formula: (3+4+4+5)/4 = 16/4 = 4. Arithmetic mean weighted has the following form:

Where f is the number of values ​​with the same X value (frequency). >For example, a student passed 4 exams and received the following grades: 3, 4, 4 and 5. Calculate the average score using the arithmetic weighted average formula: (3*1 + 4*2 + 5*1)/4 = 16/4 = 4 . If the X values ​​are given as intervals, then the midpoints of the X intervals are used for calculations, which are defined as half the sum of the upper and lower boundaries of the interval. And if the interval X does not have a lower or upper limit (open interval), then to find it, the range (the difference between the upper and lower limits) of the adjacent interval X is used. For example, at the enterprise there are 10 employees with work experience up to 3 years, 20 - with work experience from 3 to 5 years, 5 employees - with work experience of more than 5 years. Then we calculate the average length of service of employees using the arithmetic weighted average formula, taking as X the middle of the length of service intervals (2, 4 and 6 years): (2*10+4*20+6*5)/(10+20+5) = 3.71 years.

AVERAGE function

This function calculates the average (arithmetic) of its arguments.

AVERAGE(number1, number2, ...)

Number1, number2, ... are 1 to 30 arguments for which the average is calculated.

Arguments must be numbers or names, arrays or references containing numbers. If the argument, which is an array or a link, contains texts, booleans, or empty cells, then those values ​​are ignored; however, cells that contain null values ​​are counted.

AVERAGE function

Calculates the arithmetic mean of the values ​​given in the argument list. In addition to numbers, text and logical values, such as TRUE and FALSE, can participate in the calculation.

AVERAGE(value1, value2,...)

Value1, value2,... are 1 to 30 cells, cell ranges, or values ​​for which the average is calculated.

Arguments must be numbers, names, arrays, or references. Arrays and links containing text are interpreted as 0 (zero). Empty text ("") is interpreted as 0 (zero). Arguments containing the value TRUE are interpreted as 1, Arguments containing the value FALSE are interpreted as 0 (zero).

The arithmetic mean is used most often, but there are times when other types of averages are needed. Let's consider such cases further.

Average harmonic

Harmonic mean for determining the average sum of reciprocals;

Average harmonic is used when the original data do not contain frequencies f for individual values ​​of X, but are presented as their product Xf. Denoting Xf=w, we express f=w/X, and substituting these designations into the weighted arithmetic mean formula, we obtain the weighted harmonic mean formula:

Thus, the harmonic weighted average is used when the frequencies f are unknown, but w=Xf is known. In those cases where all w=1, that is, the individual values ​​of X occur 1 time, the harmonic simple mean formula is applied: or For example, a car was traveling from point A to point B at a speed of 90 km/h and back at a speed of 110 km/h. To determine the average speed, we apply the harmonic simple formula, since the example gives the distance w 1 \u003d w 2 (the distance from point A to point B is the same as from B to A), which is equal to the product of speed (X) and time ( f). Average speed = (1+1)/(1/90+1/110) = 99 km/h.

SRHARM function

Returns the harmonic mean of the data set. The harmonic mean is the reciprocal of the arithmetic mean of reciprocals.

SGARM(number1, number2, ...)

Number1, number2, ... are 1 to 30 arguments for which the average is calculated. You can use an array or an array reference instead of semicolon-separated arguments.

The harmonic mean is always less than the geometric mean, which is always less than the arithmetic mean.

Geometric mean

Geometric mean for estimating the average growth rate of random variables, finding the value of a feature equidistant from the minimum and maximum values;

Geometric mean used in determining average relative changes. The geometric mean value gives the most accurate averaging result if the task is to find such a value of X, which would be equidistant from both the maximum and minimum values ​​of X. For example, between 2005 and 2008inflation index in Russia was: in 2005 - 1.109; in 2006 - 1,090; in 2007 - 1,119; in 2008 - 1,133. Since the inflation index is a relative change (dynamic index), then you need to calculate the average value using the geometric mean: (1.109 * 1.090 * 1.119 * 1.133) ^ (1/4) = 1.1126, that is, for the period from 2005 to 2008 annually prices grew by an average of 11.26%. An erroneous calculation on the arithmetic mean would give an incorrect result of 11.28%.

SRGEOM function

Returns the geometric mean of an array or range of positive numbers. For example, the CAGEOM function can be used to calculate the average growth rate if compound income with variable rates is given.

SRGEOM(number1; number2; ...)

Number1, number2, ... are 1 to 30 arguments for which the geometric mean is calculated. You can use an array or an array reference instead of semicolon-separated arguments.

root mean square

The root mean square is the initial moment of the second order.

root mean square is used when the initial values ​​of X can be both positive and negative, for example, when calculating average deviations. The main use of the quadratic mean is to measure the variation in X values.

Average cubic

The average cubic is the initial moment of the third order.

Average cubic is used extremely rarely, for example, when calculating poverty indices for developing countries (HPI-1) and for developed countries (HPI-2), proposed and calculated by the UN.

In the process of studying mathematics, students get acquainted with the concept of the arithmetic mean. In the future, in statistics and some other sciences, students are faced with the calculation of others. What can they be and how do they differ from each other?

meaning and difference

Not always accurate indicators give an understanding of the situation. In order to assess this or that situation, it is sometimes necessary to analyze a huge number of figures. And then averages come to the rescue. They allow you to assess the situation in general.

Since school days, many adults remember the existence of the arithmetic mean. It is very easy to calculate - the sum of a sequence of n terms is divisible by n. That is, if you need to calculate the arithmetic mean in the sequence of values ​​27, 22, 34 and 37, then you need to solve the expression (27 + 22 + 34 + 37) / 4, since 4 values ​​\u200b\u200bare used in the calculations. In this case, the desired value will be equal to 30.

Often, as part of the school course, the geometric mean is also studied. The calculation of this value is based on extracting the root of the nth degree from the product of n terms. If we take the same numbers: 27, 22, 34 and 37, then the result of the calculations will be 29.4.

The harmonic mean in a general education school is usually not the subject of study. However, it is used quite often. This value is the reciprocal of the arithmetic mean and is calculated as a quotient of n - the number of values ​​and the sum 1/a 1 +1/a 2 +...+1/a n . If we again take the same for calculation, then the harmonic will be 29.6.

Weighted Average: Features

However, all of the above values ​​may not be used everywhere. For example, in statistics, when calculating some, the "weight" of each number used in calculations plays an important role. The results are more revealing and correct because they take into account more information. This group of values ​​is collectively referred to as the "weighted average". They are not passed at school, so it is worth dwelling on them in more detail.

First of all, it is worth explaining what is meant by the "weight" of a particular value. The easiest way to explain this is with a specific example. The body temperature of each patient is measured twice a day in the hospital. Of the 100 patients in different departments of the hospital, 44 will have a normal temperature - 36.6 degrees. Another 30 will have an increased value - 37.2, 14 - 38, 7 - 38.5, 3 - 39, and the remaining two - 40. And if we take the arithmetic mean, then this value in general for the hospital will be over 38 degrees! But almost half of the patients have absolutely And here it would be more correct to use the weighted average, and the "weight" of each value will be the number of people. In this case, the result of the calculation will be 37.25 degrees. The difference is obvious.

In the case of weighted average calculations, the "weight" can be taken as the number of shipments, the number of people working on a given day, in general, anything that can be measured and affect the final result.

Varieties

The weighted average corresponds to the arithmetic average discussed at the beginning of the article. However, the first value, as already mentioned, also takes into account the weight of each number used in the calculations. In addition, there are also weighted geometric and harmonic values.

There is another interesting variety used in series of numbers. This is a weighted moving average. It is on its basis that trends are calculated. In addition to the values ​​themselves and their weight, periodicity is also used there. And when calculating the average value at some point in time, values ​​​​for previous time periods are also taken into account.

Calculating all these values ​​is not that difficult, but in practice, only the usual weighted average is usually used.

Calculation methods

In the age of computerization, there is no need to manually calculate the weighted average. However, it would be useful to know the calculation formula so that you can check and, if necessary, correct the results obtained.

It will be easiest to consider the calculation on a specific example.

It is necessary to find out what is the average wage at this enterprise, taking into account the number of workers receiving a particular salary.

So, the calculation of the weighted average is carried out using the following formula:

x = (a 1 *w 1 +a 2 *w 2 +...+a n *w n)/(w 1 +w 2 +...+w n)

For example, the calculation would be:

x = (32*20+33*35+34*14+40*6)/(20+35+14+6) = (640+1155+476+240)/75 = 33.48

Obviously, there is no particular difficulty in manually calculating the weighted average. The formula for calculating this value in one of the most popular applications with formulas - Excel - looks like the SUMPRODUCT (series of numbers; series of weights) / SUM (series of weights) function.

The average value is a generalizing indicator that characterizes the typical level of the phenomenon. It expresses the value of the attribute, related to the unit of the population.

The average value is:

1) the most typical value of the attribute for the population;

2) the volume of the sign of the population, 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 the trait, i.e. in averages, individual differences in the units of the population due to random circumstances are canceled out. In contrast to the average, the absolute value that characterizes the level of a feature of an individual unit of the population does not allow comparing the values ​​of the feature for 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 wages of the workers selected for comparison may not be typical for these enterprises. If we compare the size of wage funds at the enterprises under consideration, then 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 averages can be compared, i.e. How much does one worker earn on average in each company? Thus, there is a need to calculate the average value as a generalizing characteristic of the population.

It is important to note that in the process of averaging, the aggregate value of the attribute levels or its final value (in the case of calculating average levels in a time series) must remain unchanged. In other words, when calculating the average value, the volume of the trait under study should not be distorted, and the expressions made when calculating the average must necessarily make sense.

Calculating the average is one common generalization technique; the average indicator denies the general that is typical (typical) for all units of the studied population, at the same time it ignores the differences between individual units. In every phenomenon and its development there is a combination of chance and necessity. When calculating averages, due to the operation of the law of large numbers, randomness cancels each other out, balances out, so it is possible to abstract from the insignificant features of the phenomenon, from the quantitative values ​​of the attribute in each specific case. In the ability to abstract from the randomness of individual values, fluctuations lies the scientific value of averages as generalizing characteristics of aggregates.

In order for the average to be truly typifying, it must be calculated taking into account certain principles.

Let us dwell on some general principles for the application of averages.

1. The average should be determined for populations consisting of qualitatively homogeneous units.

2. The average should be calculated for a population consisting of a sufficiently large number of units.

3. The average should be calculated for the population, the units of which 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 averages, the features of their calculation and areas of application. Average values ​​are divided into two large classes: power averages, structural averages.

Power-law averages include the most well-known and commonly used types, such as geometric mean, arithmetic mean, and mean square.

The mode and median are considered as structural averages.

Let us dwell on power averages. Power averages, depending on the presentation of the initial data, can be simple and weighted. simple average is calculated from ungrouped data and has the following general form:

,

where X i is the variant (value) of the averaged feature;

n is the number of options.

Weighted average is calculated by grouped data and has a general form

,

where X i is the variant (value) of the averaged feature or the middle value of the interval in which the variant is measured;

m is the exponent of the mean;

f i - frequency showing how many times the i-e value of the averaged feature occurs.

If we calculate all types of averages for the same initial data, then their values ​​will not be the same. Here the rule of majorance of averages applies: with an increase in the exponent m, the corresponding average value also increases:

In statistical practice, more often than other types of weighted averages, arithmetic and harmonic weighted averages are used.

Types of Power Means

Type of power middle	Index degrees (m)	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 the weights are not the units of the population - the carriers of the trait, but the products of these units and the values ​​of the trait (i.e. m = Xf). The average harmonic downtime should be used in cases of determining, for example, the average costs of labor, time, materials per unit of production, per 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 breaking the connection between individual and summary indicators. In other words, the average value should 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 result is called determining since the nature of its relationship with individual values ​​determines the specific formula for calculating the average value. Let's show this rule on the example of the geometric mean.

Geometric mean formula

most often used when calculating the average value of individual relative values ​​of the dynamics.

The geometric mean is used if a sequence of chain relative values ​​of dynamics is given, indicating, for example, an increase in production 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 growth over the years:

q n =q 0 × i 1 × i 2 ×…×i n .

Taking q n as a defining 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 series of distribution 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 there were no data in the considered example). and on the volume of production, and on the amount of costs by groups of enterprises).

Indicators are most often used as structural averages. fashion - the most frequently repeated feature value - and median - the value of a feature that divides the ordered sequence of its values ​​into two parts equal in number. As a result, in one half of the population units, the value of the attribute does not exceed the median level, and in the other half it is not less than it.

If the feature under study has discrete values, then there are no particular difficulties in calculating the mode and median. If the data on the values ​​of the 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 parts equal in number, it ends up in one of the intervals of the feature X. Using interpolation, the median value is found in this median interval:

,

where X Me is the lower limit of the median interval;

h Me is its value;

(Sum m) / 2 - half of the total number of observations or half of 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 is the sum of observations (or the volume of the weighting feature) accumulated before the beginning of the median interval;

m Me is the number of observations or the volume of the weighting feature in the median interval (also in absolute or relative terms).

When calculating the modal value of a feature according to the data of the interval series, it is necessary to pay attention to the fact that the intervals are the same, since the indicator of the frequency of feature values ​​X depends on this. For an interval series with equal intervals, the mode value is determined as

,

where X Mo is the lower value of the modal interval;

m Mo is the number of observations or the volume of the weighting feature in the modal interval (in absolute or relative terms);

m Mo-1 - the same for the interval preceding the modal;

m Mo+1 - the same for the interval following the modal;

h is the value of the interval of change of the trait in groups.

TASK 1

The following data are available for the group of industrial enterprises for the reporting year

№ enterprises	Production volume, million rubles		Average number of employees, pers.	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 perform a grouping of enterprises for the exchange of products, taking the following intervals:

up to 200 million rubles


from 200 to 400 million rubles


1. from 400 to 600 million rubles
   

   For each group and for all together, determine the number of enterprises, the volume of production, the average number of employees, the average output per employee. The grouping results should be presented in the form of a statistical table. Formulate a conclusion.
   

   SOLUTION
   

   Let's make a grouping of enterprises for the exchange of products, the calculation of the number of enterprises, the volume of production, the average number of employees according to the formula of a simple average. The results of grouping and calculations are summarized in a table.
   

   Groups by production volume	№ enterprises	Production volume, million rubles	Average annual cost of fixed assets, million rubles	average sleep juicy number of employees, pers.	Profit, thousand rubles	Average output per worker
   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
   2 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
   Aggregate average		379,6	59,9	1223,6	79,5

   Conclusion. Thus, in the aggregate under consideration, the largest number of enterprises in terms of output fell into the third group - seven, or half of the enterprises. The value of the average annual value of fixed assets is also in this group, as well as the large value of the average number of employees - 9974 people, the enterprises of the first group are the least profitable.
   

   TASK 2
   

   We have the following data on the enterprises of the company
   

   Number of the enterprise belonging to the company	I quarter		II quarter
   	Output, thousand rubles		Worked by working man-days	Average output per worker per day, rub.
   	59390,13