Most of all in eq. In practice, one has to use the arithmetic mean, which can be calculated as the simple and weighted arithmetic mean.

Arithmetic mean (CA)-n the most common type of medium. It is used in cases where the volume of a variable attribute for the entire population is the sum of the values ​​of the attributes of its individual units. Social phenomena are characterized by the additivity (summation) of the volumes of the varying attribute, this determines the scope of the SA and explains its prevalence as a generalizing indicator, for example: the general salary fund is the sum of the salary of all employees.

To calculate SA, you need to divide the sum of all feature values ​​by their number. SA is used in 2 forms.

Consider first the simple arithmetic mean.

1-CA simple (initial, defining form) is equal to the simple sum of the individual values ​​of the averaged feature, divided by the total number of these values ​​(used when there are ungrouped index values ​​of the feature):

The calculations made can be summarized in the following formula:

(1)

where - the average value of the variable attribute, i.e., the simple arithmetic mean;

means summation, i.e., the addition of individual features;

x- individual values ​​of a variable attribute, which are called variants;

n - number of population units

Example1, it is required to find the average output of one worker (locksmith), if it is known how many parts each of the 15 workers produced, i.e. given a number of ind. trait values, pcs.: 21; twenty; twenty; 19; 21; 19; eighteen; 22; 19; twenty; 21; twenty; eighteen; 19; twenty.

SA simple is calculated by the formula (1), pcs.:

Example2. Let us calculate SA based on conditional data for 20 stores that are part of a trading company (Table 1). Table 1

Distribution of shops of the trading company "Vesna" by trading area, sq. M

store number		store number

To calculate the average store area ( ) it is necessary to add up the areas of all stores and divide the result by the number of stores:

Thus, the average store area for this group of trade enterprises is 71 sq.m.

Therefore, in order to determine the SA is simple, it is necessary to divide the sum of all values ​​of a given attribute by the number of units that have this attribute.

2

where f 1 , f 2 , … ,f n – weight (frequency of repetition of the same features);

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

is the total number of population units.

- SA weighted - With the middle of the options, which are repeated a different number of times, or are said to have different weights. The weights are the number of units in different groups aggregates (the same options are combined into a group). SA weighted – average of grouped values x 1 , x 2 , .., x n – calculated: (2)

Where X- options;

f- frequency (weight).

SA weighted is the quotient of dividing the sum of the products of the variants and their corresponding frequencies by the sum of all frequencies. Frequencies ( f) appearing in the SA formula are usually called scales, as a result of which the SA calculated taking into account the weights is called the weighted SA.

We will illustrate the technique for calculating weighted SA using the example 1 considered above. To do this, we group the initial data and place them in Table.

The average of the grouped data is determined as follows: first, the options are multiplied by the frequencies, then the products are added and the resulting sum is divided by the sum of the frequencies.

According to formula (2), the weighted SA is, pcs.:

The distribution of workers for the development of parts

P

the data given in the previous example 2 can be combined into homogeneous groups, which are presented in table. Table

Distribution of Vesna stores by retail space, sq. m

Thus, the result is the same. However, this will already be the arithmetic weighted average.

In the previous example, we computed the arithmetic average, provided that the absolute frequencies (number of stores) are known. However, in some cases there are no absolute frequencies, but relative frequencies are known, or, as they are commonly called, frequencies that show the proportion or the proportion of frequencies in the entire population.

When calculating SA weighted use frequencies allows you to simplify calculations when the frequency is expressed in large, multi-digit numbers. The calculation is made in the same way, however, since the average value is increased by 100 times, the result should be divided by 100.

Then the formula for the arithmetic weighted average will look like:

where d– frequency, i.e. the share of each frequency in the total sum of all frequencies.

(3)

In our example 2, we first determine the share of stores by groups in the total number of stores of the company "Spring". So, for the first group, the specific gravity corresponds to 10%
. We get the following data Table3

In preparation for successfully completing Problem 19 of Part 3, you need to know some Excel functions. One of these functions is AVERAGE. Let's consider it in more detail.

excel allows you to find the arithmetic mean of the arguments. The syntax for this function is:

AVERAGE(number1, [number2],…)

Do not forget that entering a formula into a cell begins with the "=" sign.

In parentheses, we can list the numbers whose average we want to find. For example, if we write in a cell =AVERAGE(1, 2, -7, 10, 7, 5, 9), then we get 3.857142857. This is easy to check - if we add all the numbers in brackets (1 + 2 + (-7) + 10 + 7 + 5 + 9 = 27) and divide by their number (7), we get 3.857142857142857.

Notice the numbers in parentheses separated by a semicolon (; ). Thus, we can specify up to 255 numbers.

For examples, I'm using Microsort Excel 2010.

In addition, with the help AVERAGE functions we can find average value of a range of cells. Suppose we have some numbers stored in the range A1:A7, and we want to find their arithmetic mean.

Let's put in cell B1 the arithmetic mean of the range A1:A7. To do this, place the cursor in cell B1 and write =AVERAGE(A1:A7). In parentheses, I indicated the range of cells. Note that the delimiter is the character colon (: ). It would be even easier to do - write in cell B1 =AVERAGE(, and then select the desired range with the mouse.

As a result, in cell B1 we will get the number 15.85714286 - this is the arithmetic mean of the range A1:A7.

As a warm-up, I propose to find the average value of numbers from 1 to 100 (1, 2, 3, etc. up to 100). The first person to answer correctly in the comments will receive 50 rubles to the phone We are working.

The concept of arithmetic average means the result of a simple sequence of calculations of the average value for a series of numbers that are predetermined. It should be noted that this value is currently widely used by specialists in a number of industries. For example, formulas are known when performing calculations by economists or employees of the statistical industry, where it is required to have a value of this type. In addition, this indicator is actively used in a number of other industries that are related to the above.

One of the features of calculating this value is the simplicity of the procedure. Carry out calculations anyone can. For this you do not need to have special education. Often there is no need to use computer technology.

As an answer to the question of how to find the arithmetic mean, consider a number of situations.

The simplest way to calculate this value is to calculate it for two numbers. The calculation procedure in this case is very simple:

1. Initially, it is required to carry out the operation of adding the selected numbers. This can often be done, as they say, manually, without using electronic equipment.
2. After the addition is made and its result is obtained, it is necessary to divide. This operation involves dividing the sum of two added numbers by two - the number of added numbers. It is this action that will allow you to get the required value.

Formula

Thus, the formula for calculating the required value in the case of two will look like this:

(A+B)/2

This formula uses the following notation:

A and B are pre-selected numbers for which you need to find a value.

Finding a value for three

The calculation of this value in a situation where three numbers are selected will not differ much from the previous option:

1. To do this, select the numbers needed in the calculation and add them to get the total.
2. After this sum of three is found, it is required to perform the division procedure again. In this case, the resulting amount must be divided by three, which corresponds to the number of selected numbers.

Formula

Thus, the formula required when calculating the arithmetic three will look like this:

(A+B+C)/3

In this formula the following notation has been adopted:

A, B and C are the numbers to which it will be necessary to find the arithmetic mean.

Calculating the arithmetic mean of four

As already seen by analogy with the previous options, the calculation of this value for an amount equal to four will be of the following order:

1. Four digits are selected for which the arithmetic mean is to be calculated. Next, the summation and finding the final result of this procedure is carried out.
2. Now, to get the final result, you should take the resulting sum of four and divide it by four. The received data will be the required value.

Formula

From the sequence of actions described above for finding the arithmetic mean for four, you can get the following formula:

(A+B+C+E)/4

In this formula variables have the following meaning:

A, B, C and E are those for which you need to find the value of the arithmetic mean.

Using this formula, it will always be possible to calculate the required value for a given number of numbers.

Calculating the arithmetic mean of five

Performing this operation will require a certain algorithm of actions.

1. First of all, you need to select five numbers for which the arithmetic mean will be calculated. After this selection, these numbers, as in the previous options, you just need to add up and get the final amount.
2. The resulting amount will need to be divided by their number by five, which will allow you to get the required value.

Formula

Thus, similarly to the previously considered options, we obtain the following formula for calculating the arithmetic mean:

(A+B+C+E+P)/5

In this formula, the variables have the following notation:

A, B, C, E and P are the numbers for which you want to get the arithmetic mean.

Universal Calculation Formula

Conducting a review various options formulas to calculate the arithmetic mean, you can pay attention to the fact that they have a common pattern.

Therefore, it will be more practical to apply the general formula for finding the arithmetic mean. After all, there are situations when the number and size of calculations can be very large. Therefore, it would be wiser to use a universal formula and not deduce an individual technology each time to calculate this value.

The main thing in determining the formula is the principle of calculating the arithmetic mean about.

This principle, as it was seen from the above examples, looks like this:

1. The number of numbers that are specified to obtain the required value is counted. This operation can be carried out both manually with a small number of numbers, and with the help of computer technology.
2. The selected numbers are summed. This operation in most situations is performed using computer technology, since numbers can consist of two, three or more digits.
3. The amount obtained by adding the selected numbers must be divided by their number. This value is determined at the initial stage of calculating the arithmetic mean.

Thus, the general formula for calculating the arithmetic mean of a series of selected numbers will look like this:

(А+В+…+N)/N

This formula contains the following variables:

A and B are numbers that are chosen in advance to calculate their arithmetic mean.

N is the number of numbers that were taken in order to calculate the required value.

Substituting the selected numbers into this formula each time, we can always get the required value of the arithmetic mean.

As seen, finding the arithmetic mean is an easy procedure. However, one must be attentive to the calculations and check the result obtained. This approach is explained by the fact that even in the simplest situations, there is a possibility of getting an error, which can then affect further calculations. In this regard, it is recommended to use computer technology that is capable of making calculations of any complexity.

The most common type of average is the arithmetic average.

simple arithmetic mean

The simple arithmetic mean is the average term, in determining which the total volume of a given attribute in the data is equally distributed among all units included in this population. Thus, the average annual output per worker is such a value of the volume of output that would fall on each employee if the entire volume of output was equally distributed among all employees of the organization. The arithmetic mean simple value is calculated by the formula:

simple arithmetic mean— Equal to the ratio of the sum of individual values ​​of a feature to the number of features in the aggregate

Example 1 . A team of 6 workers receives 3 3.2 3.3 3.5 3.8 3.1 thousand rubles per month.

Find the average salary
Solution: (3 + 3.2 + 3.3 +3.5 + 3.8 + 3.1) / 6 = 3.32 thousand rubles.

Arithmetic weighted average

If the volume of the data set is large and represents a distribution series, then a weighted arithmetic mean is calculated. This is how the weighted average price per unit of production is determined: the total cost of production (the sum of the products of its quantity and the price of a unit of production) is divided by the total quantity of production.

We represent this in the form of the following formula:

Weighted arithmetic mean- is equal to the ratio (the sum of the products of the attribute value to the frequency of repetition of this attribute) to (the sum of the frequencies of all attributes). It is used when the variants of the studied population occur an unequal number of times.

Example 2 . Find the average wages of shop workers per month

The average wage can be obtained by dividing the total wages for the total number of workers:

Answer: 3.35 thousand rubles.

Arithmetic mean for an interval series

When calculating the arithmetic mean for an interval variation series, the average for each interval is first determined as the half-sum of the upper and lower limits, and then the average of the entire series. In the case of open intervals, the value of the lower or upper interval is determined by the value of the intervals adjacent to them.

Averages calculated from interval series are approximate.

Example 3. Determine the average age of students in the evening department.

Averages calculated from interval series are approximate. The degree of their approximation depends on the extent to which the actual distribution of population units within the interval approaches uniform.

When calculating averages, not only absolute, but also relative values ​​(frequency) can be used as weights:

The arithmetic mean has a number of properties that more fully reveal its essence and simplify the calculation:

1. The product of the average and the sum of the frequencies is always equal to the sum of the products of the variant and the frequencies, i.e.

2. The arithmetic mean of the sum of the varying values ​​is equal to the sum of the arithmetic means of these values:

3. The algebraic sum of the deviations of the individual values ​​of the attribute from the average is zero:

4. The sum of the squared deviations of the options from the mean is less than the sum of the squared deviations from any other arbitrary value, i.e.

Let's assume that you need to find the average number of days for tasks to be completed by different employees. Or you want to calculate a time interval of 10 years Average temperature on a particular day. Calculating the average value of a series of numbers in several ways.

The mean is a function of the measure of central tendency, which is the center of a series of numbers in a statistical distribution. The three most common criteria for the central trend are.

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 has 30 divided by 6, 5;

Median The middle number of a series of numbers. Half of the numbers have values ​​that are greater than the Median, and half of 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 frequently occurring number in a group of numbers. For example mode 2, 3, 3, 5, 7 and 10 - 3.

These three measures of the central tendency of the symmetrical distribution of a series of numbers are one and the same. In an asymmetric distribution of a number of numbers, they can be different.

Calculate the average value of cells located continuously in one row or one column

Do the following.

Calculating the Average of Scattered Cells

To accomplish this task, use the function AVERAGE. Copy the table below onto a blank sheet.

Calculating the weighted average

SUMPRODUCT and amounts. The vThis example calculates the average unit price paid across three purchases, where each purchase is for a different number of units of measure at different unit prices.

Copy the table below onto a blank sheet.

Calculating the average value of numbers, ignoring zero values

To accomplish 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.