Calculate the arithmetic mean. Arithmetic mean formula

In mathematics, the arithmetic mean of numbers (or just the average) is the sum of all numbers in a given set, divided by their number. This is the most generalized and widespread concept. average size... As you already understood, in order to find the average value, you need to sum all the numbers given to you, and divide the result by the number of terms.

What is arithmetic mean?

Let's take an example.

Example 1... Given numbers: 6, 7, 11. You need to find their average value.

Solution.

First, let's find the sum of all these numbers.

Now let's divide the resulting sum by the number of terms. Since we have three terms, respectively, we will divide by three.

Therefore, the average of the numbers 6, 7 and 11 is 8. Why exactly 8? Because the sum of 6, 7 and 11 will be the same as three eights. This is clearly seen in the illustration.

The average is somewhat similar to the "alignment" of a series of numbers. As you can see, the piles of pencils have become one level.

Let's consider another example to consolidate the knowledge gained.

Example 2. Given numbers: 3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29. You need to find their arithmetic mean.

Solution.

We find the amount.

3 + 7 + 5 + 13 + 20 + 23 + 39 + 23 + 40 + 23 + 14 + 12 + 56 + 23 + 29 = 330

Divide by the number of terms (in this case - 15).

Therefore, the average value of this series of numbers is 22.

Now let's look at negative numbers. Let's remember how to summarize them. For example, you have two numbers 1 and -4. Let's find their sum.

1 + (-4) = 1 – 4 = -3

With this in mind, consider another example.

Example 3. Find the average value of a series of numbers: 3, -7, 5, 13, -2.

Solution.

Find the sum of the numbers.

3 + (-7) + 5 + 13 + (-2) = 12

Since there are 5 terms, we divide the resulting sum by 5.

Therefore, the arithmetic mean of the numbers 3, -7, 5, 13, -2 is 2.4.

In our time of technological progress, it is much more convenient to use to find the average value computer programs... Microsoft Office Excel is one of them. Finding the average in Excel is quick and easy. Moreover, this program is included in the Microsoft Office software package. Let's look at a quick guide on how to find the arithmetic mean using this program.

In order to calculate the average value of a series of numbers, you need to use the AVERAGE function. The syntax for this function is:
= Average (argument1, argument2, ... argument255)
where argument1, argument2, ... argument255 are either numbers or cell references (cells mean ranges and arrays).

To make it clearer, let's try out the knowledge gained.

Enter the numbers 11, 12, 13, 14, 15, 16 in cells C1 - C6.
Select cell C7 by clicking on it. In this cell, we will display the average value.
Click on the Formulas tab.
Choose More Functions> Statistical to open the drop-down list.
Select AVERAGE. After that, a dialog box should open.
Select and drag cells C1-C6 there to set the range in the dialog box.
Confirm your actions with the "OK" key.
If you did everything correctly, in cell C7 you should have the answer - 13.7. When you click on cell C7, the function (= Average (C1: C6)) will be displayed in the formula bar.

It is very convenient to use this function for accounting, invoicing, or when you just need to find the average of a very long series of numbers. Therefore, it is often used in offices and large companies. This allows you to keep the records in order and makes it possible to quickly calculate something (for example, the average income per month). Also, using Excel, you can find the average value of the function.

Average

This term has other meanings, see mean.

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

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

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

Introduction

Let us denote the data set X = (x 1 , x 2 , …, x n), then the sample mean is usually indicated by a horizontal bar above 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 mean 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 collection 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 the sample rather than the entire population. Therefore, if the sample is presented at random (in terms of probability theory), then x ¯ (\ displaystyle (\ bar (x))) (but not μ) can be treated as a random variable having a probability distribution over the sample (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 mathematical expectation.

It is proved 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 "mean" values, including power mean, Kolmogorov mean, harmonic mean, arithmetic-geometric mean, and various weighted averages (eg, weighted arithmetic mean, weighted geometric mean, weighted harmonic mean).

Examples of

For three numbers, add them up and divide by 3:

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

For four numbers, 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 more simply 5 + 5 = 10, 10: 2. Because we added 2 numbers, which means how many numbers we add, we divide by so many.

Continuous random variable

For a continuously distributed quantity f (x) (\ displaystyle f (x)), the arithmetic mean over the segment [a; b] (\ displaystyle) is defined in terms of the definite integral:

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

Some problems of using the mean

Lack of robustness

Main article: Robustness in statistics

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

A classic example is calculating the average income. The arithmetic mean can be misinterpreted as the median, which can lead to the conclusion that there are more people with higher incomes than they actually are. “Average” income is interpreted in such a way that the income of most people is close to this number. This “average” (in the sense of the arithmetic mean) income is higher than the income of most people, since high income with a large deviation from the mean makes the arithmetic mean strongly skewed (in contrast, the median income “resists” such bias). However, this “average” income says nothing about the number of people near the median income (and does not say anything about the number of people near the modal income). Nevertheless, if you take lightly the concepts of "average" and "majority of the people", then you can make the wrong conclusion that most people have incomes higher than they really are. For example, a report on "average" net income in Medina, Washington, calculated as the arithmetic average of all residents' annual net incomes, would yield a surprisingly large number because of 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 average.

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 stocks fell by 10% in the first year and increased by 30% in the second year, then it is incorrect to calculate the “average” increase over these two years as the arithmetic mean (-10% + 30%) / 2 = 10%; the correct average value in this case is given by the cumulative annual growth rate, at which the annual growth is 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 was at $ 30 in the beginning and fell 10%, it is at $ 27 in the beginning of the second year. If the stock is up 30%, it is worth $ 35.1 at the end of the second year. The arithmetic average of this growth is 10%, but since the stock has only gained $ 5.1 in 2 years, average height 8.2% gives final result $35.1:

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

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

Directions

Main article: Destination statistics

Special care should be taken when calculating the arithmetic mean of some variable that changes cyclically (for example, phase or angle). 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 standards are only defined for the range 0 ° to 360 ° (or 0 to 2π when measured in radians). Thus, the same pair of numbers could be written as (1 ° and −1 °) or as (1 ° and 719 °). The average 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, 0 ° (equivalent to 360 °) would be the geometrically better mean since the numbers deviate less from 0 ° than from any other value (0 ° has the least 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 the cyclic variable, calculated using the above formula, will be artificially shifted from the real average towards the middle of the numeric range. Because of this, the mean is calculated in a different way, namely, the number with the least variance (center point) is chosen as the mean. Also, instead of subtracting, the modular distance (that is, the circumferential distance) is used. For example, the modular distance between 1 ° and 359 ° is 2 °, not 358 ° (on a circle between 359 ° and 360 ° == 0 ° - one degree, between 0 ° and 1 ° - also 1 °, in total - 2 °).

Weighted average - what is it and how to calculate it?

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

Average values: meaning and differences

Not always accurate indicators give an understanding of the situation. In order to assess a particular situation, it is sometimes necessary to analyze a huge number of figures. And then averages come to the rescue. They make it possible to assess the situation as a whole.

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 members is divisible by n. That is, if you need to calculate the arithmetic mean in a sequence of values 27, 22, 34 and 37, then you need to solve the expression (27 + 22 + 34 + 37) / 4, since 4 values are used in the calculations. In this case, the required value will be equal to 30.

Often within school course study and geometric mean. The calculation of this value is based on extracting the nth root of 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.

Harmonic mean in comprehensive school usually not a subject of study. Nevertheless, 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 series of numbers 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 average values, the "weight" of each number used in the calculations plays an important role. The results are more indicative and correct because they take into account more information. This group of values is collectively referred to as "weighted average". They do not pass at school, so it is worth dwelling on them in more detail.

First of all, it is worth telling what is meant by "weight" of this or that value. The easiest way to explain this is with a specific example. Every patient's body temperature is measured twice a day in the hospital. Out of 100 patients in different departments of the hospital, 44 will have a normal temperature of 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 more than 38 degrees! But almost half of the patients have a completely normal temperature. And here it will be more correct to use the weighted average value, 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 geometric and harmonic weighted mean values.

There is another interesting variation used in the 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 weights, periodicity is also used there. And when calculating the average value at some point in time, the values for the previous time intervals are also taken into account.

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

Calculation methods

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

The easiest way to consider the computation is with 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 this or that earnings.

So, the weighted average is calculated 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 will be like this:

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.

How to find the average in excel?

how to find arithmetic mean in excel?

Vladimir09854

Easy peasy. It only takes 3 cells to find the average in excel. In the first we will write one number, in the second - another. And in the third cell, we will hammer in a formula that will give us the average value between these two numbers from the first and second cells. If cell number 1 is called A1, cell number 2 is called B1, then in the cell with the formula you need to write as follows:

This formula calculates the arithmetic mean of two numbers.

For the beauty of our calculations, you can select cells with lines, in the form of a plate.

There is also a function for determining the average value in the Excel itself, but I use the old-fashioned method and enter the formula I need. Thus, I am sure that the Excel will calculate exactly as I need it, and will not come up with some kind of rounding of its own.

M3sergey

It is very easy if the data has already been entered into the cells. If you are just interested in a number, it is enough to select the required range / ranges, and the value of the sum of these numbers, their arithmetic mean and their number will appear in the bottom right of the status bar.

You can select an empty cell, click on the triangle (drop-down list) "AutoSum" and select "Average" there, and then agree with the proposed range for calculation, or choose your own.

Finally, you can use formulas directly by clicking Insert Function next to the formula bar and cell address. The AVERAGE function is located in the "Statistical" category, and accepts as arguments both numbers and cell references, etc. There you can also choose more complex options, for example, AVERAGEIF - calculating the average by condition.

Find average in excel is a fairly straightforward task. Here you need to understand whether you want to use this average value in some formulas or not.

If you need to get only the value, then it is enough to select the required range of numbers, after which excel will automatically calculate the average value - it will be displayed in the status bar, heading "Average".

In the case when you want to use the obtained result in formulas, you can do this:

1) Sum the cells using the SUM function and divide it all by the number of numbers.

2) A more correct option is to use a special function called AVERAGE. The arguments to this function can be numbers specified sequentially, or a range of numbers.

Vladimir tikhonov

circle the values that will participate in the calculation, click the "Formulas" tab, there you will see "AutoSum" on the left and next to it a triangle pointing down. click on this triangle and choose "Average". Voila, done) at the bottom of the bar you will see the average :)

Ekaterina mutalapova

Let's start at the beginning and in order. What does mean mean?

Average is a value that is the arithmetic mean, i.e. is calculated by adding a set of numbers and then dividing the entire sum of the numbers by their number. For example, for the numbers 2, 3, 6, 7, 2 there will be 4 (the sum of the numbers 20 is divided by their number 5)

In an Excel spreadsheet for me personally, the easiest way was to use the formula = AVERAGE. To calculate the average value, you need to enter data into the table, write the function = AVERAGE () under the data column, and in parentheses indicate the range of numbers in the cells, highlighting the data column. After that, press ENTER, or simply left-click on any cell. The result will be displayed in the cell below the column. It looks incomprehensible, but in fact it is a matter of minutes.

Adventurer 2000

Ecxel's program is diverse, so there are several options that will allow you to find the average:

First option. You simply add up all the cells and divide by their number;

Second option. Use a special command, write in the required cell the formula "= AVERAGE (and then specify the range of cells)";

The third option. If you select the required range, then note that on the page below, the average value in these cells is also displayed.

Thus, there are a lot of ways to find the average value, you just need to choose the best one for you and use it constantly.

In Excel, using the AVERAGE function, you can calculate the arithmetic prime mean. To do this, you need to drive in a number of values. Press equals and select in the Statistical Category, among which select the AVERAGE function

Also, using statistical formulas, you can calculate the weighted arithmetic mean, which is considered more accurate. To calculate it, we need the indicator values and frequency.

How to find the average in Excel?

The situation is as follows. There is the following table:

The bars shaded in red contain the numerical values of the grades for the subjects. In the column " Average score"it is required to calculate their average value.
The problem is this: there are 60-70 items in total and some of them are on another sheet.
I looked in another document, the average was already calculated, and in the cell there is a formula like
= "sheet name"! | E12
but it was done by some programmer who was fired.
Please tell me who understands this.

Hector

In the line of functions you insert from the offered functions "AVERAGE" and choose from where they need to be calculated (B6: N6) for Ivanov, for example. I don't know exactly about the neighboring sheets, but for sure it is contained in the standard Windows help

Tell me how to calculate the average value in a Word

Please tell me how to calculate the average value in the Word. Namely, the average of the ratings, not the number of people who received the ratings.

Julia pavlova

Word can do a lot with macros. Press ALT + F11 and write a macro program ..
In addition, Insert-Object ... will allow you to use other programs, even Excel, to create a sheet with a table inside a Word document.
But in this case, you need to write down your numbers in the column of the table, and enter the average in the bottom cell of the same column, right?
To do this, insert a field into the bottom cell.
Insert-Field ... -Formula
Field content
[= AVERAGE (ABOVE)]
gives the average of the sum of the above lying cells.
If the field is selected and the right mouse button is pressed, then it can be Refreshed if the numbers have changed,
view the code or value of the field, change the code directly in the field.
If something goes wrong, delete the entire field in the cell and re-create it.
AVERAGE means average, ABOVE means about, that is, the row of cells above.
I didn't know all this myself, but I easily found it in HELP, of course, thinking a little.

Average values are widespread in statistics. Average values characterize the qualitative indicators of commercial activity: distribution costs, profit, profitability, etc.

Average is one of the common generalization techniques. A correct understanding of the essence of the average determines its special significance in the conditions of a market economy, when the average, through the single and random, makes it possible to identify the general and necessary, to reveal the tendency of regularities economic development.

average value - these are generalizing indicators in which the action of general conditions, patterns of the phenomenon under study are expressed.

Statistical averages are calculated on the basis of mass data of correctly statistically organized mass observation (continuous and selective). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). For example, if you calculate the average wages in cooperatives and state-owned enterprises, and extend the result to the entire population, then the average is fictitious, since it is calculated over a heterogeneous population, and such an average loses all meaning.

With the help of the average, there is, as it were, smoothing out the differences in the value of the attribute, which arise for one reason or another in individual units of observation.

For example, the average output of a salesperson depends on many reasons: qualifications, length of service, age, form of service, health, etc.

Average output reflects the general property of the entire population.

The average value is a reflection of the values of the trait under study, therefore, it is measured in the same dimension as this trait.

Each average value characterizes the studied population for any one attribute. In order to get a complete and comprehensive picture of the population under study in terms of a number of essential features, in general, it is necessary to have a system of average values that can describe the phenomenon from different angles.

There are various averages:

arithmetic mean;

geometric mean;

average harmonic;

root mean square;

average chronological.

Let's consider some types of averages that are most often used in statistics.

Arithmetic mean

The simple arithmetic mean (unweighted) is equal to the sum of the individual values of the attribute, divided by the number of these values.

Individual values of a feature are called variants and are denoted by x (); the number of units in the population is denoted by n, the average value of the feature is denoted by ... Therefore, the simple arithmetic mean is:

According to the data of the discrete distribution series, it can be seen that the same values of the attribute (variants) are repeated several times. So, option x occurs in aggregate 2 times, and option x - 16 times, etc.

The number of identical values of a feature in the distribution series is called the frequency or weight and is denoted by the symbol n.

Let's calculate the average wage of one worker in rubles:

The wage bill for each group of workers is equal to the product of the options by the frequency, and the sum of these products gives the total wage bill of all workers.

In accordance with this, the calculations can be presented in general form:

The resulting formula is called the weighted arithmetic mean.

The statistical material as a result of processing can be presented not only in the form of discrete distribution series, but also in the form of interval variation series with closed or open intervals.

The calculation of the average for the grouped data is made according to the formula of the arithmetic weighted average:

In the practice of economic statistics, sometimes it is necessary to calculate the average by group means or by means of individual parts of the population (private means). In such cases, group or partial averages are taken as options (x), on the basis of which the total average is calculated as the usual weighted arithmetic mean.

Basic properties of the arithmetic mean .

The arithmetic mean has a number of properties:

1. From a decrease or increase in the frequencies of each value of the attribute x in n times, the value of the arithmetic mean will not change.

If all frequencies are divided or multiplied by any number, then the value of the average will not change.

2. The common factor of individual values of the attribute can be taken out of the mean sign:

3. The average of the sum (difference) of two or more values is equal to the sum (difference) of their average:

4. If x = c, where c is a constant, then
.

5. The sum of the deviations of the values of the attribute X from the arithmetic mean x is equal to zero:

Average harmonic.

The characteristics of the variation series, along with the mean, are the mode and the median.

Fashion - This is the value of a feature (option), which is most often repeated in the studied population. For discrete distribution series, the mode will be the value of the variant with the highest frequency.

For interval series of distribution with equal intervals, the mode is determined by the formula:

where
- the initial value of the interval containing the mode;

- the value of the modal interval;

- the frequency of the modal interval;

- the frequency of the interval preceding the modal;

is the frequency of the interval following the modal.

Median - this is a variant located in the middle of the variation series. If the distribution series is discrete and has odd number members, then the median will be the option located in the middle of the ordered row (an ordered row is the arrangement of the units of the population in ascending or descending order).

What is arithmetic mean

The arithmetic mean of several quantities is the ratio of the sum of these quantities to their number.

The arithmetic mean of a certain series of numbers is the sum of all these numbers, divided by the number of terms. Thus, the arithmetic mean is the average of a number series.

What is the arithmetic mean of several numbers? And they are equal to the sum of these numbers, which is divided by the number of terms in this sum.

How to find the arithmetic mean

There is nothing difficult in calculating or finding the arithmetic mean of several numbers, it is enough to add up all the presented numbers, and divide the resulting sum by the number of terms. The resulting result will be the arithmetic mean of these numbers.

Let's take a closer look at this process. What do we need to do to calculate the arithmetic mean and get the final result of this number.

First, to calculate it, you need to determine a set of numbers or their number. This set can include large and small numbers, and their number can be anything.

Secondly, all these numbers need to be added to get their sum. Naturally, if the numbers are simple and their number is small, then the calculations can be made by writing it down by hand. And if the set of numbers is impressive, then it is better to use a calculator or spreadsheet.

And, fourthly, the sum obtained from the addition must be divided by the number of numbers. As a result, we will get the result, which will be the arithmetic mean of this series.

What is the arithmetic mean for?

The arithmetic mean can be useful not only for solving examples and problems in mathematics lessons, but for other purposes necessary in Everyday life person. Such purposes can be the calculation of the arithmetic average to calculate the average financial expense per month, or to calculate the time you spend on the road, also in order to find out the attendance, productivity, speed of movement, yield and much more.

So, for example, let's try to calculate how much time you spend getting to school. Going to school or returning home, you spend every time on the road different time because when you are in a hurry, you go faster and therefore the journey takes less time. But, returning home, you can go slowly, communicating with classmates, admiring nature, and therefore it will take more time on the road.

Therefore, you will not be able to accurately determine the time spent on the road, but thanks to the arithmetic mean, you can approximately find out the time that you spend on the road.

Let's say that on the first day after the weekend, you spent fifteen minutes on the way from home to school, on the second day your journey took twenty minutes, on Wednesday you covered the distance in twenty-five minutes, in the same time you made your way on Thursday, and on Friday you were in no hurry and returned for half an hour.

Let's find the arithmetic mean, adding time, for all five days. So,

15 + 20 + 25 + 25 + 30 = 115

Now let's divide this amount by the number of days

Through this method, you learned that the journey from home to school takes approximately twenty-three minutes of your time.

Homework

1. Find the average by some simple calculations. arithmetic number your class's attendance per week.

2. Find the arithmetic mean:

3. Solve the problem:

This term has other meanings, see mean.

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

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

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

Introduction

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)).)

Examples of

For three numbers, add them up and divide by 3:

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

For four numbers, 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 more simply 5 + 5 = 10, 10: 2. Because we added 2 numbers, which means how many numbers we add, we divide by so many.

Continuous random variable

For a continuously distributed quantity f (x) (\ displaystyle f (x)), the arithmetic mean over the segment [a; b] (\ displaystyle) is defined in terms of the definite integral:

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

Some problems of using the mean

Lack of robustness

Main article: Robustness in statistics

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.

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 was at $ 30 in the beginning and fell 10%, it is at $ 27 in the beginning of the second year. If the stock is up 30%, it is worth $ 35.1 at the end of the second year. The arithmetic average of this growth is 10%, but since the stock is only $ 5.1 in 2 years, an average 8.2% rise gives the 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 mean of 10% in the same way, we will not get the actual value: [$ 30 (1 + 0.1) (1 + 0.1) = $ 36.3].

Directions

Main article: Destination statistics

First, angular standards are only defined for the range 0 ° to 360 ° (or 0 to 2π when measured in radians). Thus, the same pair of numbers could be written as (1 ° and −1 °) or as (1 ° and 719 °). The average 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, 0 ° (equivalent to 360 °) would be the geometrically better mean since the numbers deviate less from 0 ° than from any other value (0 ° has the least variance). Compare:
- the number 1 ° deviates from 0 ° by only 1 °;
- the number 1 ° deviates from the calculated average of 180 ° by 179 °.

4.3. Average values. Essence and meaning of averages

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

For example, a generalizing 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 social payments for the period under review (year, quarter, month) to the number of workers in the JSC.

Calculating the average is one of the common generalization techniques; the average reflects what is common, which 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 accidents and necessity. When calculating averages, due to the action of the law of large numbers, chances are canceled out and balanced, so one can abstract from the insignificant 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 and the scientific value of averages as generalizing characteristics of aggregates.

Where there is a need for generalization, the calculation of such characteristics leads to the replacement of many different individual values of the characteristic average an indicator characterizing the entire totality 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 studied phenomena, characterizes these levels and their changes in time and space.

Average is a summary characteristic of the regularities of the process in the conditions in which it takes place.

4.4. Types of averages and how to calculate them

The choice of the type of average is determined by the economic content of a certain indicator and initial data. In each case, one of the average values is applied: arithmetic, garmonic, geometric, quadratic, cubic etc. The listed averages belong to the class power-law medium.

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

Let us dwell in more detail on power averages.

Arithmetic mean

The most common type of medium is average arithmetic. It is used in cases where the volume of a variable characteristic for the entire population is the sum of the values of the characteristics of its individual units. Social phenomena are characterized by the additivity (summation) of the volumes of the varying attribute, this determines the area of application of the arithmetic mean and explains its prevalence as a generalizing indicator, for example: the total wage fund is the sum of the wages of all workers, the gross harvest is the sum of the products produced from the entire sowing area.

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

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

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

where
- individual values of the variable (options); 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 (locksmith) if you know how many parts each of the 15 workers made, i.e. a number of individual values of the characteristic are given, pieces:

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

The simple arithmetic mean is calculated by the formula (4.1), 1 piece:

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

Weighted arithmetic mean- the average of the grouped values, - is calculated by the formula:

, (4.2)

where
- weight (frequency of repetition of the same signs);

- the sum of the products of the magnitude of the features by their frequency;

- total number units of the population.

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

Table 4.1

Distribution of workers for the production of parts

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

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

where
- particular, i.e. the share of each frequency in the total sum of all

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

Calculating the weighted arithmetic mean from group means is carried out according to the formula:

where f- the number of units in each group.

The results of calculating the arithmetic mean of the 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 the distribution series

If the values of the averaged feature are specified in the form of intervals ("from - to"), i.e. interval series of distribution, then when calculating the arithmetic mean, the midpoints of these intervals are taken as the values of the attributes in the groups, as a result of which a discrete series is formed. Consider the following example (Table 4.3).

We pass from the interval series to the discrete one by replacing the interval values with their mean values / (simple mean

Table 4.3

Distribution of JSC workers by the level of monthly wages

Worker groups	Number of workers	The middle of the interval,
wages, rub.	people, f	rub., NS





900 and more

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

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

After the middle of the intervals are found, the calculations are done in the same way as in the 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 the wages of the workers of the AO is 729 rubles. per month.

Calculating the arithmetic mean is often time-consuming and labor-intensive. However, in some cases, the procedure for calculating the average can be simplified and facilitated by using its properties. Let us present (without proof) some of the basic properties of the arithmetic mean.

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

Property 2. If all variants of the averaged feature decreasesew or increase by the 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.

Instead of absolute indicators, weights in the total total (shares or percentages) can be used as weights of the average. This simplifies the calculations of the average.

To simplify the calculations of the average, they follow the path of decreasing the values of variants and frequencies. The greatest simplification is achieved when the quality BUT the value of one of the central variants with the highest frequency is selected, as / is the value of the interval (for rows with equal intervals). The quantity Л is called the origin, therefore this method of calculating the average is called the "method of counting from the conditional zero" or "The way of the moments."

Let's assume that all options NS first reduced by the same number A, and then reduced by i once. We get a new variation series of the distribution of new options .

Then new options will be expressed:

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

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

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

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

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

Table 4.4

Distribution of small enterprises in the region by the value of fixed assets (OPF) in 2000

Groups of enterprises at the cost of OPF, thousand rubles	Number of enterprises f	The middle of the intervals, x
14-16 16-18 18-20 20-22 22-24

Find the moment of the first order

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

Types of averages and methods for their calculation

At the stage of statistical processing, a variety of research tasks 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 by the following rule: the values that represent the numerator and denominator of the average must be logically related.

power averages;
structural averages.

Let us introduce the following conventions:

The values for which the average is calculated;

Average, where the line above indicates that there is averaging of individual values;

Frequency (repeatability of individual values of a feature).

Various averages are derived from general formula power-law mean:

(5.1)

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

Average values are simple and weighted. Weighted averages they call the values that take into account that some variants of the values of the trait may have different numbers, in connection with which each option has to be multiplied by this number. In other words, the "weights" are the numbers of units of the population 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 medium. It is used when the calculation is carried out on non-grouped statistical data, where you want to get the average term. The arithmetic mean is such an average value of a feature, upon receipt of which the total volume of a feature in the aggregate remains unchanged.

The arithmetic mean formula ( simple) has the form

where n is the population size.

For example, the average wage of employees of an enterprise is calculated as the arithmetic mean:

The defining indicators here are the wages 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, as it were, among all workers equally. For example, you need to calculate the average salary of employees of a small company where 8 people are employed:

When calculating the average values, the individual values of the attribute, which is averaged, can be repeated, therefore, the average value is calculated according to the grouped data. In this case it comes about using weighted arithmetic mean which has the form

(5.3)

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

1 - 800 ac. - 1010 rubles.

2 - 650 ac. - 990 rubles.

3 - 700 ac. - 1015 rubles.

4 - 550 ac. - 900 rubles.

5 - 850 ac. - 1150 rubles.

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

By discipline: Statistics

Option number 2

Averages used in statistics

Introduction ………………………………………………………………………… .3

Theoretical task

Average value in statistics, its essence and conditions of use.

1.1. The essence of the average size and conditions of use ...................... 4

1.2. Types of average values …………………………………………… 8

Practical task

Task 1,2,3 …………………………………………………………………… 14

Conclusion ……………………………………………………………………… .21

List of used literature …………………………………………… ... 23

Introduction

This test consists of two parts - theoretical and practical. In the theoretical part, such an important statistical category as the average will be considered in detail in order to identify its essence and conditions of use, as well as highlight the types of averages and methods for their calculation.

Statistics, as you know, studies mass socio-economic phenomena. Each of these phenomena can have a different quantitative expression of the same attribute. For example, the wages of the same profession of workers or the prices in the market for the same product, etc. Average values characterize the qualitative indicators of commercial activity: distribution costs, profit, profitability, etc.

To study any set of varying (quantitatively changing) characteristics, statistics use averages.

Medium Essence

The average value is a generalizing quantitative characteristic of a set of phenomena of the same type according to one varying feature. In economic practice, a wide range of indicators are used, calculated as averages.

The most important property of the average is that it represents the value of a certain feature in the entire set by one number, despite its quantitative differences in individual units of the set, and expresses the general that is inherent in all units of the studied set. Thus, through the characteristics of a unit of the population, it characterizes the entire population as a whole.

Average values are associated with the law of large numbers. The essence of this connection lies in the fact that during averaging, random deviations of individual values are canceled out due to the action of the law of large numbers, and in the mean, the main development trend, necessity, and regularity are revealed. Averages allow you to compare indicators related to populations with different numbers of units.

In modern conditions of the development of market relations in the economy, averages serve as a tool for studying the objective laws of socio-economic phenomena. However, economic analysis should not be limited to only averages, since the general favorable averages may hide both major serious shortcomings in the activities of individual economic entities, and the sprouts of a new, progressive one. For example, the distribution of the population by income makes it possible to identify the formation of new social groups. Therefore, along with the average statistical data, it is necessary to take into account the characteristics of individual units of the population.

The average value is the resultant of all factors influencing the phenomenon under study. That is, when calculating the average values, the influence of random (perturbation, individual) factors is canceled out and, thus, it is possible to determine the regularity inherent in the phenomenon under study. Adolphe Quetelet emphasized that the significance of the method of average values lies in the possibility of transition from the single to the general, from the accidental to the regular, and the existence of average values is a category of objective reality.

Statistics studies mass phenomena and processes. Each of these phenomena has both common for the entire set and special, individual properties. The distinction between individual phenomena is called variation. Another property of mass phenomena is their inherent closeness of the characteristics of individual phenomena. So, the interaction of the elements of a set leads to a limitation of the variation of at least some of their properties. This tendency exists objectively. It is precisely in its objectivity that the reason for the broadest application of averages in practice and in theory lies.

The average value in statistics is called a generalizing indicator that characterizes the typical level of a phenomenon in specific conditions of place and time, reflecting the value of a variable attribute per unit of a qualitatively homogeneous population.

In economic practice, a wide range of indicators are used, calculated as averages.

Using the method of averages, statistics solve many problems.

The main meaning of averages consists in their generalizing function, that is, replacing many different individual values of a feature with an average that characterizes the entire set of phenomena.

If the average summarizes the qualitatively homogeneous values of a feature, then it is a typical characteristic of a feature in a given population.

However, it is wrong to reduce the role of average values only to the characteristic of typical values of attributes in populations homogeneous for a given attribute. In practice, modern statistics much more often uses averages that generalize clearly homogeneous phenomena.

The average value of the national income per capita, the average yield of grain crops throughout the country, the average consumption of various food products - these are the characteristics of the state as a single national economic system, these are the so-called system averages.

System averages can characterize both spatial or object systems that exist simultaneously (state, industry, region, planet Earth, etc.), and dynamic systems that are extended in time (year, decade, season, etc.).

The most important property of the average is that it reflects the general that is inherent in all units of the studied population. The values of the attribute of individual units of the population fluctuate in one direction or another under the influence of many factors, among which there can be both basic and random ones. For example, the stock price of a corporation as a whole is determined by its financial position. At the same time, on certain days and on certain stock exchanges, due to the current circumstances, these shares may be sold at a higher or lower rate. The essence of the average lies in the fact that it cancels out deviations in the values of the attribute of individual units of the population, caused by the action of random factors, and takes into account the changes caused by the action of the main factors. This allows the average to reflect the typical level of the trait and abstract from the individual characteristics inherent in individual units.

Average is a summary characteristic of the regularities of the process in the conditions in which it takes place.

Each average characterizes the studied population according to any one criterion, but a system of average indicators is needed to characterize any population, describe its typical features and qualitative features. Therefore, in the practice of domestic statistics for the study of socio-economic phenomena, as a rule, a system of average indicators is calculated. So, for example, the indicator of the average wage is assessed together with the indicators of the average output, capital-labor ratio and power-to-labor ratio, the degree of mechanization and automation of work, etc.

The average should be calculated taking into account the economic content of the indicator under study. Therefore, for a specific indicator used in socio-economic analysis, only one true value of the average can be calculated based on the scientific method of calculation.

The average value is one of the most important generalizing statistical indicators that characterize the totality of phenomena of the same type according to some quantitatively varying attribute. Averages in statistics are generalizing indicators, numbers that express the typical characteristic dimensions of social phenomena according to one quantitatively varying attribute.

Types of averages

The types of average values differ primarily in which property, which parameter of the initial varying mass of individual values of the attribute should be kept unchanged.

Arithmetic mean

The arithmetic mean is such an average value of a feature, when calculating which the total amount of a feature in the aggregate remains unchanged. Otherwise, we can say that the arithmetic mean is the mean term. When calculating it, the total volume of the attribute is mentally distributed equally among all units of the population.

The arithmetic mean is used if the values of the averaged attribute (x) and the number of units of the population with a certain value of the attribute (f) are known.

The arithmetic mean is simple and weighted.

Simple arithmetic mean

Simple is used if each value of the attribute x occurs once, i.e. for each x the value of the feature f = 1, or if the initial data is not ordered and it is not known how many units have certain feature values.

The formula for the simple arithmetic mean is:

where is the average value; x is the value of the averaged attribute (variant), is the number of units of the studied population.

Weighted arithmetic mean

Unlike a simple average, the arithmetic weighted average is used if each value of the attribute x occurs several times, i.e. for each attribute value f ≠ 1. This average is widely used in calculating the average based on a discrete distribution series:

where is the number of groups, x is the value of the averaged feature, f is the weight of the feature value (frequency, if f is the number of units in the population; frequency, if f is the proportion of units with variant x in the total volume of the population).

Average harmonic

Along with the arithmetic mean, statistics use the harmonic mean, the reciprocal of the arithmetic mean of the reciprocal values of the attribute. Like the arithmetic mean, it can be simple and weighted. It is used when the required weights (f i) in the initial data are not specified directly, but are included as a factor in one of the available indicators (i.e., when the numerator of the initial ratio of the mean is known, but its denominator is unknown).

Average harmonic weighted

The product xf gives the volume of the averaged feature x for a set of units and is denoted by w. If the initial data contains the values of the averaged feature x and the volume of the averaged feature w, then the harmonic weighted is used to calculate the average:

where x is the value of the averaged feature x (option); w is the weight of variants x, the volume of the averaged feature.

Unweighted harmonic mean (simple)

This form of the average, used much less often, has the following form:

where x is the value of the averaged feature; n is the number of x values.

Those. it is the reciprocal of the arithmetic mean of the simple of the reciprocal values of the attribute.

In practice, the simple harmonic mean is rarely used when the w values for the population units are equal.

Root mean square and mean cubic

In a number of cases in economic practice, there is a need to calculate the average size of a feature, expressed in square or cubic units. Then the root-mean-square is used (for example, to calculate the average size of the side and square sections, the average diameters of pipes, trunks, etc.) and the cubic average (for example, when determining the average length of the side and cubes).

If, when replacing individual values of a feature with an average value, it is necessary to keep the sum of the squares of the original values unchanged, then the average will be the quadratic average, simple or weighted.

Mean square simple

Simple is used if each value of the attribute x occurs once, in general it has the form:

where is the square of the values of the averaged feature; - the number of units in the population.

Weighted mean square

The weighted mean square is applied if each value of the averaged attribute x occurs f times:

where f is the weight of options x.

Average cubic simple and weighted

The average cubic simple is the cube root of the quotient of dividing the sum of the cubes of individual values of the characteristic by their number:

where are the values of the feature, n is their number.

Cubic average weighted:

where f is the weight of the options x.

Root mean square and cubic mean are of limited use in the practice of statistics. The statistics of the root mean square is widely used, but not from the options themselves x , and from their deviations from the mean when calculating the indicators of variation.

The average can be calculated not for all, but for some part of the population units. An example of such an average can be the progressive average as one of the partial averages, calculated not for everyone, but only for the "best" (for example, for indicators above or below the individual average).

Geometric mean

If the values of the averaged feature are significantly distant from each other or are set by coefficients (growth rates, price indices), then the geometric mean is used for the calculation.

The geometric mean is calculated by extracting the root of the degree and from the products of individual values - variants of the attribute NS:

where n is the number of options; P is the sign of the work.

The geometric mean was most widely used to determine the average rate of change in the series of dynamics, as well as in the series of distribution.

Average values are generalizing indicators in which the action of general conditions, the regularity of the phenomenon under study, are expressed. Statistical averages are calculated on the basis of mass data of a correctly statistically organized mass observation (continuous or selective). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). The use of averages should proceed from a dialectical understanding of the categories of general and individual, mass and singular.

The combination of general means with group means makes it possible to restrict qualitatively homogeneous populations. Dividing the mass of objects that make up this or that complex phenomenon into internally homogeneous, but qualitatively different groups, characterizing each of the groups by its average, it is possible to reveal the reserves of the process of an emerging new quality. For example, the distribution of the population by income makes it possible to identify the formation of new social groups. In the analytical part, we considered a particular example of using the average. Summing up, we can say that the field of application and use of averages in statistics is quite wide.

Practical task

Problem number 1

Determine the average buying rate and the average selling rate of one and US $

Average purchase rate

Average selling rate

Problem number 2

Dynamics of the volume of own products of public catering Chelyabinsk region for 1996-2004 is presented in the table in comparable prices (million rubles)

Close rows A and B. To analyze a number of dynamics in the production of finished products, calculate:

1. Absolute increments, growth rates and increments, chain and base

2. Average annual production of finished goods

3. The average annual growth rate and increase in the company's production

4. Perform analytical alignment of a number of dynamics and calculate the forecast for 2005

5. Show graphically a series of dynamics

6. Make a conclusion based on the results of the dynamics

1) уi B = уi-у1 уi Ц = уi-у1

y2 B = 2.175 - 2.04 y2 C = 2.175 - 2.04 = 0.135

y3B = 2.505 - 2.04 y3 C = 2, 505 - 2.175 = 0.33

y4 B = 2.73 - 2.04 y4 C = 2.73 - 2.505 = 0.225

y5 B = 1.5 - 2.04 y5 C = 1.5 - 2.73 = 1.23

y6 B = 3.34 - 2.04 y6 C = 3.34 - 1.5 = 1.84

y7 B = 3.6 3 - 2.04 y7 C = 3, 6 3 - 3.34 = 0.29

y8 B = 3.96 - 2.04 y8 C = 3.96 - 3.63 = 0.33

y9 B = 4.41–2.04 y9 C = 4.41 - 3.96 = 0.45

Tr B2 Tr C2

Tr B3 Tr C3

Tr B4 Tr Ts4

Tr B5 Tr C5

Tr B6 Tr C6

Tr B7 Tr C7

Tr B8 Tr C8

Tr B9 Tr C9

Tr B = (TprB * 100%) - 100%

Tr B2 = (1.066 * 100%) - 100% = 6.6%

Tr Ts3 = (1.151 * 100%) - 100% = 15.1%

2) y million rubles - average productivity of products

2,921 + 0,294*(-4) = 2,921-1,176 = 1,745

2,921 + 0,294*(-3) = 2,921-0,882 = 2,039

(yt-y) = (1.745-2.04) = 0.087

(yt-yt) = (1.745-2.921) = 1.382

(y-yt) = (2.04-2.921) = 0.776

y2005 = 2.921 + 1.496 * 4 = 2.921 + 5.984 = 8.905

8,905+2,306*1,496=12,354

8,905-2,306*1,496=5,456

5,456 2005 12,354

Problem number 3

The statistical data of the wholesale supplies of food and non-food products and the retail trade network of the region in 2003 and 2004 are presented in the corresponding graphs.

According to tables 1 and 2, it is required

1. Find the general index of the wholesale supply of food products in actual prices;

2. Find the general index of the actual supply of food products;

3. Compare the general indices and make the appropriate conclusion;

4. Find the general index of the supply of non-food products in actual prices;

5. Find the general index of the physical volume of supply of non-food products;

6. Compare the obtained indices and make a conclusion on non-food products;

7. Find the consolidated general indices of the supply of the entire mass of commodities in actual prices;

8. Find the consolidated general index of physical volume (for the entire mass of commodities);

9. Compare the resulting composite indices and make the appropriate conclusion.

Base period		Reporting period (2004)	Deliveries of the reporting period in the prices of the base period




	1,291-0,681=0,61= - 39 Conclusion In conclusion, let's summarize. Average values are generalizing indicators in which the action of general conditions, the regularity of the phenomenon under study, are expressed. Statistical averages are calculated on the basis of mass data of a correctly statistically organized mass observation (continuous or selective). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). The use of averages should proceed from a dialectical understanding of the categories of general and individual, mass and singular. The average reflects the total that is added up in each separate, single object, thanks to this, the average receives great importance to identify patterns inherent in mass social phenomena and imperceptible in isolated phenomena. The deviation of the individual from the general is a manifestation of the development process. In some isolated cases, elements of a new, advanced one can be laid. In this case, it is the specific factor taken against the background of average values that characterizes the development process. Therefore, the average reflects the characteristic, typical, real level of the studied phenomena. The characteristics of these levels and their changes in time and space are one of the main tasks of averages. So, through averages, it is manifested, for example, characteristic of enterprises in a certain stage economic development; the change in the well-being of the population is reflected in the average indicators of wages, family income as a whole and by individual social groups, the level of consumption of products, goods and services. The average indicator is a typical value (ordinary, normal, prevailing in general), but it is such by what is formed in the normal, natural conditions of the existence of a particular mass phenomenon, considered as a whole. The average reflects the objective property of the phenomenon. In reality, only deviating phenomena often exist, and the average as a phenomenon may not exist, although the concept of the typicality of a phenomenon is borrowed from reality. The average value is a reflection of the value of the trait under study and, therefore, is measured in the same dimension as this trait. However, there are various ways to approximate the level of distribution of the population for comparing summary indicators that are not directly comparable with each other, for example, the average population in relation to the territory (average population density). Depending on which factor needs to be eliminated, the content of the average will also be found. The combination of general means with group means makes it possible to restrict qualitatively homogeneous populations. Dividing the mass of objects that make up this or that complex phenomenon into internally homogeneous, but qualitatively different groups, characterizing each of the groups by its average, it is possible to reveal the reserves of the process of an emerging new quality. For example, the distribution of the population by income makes it possible to identify the formation of new social groups. In the analytical part, we considered a particular example of using the average. Summing up, we can say that the scope and use of averages in statistics is quite wide. Bibliography 1. Gusarov, V.M. The theory of statistics by quality [Text]: textbook. allowance / V.M. Gusarov manual for universities. - M., 1998 2. Edronova, N.N. General theory of statistics [Text]: textbook / Ed. N.N. Edronova - M .: Finance and statistics 2001 - 648 p. 3. Eliseeva I.I., Yuzbashev M.M. General theory of statistics [Text]: Textbook / Ed. Corresponding Member RAS I. I. Eliseeva. - 4th ed., Rev. and add. - M .: Finance and statistics, 1999. - 480p .: ill. 4. Efimova M.R., Petrova E.V., Rumyantsev V.N. General theory of statistics: [Text]: Textbook. - M .: INFRA-M, 1996 .-- 416s. 5. Ryauzova, N.N. General theory of statistics [Text]: textbook / Ed. N.N. Ryauzov - M .: Finance and Statistics, 1984. Gusarov V.M. Theory of statistics: Textbook. Manual for universities. - M., 1998.-p. 60. Eliseeva I.I., Yuzbashev M.M. General theory of statistics. - M., 1999.-P.76. Gusarov V.M. Theory of statistics: Textbook. Manual for universities. -M., 1998.-P.61.