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.

Helpful advice

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 fall outside the range defined by the largest and smallest values ​​in that set of numbers. Average arithmetic value- the most commonly used type of medium.

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 the Run command 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 the Microsoft Excel spreadsheet editor for the same purpose. In this case, launch the editor and enter all the values ​​of the sequence of numbers into the adjacent cells. If, after entering 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 desired formula for calculating the arithmetic mean into the selected cell. Press the Enter key and the value will be calculated.

The arithmetic mean is one of the measures of central 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

Please note that in general case The geometric mean of numbers is found by multiplying these numbers and taking from them the root of the power that corresponds to the number of numbers. For example, if you need to find the geometric mean of five numbers, then you will need to extract the 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 whole root is not extracted, round the result to the required 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 power of 1/3, which corresponds to the third root. Get 512^1/3=8. This is the geometric mean of the numbers 2.4 and 64.

By using 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 a number equal to the sum of the decimal logarithms of the numbers 2, 4 and 64. Divide the resulting number by 3, since this is the number of numbers 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.

The average value is a general indicator characterizing the typical level of a phenomenon. It expresses the value of a characteristic per unit of the population.

The average value is:

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

2) the volume of the population attribute, distributed equally among the units of the population.

The characteristic for which it is calculated average value, in statistics is called “averaged”.

The average always generalizes the quantitative variation of a trait, i.e. in average values, individual differences between units in the population due to random circumstances are eliminated. Unlike the average, the absolute value characterizing the level of a characteristic of an individual unit of a population does not allow one to compare the values ​​of a characteristic among units belonging to different populations. So, if you need to compare the levels of remuneration of workers at two enterprises, then you cannot compare this characteristic two workers from different companies. The compensation of workers selected for comparison may not be typical for these enterprises. If we compare the size of wage funds at the enterprises under consideration, the number of employees is not taken into account and, therefore, it is impossible to determine where the level of wages is higher. Ultimately, only average indicators can be compared, i.e. How much does one employee earn on average at each enterprise? Thus, there is a need to calculate the average value as a generalizing characteristic of the population.

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

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

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

Let's look at some general principles application of average values.

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

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

3. The average must be calculated for a population whose units are in a normal, natural state.

4. The average should be calculated taking into account the economic content of the indicator under study.

5.2. Types of averages and methods for calculating them

Let us now consider the types of average values, features of their calculation and areas of application. Average values ​​are divided into two large classes: power averages, structural averages.

Power means include the most well-known and frequently used types, such as geometric mean, arithmetic mean and square mean.

The mode and median are considered as structural averages.

Let's focus on power averages. Power averages, depending on the presentation of the source data, can be simple or weighted. Simple average It is calculated based on ungrouped data and has the following general form:

,

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

n – number option.

Weighted average is calculated based on grouped data and has a general appearance

,

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

m – average degree index;

f i – frequency showing how many times it occurs i-e value averaging characteristic.

If you calculate all types of averages for the same initial data, then their values ​​will turn out to be different. The rule of majority of averages applies here: as the exponent m increases, the corresponding average value also increases:

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

Types of power means

Kind of power average	Index degree (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 not the units of the population - the carriers of the characteristic - are used as weights, but the product of these units by the values ​​of the characteristic (i.e. m = Xf). The average harmonic simple should be resorted to in cases of determining, for example, the average cost of labor, time, materials per unit of production, per one part for two (three, four, etc.) enterprises, workers engaged in the manufacture of the same type of product , the same part, product.

The main requirement for the formula for calculating the average value is that all stages of the calculation have a real meaningful justification; the resulting average value should replace the individual values ​​of the attribute for each object without disrupting the connection between the individual and summary indicators. In other words, the average value must be calculated so that when each individual value of the averaged indicator is replaced by its average value, some final summary indicator remains unchanged, related topic or in another way with the one being averaged. This total is called defining since the nature of its relationship with individual values ​​determines the specific formula for calculating the average value. Let us demonstrate this rule using the example of the geometric mean.

Geometric mean formula

used most often when calculating the average value based on individual relative dynamics.

The geometric mean is used if a sequence of chain relative dynamics is given, indicating, for example, an increase in production compared to the level of the previous year: i 1, i 2, i 3,…, i n. Obviously, the volume of production in the last year is determined by its initial level (q 0) and subsequent increase over the years:

q n =q 0 × i 1 × i 2 ×…×i n .

Taking q n as the determining indicator and replacing the individual values ​​of the dynamics indicators with average ones, we arrive at the relation

From here

A special type of averages - structural averages - is used to study internal structure series of distribution of attribute values, as well as for estimating the average value (power type), if its calculation cannot be carried out according to the available statistical data (for example, if in the example considered there were no data on both the volume of production and the amount of costs for groups of enterprises) .

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

If the characteristic being studied has discrete values, then there are no particular difficulties in calculating the mode and median. If data on the values ​​of attribute X are presented in the form of ordered intervals of its change (interval series), the calculation of the mode and median becomes somewhat more complicated. Since the median value divides the entire population into two equal parts, it ends up in one of the intervals of characteristic X. Using interpolation, the value of the median is found in this median interval:

,

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

h Me – its value;

(Sum m)/2 – half of the total number of observations or half the volume of the indicator that is used as a weighting in the formulas for calculating the average value (in absolute or relative terms);

S Me-1 – the sum of observations (or the volume of the weighting attribute) accumulated before the beginning of the median interval;

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

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

,

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

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

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

m Mo+1 – the same for the interval following the modal one;

h – the value of the interval of change of the characteristic in groups.

TASK 1

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

№ enterprises	Product volume, million rubles.		Average number of employees, people.	Profit, thousand rubles
	197,7	10,0		13,5
		22,8	1500	136,2
	465,5	18,4	1412	97,6
	296,2	12,6	1200	44,4
	584,1	22,0	1485	146,0
	480,0	119,0	1420	110,4
	57805	21,6	1390	138,7
	204,7			30,6
	466,8	19,4	1375	111,8
	292,2	113,6	1200	49,6
	423,1	17,6	1365	105,8
	192,6			30,7
	360,5	14,0	1290	64,8
	280,3	10,2		33,3

It is required to group enterprises for the exchange of products, taking the following intervals:

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, volume of production, average number of employees, average output per employee. Present the grouping results in the form of a statistical table. Formulate a conclusion.
   

   SOLUTION
   

   We will group enterprises by product exchange, calculate the number of enterprises, volume of production, and the average number of employees using the simple average formula. The results of grouping and calculations are summarized in a table.
   

   Groups by product volume	№ enterprises	Product volume, million rubles.	Average annual cost of fixed assets, million rubles.	Medium sleep juicy number of employees, people.	Profit, thousand rubles	Average output per employee
   1 group up to 200 million rubles	1,8,12	197,7 204,7 192,6	10,0 9,4 8,8	900 817	13,5 30,6 30,7
   			28,2	2567	74,8	0,23
   Average level		198,3			24,9
   2nd group from 200 to 400 million rubles.	4,10,13,14	196,2 292,2 360,5 280,3	12,6 113,6 14,0 10,2	1200 1200 1290	44,4 49,6 64,8 33,3
   		1129,2	150,4	4590	192,1	0,25
   Average level		282,3	37,6	1530	64,0
   3 group from 400 to 600 million	2,3,5,6,7,9,11	592 465,5 584,1 480,0 578,5 466,8 423,1	22,8 18,4 22,0 119,0 21,6 19,4 17,6	1500 1412 1485 1420 1390 1375 1365	136,2 97,6 146,0 110,4 138,7 111,8 105,8
   		3590	240,8	9974	846,5	0,36
   Average level		512,9	34,4	1421	120,9
   Total in aggregate		5314,2	419,4	17131	1113,4	0,31
   On average		379,6	59,9	1223,6	79,5

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

   TASK 2
   

   The following data is available on the company's enterprises
   

   Number of the enterprise included in the company	I quarter		II quarter
   	Product output, thousand rubles.		Man-days worked by workers	Average output per worker per day, rub.
   	59390,13

This term has other meanings, see average meaning.

Average(in mathematics and statistics) sets of numbers - the sum of all numbers divided by their number. It is one of the most common measures of central tendency.

It was proposed (along with the geometric mean and harmonic mean) by the Pythagoreans.

Special cases of the arithmetic mean are the mean (general population) and the sample mean (sample).

Introduction

Let us denote the set of data X = (x 1 , x 2 , …, x n), then the sample mean is usually indicated by a horizontal bar over the variable (x ¯ (\displaystyle (\bar (x))), pronounced " x with a line").

The Greek letter μ is used to denote the arithmetic mean of the entire population. For a random variable for which the mean value is determined, μ is probabilistic average or the mathematical expectation of a random variable. If the set X is a collection of random numbers with a probabilistic mean μ, then for any sample x i from this set μ = E( x i) is the mathematical expectation of this sample.

In practice, the difference between μ and x ¯ (\displaystyle (\bar (x))) is that μ is a typical variable because you can see a sample rather than the entire population. Therefore, if the sample is represented randomly (in terms of probability theory), then x ¯ (\displaystyle (\bar (x))) (but not μ) can be treated as a random variable having a probability distribution on the sample (the probability distribution of the mean).

Both of these quantities are calculated in the same way:

X ¯ = 1 n ∑ i = 1 n x i = 1 n (x 1 + ⋯ + x n) . (\displaystyle (\bar (x))=(\frac (1)(n))\sum _(i=1)^(n)x_(i)=(\frac (1)(n))(x_ (1)+\cdots +x_(n)).)

If X is a random variable, then the mathematical expectation X can be considered as the arithmetic mean of values ​​in repeated measurements of a quantity X. This is a manifestation of the law of large numbers. Therefore, the sample mean is used to estimate the unknown expected value.

It has been proven in elementary algebra that the mean n+ 1 numbers above average n numbers if and only if the new number is greater than the old average, less if and only if the new number is less than the average, and does not change if and only if the new number is equal to the average. The more n, the smaller the difference between the new and old averages.

Note that there are several other “averages,” including the power mean, the Kolmogorov mean, the harmonic mean, the arithmetic-geometric mean, and various weighted averages (e.g., weighted arithmetic mean, weighted geometric mean, weighted harmonic mean).

Examples

• For three numbers, you need to add them and divide by 3:

x 1 + x 2 + x 3 3 . (\displaystyle (\frac (x_(1)+x_(2)+x_(3))(3)).)

• For four numbers, you need to add them and divide by 4:

x 1 + x 2 + x 3 + x 4 4 . (\displaystyle (\frac (x_(1)+x_(2)+x_(3)+x_(4))(4)).)

Or simpler 5+5=10, 10:2. Because we were adding 2 numbers, which means how many numbers we add, we divide by that many.

Continuous random variable

For a continuously distributed quantity f (x) (\displaystyle f(x)), the arithmetic mean on the interval [ a ; b ] (\displaystyle ) is determined through a definite integral:

F (x) ¯ [ a ; b ] = 1 b − a ∫ a b f (x) d x (\displaystyle (\overline (f(x)))_()=(\frac (1)(b-a))\int _(a)^(b) f(x)dx)

Some problems of using the average

Lack of robustness

Main article: Robustness in statistics

Although arithmetic means are often used as averages or central tendencies, this concept is not a robust statistic, meaning that the arithmetic mean is heavily influenced by "large deviations." It is noteworthy that for distributions with a large coefficient of skewness, the arithmetic mean may not correspond to the concept of “mean”, and the values ​​of the mean from robust statistics (for example, the median) may better describe the central tendency.

A classic example is calculating average income. The arithmetic mean can be misinterpreted as a median, which may lead to the conclusion that there are more people with higher incomes than there actually are. “Average” income is interpreted to mean that most people have incomes around this number. This “average” (in the sense of the arithmetic mean) income is higher than the incomes of most people, since a high income with a large deviation from the average makes the arithmetic mean highly skewed (in contrast, the average income at the median “resists” such skew). However, this "average" income says nothing about the number of people near the median income (and says nothing about the number of people near the modal income). However, if you take the concepts of “average” and “most people” lightly, you can draw the incorrect conclusion that most people have incomes higher than they actually are. For example, a report of the "average" net income in Medina, Washington, calculated as the arithmetic average of all annual net incomes of residents, would produce a surprisingly large number due to Bill Gates. Consider the sample (1, 2, 2, 2, 3, 9). The arithmetic mean is 3.17, but five out of six values ​​are below this mean.

Compound interest

Main article: Return on Investment

If the numbers multiply, but not fold, you need to use the geometric mean, not the arithmetic mean. Most often this incident occurs when calculating the return on investment in finance.

For example, if a stock fell 10% in the first year and rose 30% in the second year, then it is incorrect to calculate the “average” increase over those two years as the arithmetic mean (−10% + 30%) / 2 = 10%; the correct average in this case is given by the compound annual growth rate, which gives an annual growth rate of only about 8.16653826392% ≈ 8.2%.

The reason for this is that percentages have a new starting point each time: 30% is 30% from a number less than the price at the beginning of the first year: if the stock started out at $30 and fell 10%, it is worth $27 at the start of the second year. If the stock rose 30%, it would be worth $35.1 at the end of the second year. The arithmetic average of this growth is 10%, but since the stock has only risen by $5.1 over 2 years, the average growth of 8.2% gives a final result of $35.1:

[$30 (1 - 0.1) (1 + 0.3) = $30 (1 + 0.082) (1 + 0.082) = $35.1]. If we use the arithmetic average of 10% in the same way, we will not get the actual value: [$30 (1 + 0.1) (1 + 0.1) = $36.3].

Compound interest at the end of 2 years: 90% * 130% = 117%, that is, the total increase is 17%, and the average annual compound interest is 117% ≈ 108.2% (\displaystyle (\sqrt (117\%))\approx 108.2\%) , that is, an average annual increase of 8.2%.

Directions

Main article: Destination statistics

When calculating the arithmetic mean of some variable that changes cyclically (such as phase or angle), special care must be taken. For example, the average of 1° and 359° would be 1 ∘ + 359 ∘ 2 = (\displaystyle (\frac (1^(\circ )+359^(\circ ))(2))=) 180°. This number is incorrect for two reasons.

• First, angular measures are defined only for the range from 0° to 360° (or from 0 to 2π when measured in radians). So the same pair of numbers could be written as (1° and −1°) or as (1° and 719°). The average values ​​of each pair will be different: 1 ∘ + (− 1 ∘) 2 = 0 ∘ (\displaystyle (\frac (1^(\circ )+(-1^(\circ )))(2))=0 ^(\circ )) , 1 ∘ + 719 ∘ 2 = 360 ∘ (\displaystyle (\frac (1^(\circ )+719^(\circ ))(2))=360^(\circ )) .
• Second, in this case, a value of 0° (equivalent to 360°) will be a geometrically better average value, since the numbers deviate less from 0° than from any other value (the value 0° has the smallest variance). Compare:
  • the number 1° deviates from 0° by only 1°;
  • the number 1° deviates from the calculated average of 180° by 179°.

The average value for a cyclic variable calculated using the above formula will be artificially shifted relative to the real average towards the middle of the numerical range. Because of this, the average is calculated in a different way, namely, the number with the smallest variance (the center point) is selected as the average value. Also, instead of subtraction, the modular distance (that is, the circumferential distance) is used. For example, the modular distance between 1° and 359° is 2°, not 358° (on the circle between 359° and 360°==0° - one degree, between 0° and 1° - also 1°, in total - 2 °).

4.3. Average values. The essence and meaning of average values

Average size in statistics is a general indicator that characterizes the typical level of a phenomenon in specific conditions of place and time, reflecting the value of a varying characteristic per unit of a qualitatively homogeneous population. In economic practice, a wide range of indicators are used, calculated as average values.

For example, a general indicator of the income of workers of a joint-stock company (JSC) is the average income of one worker, determined by the ratio of the wage fund and payments social nature for the period under review (year, quarter, month) to the number of JSC workers.

Calculating the average is one of the common generalization techniques; the average indicator reflects what is common (typical) for all units of the population being studied, while at the same time it ignores the differences of individual units. In every phenomenon and its development there is a combination accidents And necessary. When calculating averages, due to the law of large numbers, the randomness cancels out and balances out, so it is possible to abstract from the unimportant features of the phenomenon, from the quantitative values ​​of the attribute in each specific case. The ability to abstract from the randomness of individual values, fluctuations lies in the scientific value of averages as generalizing characteristics of populations.

Where the need for generalization arises, the calculation of such characteristics leads to the replacement of many different individual values ​​of the attribute average an indicator that characterizes the entire set of phenomena, which makes it possible to identify patterns inherent in mass social phenomena that are invisible in individual phenomena.

The average reflects the characteristic, typical, real level of the phenomena being studied, characterizes these levels and their changes in time and space.

The average is a summary characteristic of the laws of the process in the conditions in which it occurs.

4.4. 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. In each specific case, one of the average values ​​is used: arithmetic, garmonic, geometric, quadratic, cubic etc. The listed averages belong to the class sedate average.

In addition to power averages, structural averages are used in statistical practice, which are considered to be mode and median.

Let us dwell in more detail on power averages.

Arithmetic mean

The most common type of average is average arithmetic. It is used in cases where the volume of a varying characteristic for the entire population is the sum of the characteristic values ​​of its individual units. For social phenomena the additivity (totality) of the volumes of a varying characteristic is characteristic, this determines the scope of application of the arithmetic average and explains its prevalence as a general indicator, for example: the total wage fund is the sum of wages of all workers, the gross harvest is the sum of products produced from the entire sown area.

To calculate the arithmetic mean, you need to divide the sum of all feature values ​​by their number.

The arithmetic mean is used in the form simple average and weighted average. The initial, defining form is the simple average.

Simple arithmetic mean equal to the simple sum of the individual values ​​of the characteristic being averaged, divided by total number these values ​​(it is used in cases where there are ungrouped individual characteristic values):

Where
- individual values ​​of the variable (variants); m - the number of units in the population.

Further, the summation limits will not be indicated in the formulas. For example, you need to find the average output of one worker (mechanic) if you know how many parts each of 15 workers produced, i.e. a number of individual values ​​of the characteristic are given, pcs.:

21; 20; 20; 19; 21; 19; 18; 22; 19; 20; 21; 20; 18; 19; 20.

The simple arithmetic mean is calculated using formula (4.1), 1 pc.:

The average of the options that are repeated different number times, or as they say, have different weights, is called weighted. The weights are the number of units in different groups aggregates (identical options are combined into a group).

Arithmetic average weighted- average of grouped values, - is calculated using the formula:

, (4.2)

Where
- weight (frequency of repetition of identical signs);

- the sum of the products of the magnitude of features and their frequencies;

- the total number of population units.

We illustrate the technique of calculating the arithmetic weighted average using the example discussed above. To do this, we will group the source data and place them in a table. 4.1.

Table 4.1

Distribution of workers for production of parts

According to formula (4.2), the weighted arithmetic mean is equal to, pcs.:

In some cases, weights may be presented not as absolute values, but as relative ones (in percentages or fractions of a unit). Then the formula for the arithmetic weighted average will look like:

Where
- particularity, i.e. the share of each frequency in the total sum of all

If frequencies are counted in fractions (coefficients), then
= 1, and the formula for the arithmetically weighted average has the form:

Calculation of the weighted arithmetic mean from group means carried out according to the formula:

,

Where f-number of units in each group.

The results of calculating the arithmetic mean from group means are presented in table. 4.2.

Table 4.2

Distribution of workers by average length of service

In this example, the options are not individual data on the length of service of individual workers, but the average for each workshop. Libra f are the number of workers in the shops. Hence, the average work experience of workers throughout the enterprise will be, years:

.

Calculation of the arithmetic mean in distribution series

If the values ​​of the characteristic being averaged are specified in the form of intervals (“from - to”), i.e. interval series of the distribution, then when calculating the arithmetic mean, the midpoints of these intervals are taken as the values ​​of the characteristics in the groups, resulting in the formation of a discrete series. Consider the following example (Table 4.3).

Let’s move from an interval series to a discrete series by replacing the interval values ​​with their average values/(simple average

Table 4.3

Distribution of JSC workers by monthly wage level

Groups of workers	Number of workers	The middle of the interval
wages, rub.	people, f	rub., X
900 or more

the values ​​of open intervals (first and last) are conditionally equated to the intervals adjacent to them (second and penultimate).

With this calculation of the average, some inaccuracy is allowed, since an assumption is made about the uniform distribution of units of the characteristic within the group. However, the narrower the interval and the more units in the interval, the smaller the error.

After the midpoints of the intervals have been found, calculations are done in the same way as in a discrete series - the options are multiplied by the frequencies (weights) and the sum of the products is divided by the sum of the frequencies (weights), thousand rubles:

.

So, average level remuneration for JSC workers is 729 rubles. per month.

Calculating the arithmetic mean often involves a lot of time and labor. However, in a number of cases, the procedure for calculating the average can be simplified and facilitated if you use its properties. Let us present (without proof) some basic properties of the arithmetic mean.

Property 1. If all individual values ​​of a characteristic (i.e. all options) reduce or increase in itimes, then the average value new characteristic will correspondingly decrease or increase in ionce.

Property 2. If all variants of the characteristic being averaged are reducedsew or increase by number A, then the arithmetic mean correspondswill actually decrease or increase by the same number A.

Property 3. If the weights of all averaged options are reduced or increase in To times, then the arithmetic mean will not change.

As average weights, instead of absolute indicators, you can use specific weights in the overall total (shares or percentages). This simplifies the calculations of the average.

To simplify the calculations of the average, they follow the path of reducing the values ​​of options and frequencies. The greatest simplification is achieved when, as A the value of one of the central options, which has the highest frequency, is selected as / - the value of the interval (for series with equal intervals). The quantity A is called the reference point, therefore this method of calculating the average is called the “method of counting from conditional zero” or "in the way of moments."

Let's assume that all options X first decreased by the same number A, and then decreased by i once. We obtain a new variational series of distribution of new options .

Then new options will be expressed:

,

and their new arithmetic mean , -moment first order -formula:

.

It is equal to the average of the original options, first reduced by A, and then in i once.

To obtain the real average, a first-order moment is needed m 1 , multiply by i and add A:

.

This method of calculating the arithmetic mean from a variation series is called "in the way of moments." This method is used in rows at equal intervals.

The calculation of the arithmetic mean using the method of moments is illustrated by the data in Table. 4.4.

Table 4.4

Distribution of small enterprises in the region by value of fixed production assets (FPAs) in 2000.

Groups of enterprises by OPF value, thousand rubles.	Number of enterprises f	Midpoints of intervals x
14-16 16-18 18-20 20-22 22-24

Finding the first order moment

.

Then, taking A = 19 and knowing that i= 2, calculate X, thousand roubles.:

Types of average values ​​and methods of their calculation

At the stage of statistical processing, a variety of research problems can be set, for the solution of which it is necessary to select the appropriate average. In this case, it is necessary to be guided the following rule: The quantities that represent the numerator and denominator of the average must be logically related to each other.

• power averages;
• structural averages.

Let us introduce the following conventions:

The quantities for which the average is calculated;

Average, where the bar above indicates that averaging of individual values ​​takes place;

Frequency (repeatability of individual characteristic values).

Various averages are derived from the general power average formula:

(5.1)

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

Average values ​​can be simple or weighted. Weighted averages are called quantities that take into account that some variants of attribute values ​​may have different numbers, and therefore each option has to be multiplied by this number. In other words, the “scales” are the numbers of aggregate units in different groups, i.e. Each option is “weighted” by its frequency. The frequency f is called statistical weight or average weight.

Arithmetic mean- the most common type of average. It is used when the calculation is carried out on ungrouped statistical data, where you need to obtain the average term. The arithmetic mean is the average value of a characteristic, upon obtaining which the total volume of the characteristic in the aggregate remains unchanged.

Arithmetic mean formula ( simple) has the form

where n is the population size.

For example, the average salary of an enterprise’s employees is calculated as the arithmetic average:

The determining indicators here are the salary of each employee and the number of employees of the enterprise. When calculating the average, the total amount of wages remained the same, but distributed equally among all employees. For example, you need to calculate the average salary of workers in a small company employing 8 people:

When calculating average values, individual values ​​of the characteristic that is averaged can be repeated, so the average value is calculated using grouped data. In this case we're talking about about use arithmetic average weighted, which has the form

(5.3)

So, we need to calculate average rate shares of a joint stock company traded on a stock exchange. It is known that the transactions were carried out within 5 days (5 transactions), the number of shares sold at the sales rate was distributed as follows:

1 - 800 ak. - 1010 rub.

2 - 650 ak. - 990 rub.

3 - 700 ak. - 1015 rub.

4 - 550 ak. - 900 rub.

5 - 850 ak. - 1150 rub.

The initial ratio for determining the average price of shares is the ratio of the total amount of transactions (TVA) to the number of shares sold (KPA).

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 at which the center of a series of numbers in a statistical distribution is located. 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.

The most common form of statistical indicators used in socio-economic research is the average, which is a generalized quantitative characteristics 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 average ratio (ARR) or its logical formula: . So, for example, to calculate the average wage of an enterprise's employees, 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 average wages, such a determining indicator is the wage fund. For each indicator used in socio-economic analysis, only one true initial ratio can be compiled to calculate the average. It should also be added that in order to more accurately estimate standard deviation for small samples (with the number of elements less than 30), the expression under the root should not be used in the denominator n, A n- 1.

Concept and types of averages

Average value- this is a general indicator of a statistical population that eliminates 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 include fashion And median , but most often 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 quantities, arranged in random order, using the following general power average formula (for different values ​​of k (m)):

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

Where x - average value of the phenomenon under study; x i – i-th version of the averaged characteristic;

f i – weight of the i-th option.

Where X are the values ​​of individual statistical values ​​or the middle of grouping intervals;
m is an exponent, the value of which determines the following types of power averages:
when m = -1 harmonic mean;
at m = 0 geometric mean;
with m = 1 arithmetic mean;
when m = 2 root mean square;
at m = 3 the average is cubic.

Using general formulas for simple and weighted averages for 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 tests;

The arithmetic mean is the most commonly used average value, which is obtained if you substitute general formula m=1. Arithmetic mean simple It has next view:

or

Where X are the values ​​of the quantities for which the average value must be calculated; N- total values ​​of X (the number of units in the population being studied).

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 average formula: (3+4+4+5)/4 = 16/4 = 4. Arithmetic mean weighted has the following form:

Where f is the number of quantities with the same value X (frequency). >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 weighted arithmetic average formula: (3*1 + 4*2 + 5*1)/4 = 16/4 = 4 . If the X values ​​are specified as intervals, then the midpoints of the X intervals are used for calculations, which are defined as the half-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, use the range (the difference between the upper and lower boundaries) of the adjacent interval X. For example, an enterprise has 10 employees with up to 3 years of experience, 20 with 3 to 5 years of experience, 5 employees with more than 5 years of experience. Then we calculate the average length of service of employees using the weighted arithmetic average formula, taking as X the midpoint 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 from 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 reference, contains texts, booleans, or empty cells, then such values ​​are ignored; however, cells that contain zero values ​​are counted.

AVERAGE function

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

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 average is used most often, but there are times when it is necessary to use other types of averages. Let's consider such cases further.

Harmonic mean

Harmonic mean to determine the average sum of reciprocals; Harmonic mean

is used when the source data does not contain frequencies f for individual X values, but is presented as their product Xf. Having designated Xf=w, we express f=w/X, and, substituting these notations into the formula for the arithmetic weighted average, we obtain the formula for the harmonic weighted average: Thus, the weighted harmonic mean is used when the frequencies f are unknown and w=Xf is known. In cases where all w = 1, that is, individual values ​​of X occur once, the average harmonic prime 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 formula for the average harmonic simple, since in the example the distance w 1 =w 2 is given (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.

Function SRGARM Returns the harmonic mean of a data set..

The harmonic mean is the reciprocal of the arithmetic mean

reciprocals

SRGARM(number1,number2, ...)

Number1, number2, ... are from 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.

Number1, number2, ... are from 1 to 30 arguments for which the average is calculated. You can use an array or an array reference instead of semicolon-separated arguments. Geometric mean Geometric mean for estimating the average growth rate of random variables, finding the value of a characteristic equidistant from the minimum and maximum values;inflation 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), the average value must be calculated 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 using the arithmetic mean would give an incorrect result of 11.28%.

SRGEOM function

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

SRGEOM (number1; number2; ...)

Number1, number2, ... are from 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.

Mean square

Mean square – initial moment of the second order. Mean square used in cases where the initial values ​​of X can be both positive and negative, for example, when calculating average deviations.

The main application of the quadratic average is to measure the variation of X values.

Average cubic

The main application of the quadratic average is to measure the variation of X values. The average cubic is the initial moment of the third order.