Construct a series of distributions at equal intervals. Distribution series

The grouping results of the collected statistics are usually presented as distribution series. A distribution series is an ordered distribution of population units into groups according to the trait under study.

Distribution series are divided into attributive and variational, depending on the feature underlying the grouping. If the feature is qualitative, then the distribution series is called attributive. An example of an attributive series is the distribution of enterprises and organizations by forms of ownership (see Table 3.1).

If the attribute on which the distribution series is built is quantitative, then the series is called variational.

The variation series of a distribution always consists of two parts: a variant and the corresponding frequencies (or frequencies). Variant is the value that a feature can take in population units, frequency is the number of observation units that have a given feature value. The sum of the frequencies is always equal to the volume of the population. Sometimes, instead of frequencies, frequencies are calculated - these are frequencies expressed either in fractions of one (then the sum of all frequencies is 1), or as a percentage of the total volume (the sum of frequencies will be equal to 100%).

Variational series are discrete and interval. For discrete series (Table 3.7), the options are expressed by specific numbers, most often whole ones.

Table 3.8. Distribution of employees by working time in an insurance company

Working hours in the company, full years(options)	Number of employees
	Human (frequency)	in% to total (frequency)
up to a year	15	11,6
1	17	13,2
2	19	14,7
3	26	20,2
4	10	7,8
5	18	13,9
6	24	18,6
Total	129	100,0

In interval series (see Table 3.2), the values of the indicator are set as intervals. Intervals have two boundaries: lower and upper. The intervals can be open or closed. The open ones do not have one of the borders, so, in table. 3.2 the first interval has no lower border, and the last one has no upper one. When constructing an interval series, depending on the nature of the scatter of the attribute values, both equal interval intervals and unequal intervals are used (Table 3.2 shows a variation series with at equal intervals).

If a feature takes a limited number of values, usually no more than 10, discrete distribution series are built. If the variant is larger, then the discrete series loses its clarity; in this case it is advisable to use the interval form variation series... With continuous variation of a feature, when its values within certain limits differ from each other by an arbitrarily small amount, an interval distribution series is also built.

3.3.1. Construction of discrete variation series

Let us consider the method of constructing discrete variational series using an example.

Example 3.2. There is the following data on the quantitative composition of 60 families:

In order to get an idea of the distribution of families by the number of their members, a variation series should be built. Since the feature takes a limited number of integer values, we construct a discrete variation series. To do this, it is first recommended to write out all the values of the trait (the number of members in the family) in ascending order (i.e. to rank the statistical data):

Then it is necessary to count the number of families with the same composition. The number of family members (the value of the variable feature) are options (we will denote them by x), the number of families with the same composition is the frequencies (we will denote them by f). The grouping results are presented in the form of the following discrete variational distribution series:

Table 3.11.

Number of family members (x)	Number of families (y)
1	8
2	14
3	20
4	9
5	5
6	4
Total	60

3.3.2. Construction of interval variation series

Let us show a technique for constructing interval variational distribution series using the following example.

Example 3.3. As a result of statistical observation, the following data on the average interest rate of 50 commercial banks (%) were obtained:

Table 3.12.

14,7	19,0	24,5	20,8	12,3	24,6	17,0	14,2	19,7	18,8
18,1	20,5	21,0	20,7	20,4	14,7	25,1	22,7	19,0	19,6
19,0	18,9	17,4	20,0	13,8	25,6	13,0	19,0	18,7	21,1
13,3	20,7	15,2	19,9	21,9	16,0	16,9	15,3	21,4	20,4
12,8	20,8	14,3	18,0	15,1	23,8	18,5	14,4	14,4	21,0

As you can see, it is extremely inconvenient to view such an array of data, in addition, there is no visible pattern of changes in the indicator. Let's construct an interval distribution series.

Let's define the number of intervals.
In practice, the number of intervals is often set by the researcher himself, based on the tasks of each specific observation. At the same time, it can also be calculated mathematically using the Sturgess formula

n = 1 + 3.322lgN,

where n is the number of intervals;

N is the volume of the population (the number of observation units).

For our example, we get: n = 1 + 3.322lgN = 1 + 3.322lg50 = 6.6 "7.
Let us determine the size of the intervals (i) by the formula
where x max - maximum value sign;

x min is the minimum value of the feature.

For our example

The intervals of the variation series are clear if their boundaries have "round" values, therefore, we will round the value of the interval 1.9 to 2, and the minimum value of the feature 12.3 to 12.0.
Let's define the boundaries of the intervals.
Intervals are usually recorded in such a way that the upper limit of one interval is simultaneously the lower limit of the next interval. So, for our example, we get: 12.0-14.0; 14.0-16.0; 16.0-18.0; 18.0-20.0; 20.0-22.0; 22.0-24.0; 24.0-26.0.

Such a record means that the feature is continuous. If the variants of a feature take strictly defined values, for example, only integers, but their number is too large for constructing a discrete series, then an interval series can be created where the lower boundary of the interval will not coincide with the upper boundary of the next interval (this will mean that the feature is discrete ). For example, in the distribution of employees of an enterprise by age, you can create the following interval groups of years: 18-25, 26-33, 34-41, 42-49, 50-57, 58-65, 66 and more.

Also, in our example, we could make the first and last intervals open, etc. write: up to 14.0; 24.0 and higher.

Based on the initial data, we will construct a ranged series. To do this, write in ascending order the values that the attribute takes. The results are presented in the table: Table 3.13. Ranked series of interest rates of commercial banks

Bank rate% (options)
12,3	17,0	19,9	23,8
12,8	17,4	20,0	24,5
13,0	18,0	20,0	24,6
13,3	18,1	20,4	25,1
13,8	18,5	20,4	25,6
14,2	18,7	20,5
14,3	18,8	20,7
14,4	18,9	20,7
14,7	19,0	20,8
14,7	19,0	21,0
15,1	19,0	21,0
15,2	19,0	21,1
15,3	19,0	21,4
16,0	19,6	21,9
16,9	19,7	22,7

Let's count the frequencies.
When calculating frequencies, a situation may arise when the value of a feature falls on the border of an interval. In this case, you can be guided by the rule: this unit is assigned to the interval for which its value is the upper limit. So, the value 16.0 in our example will refer to the second interval.

The grouping results obtained in our example will be presented in the table.

Table 3.14. Distribution of commercial banks by lending rate

Short rate,%	Number of banks, units (frequency)	Accumulated frequencies
12,0-14,0	5	5
14,0-16,0	9	14
16,0-18,0	4	18
18,0-20,0	15	33
20,0-22,0	11	44
22,0-24,0	2	46
24,0-26,0	4	50
Total	50	-

The last column of the table shows the accumulated frequencies, which are obtained by sequentially summing the frequencies, starting from the first (for example, for the first interval - 5, for the second interval 5 + 9 = 14, for the third interval 5 + 9 + 4 = 18, etc. .). The cumulative frequency, for example, 33, shows that 33 banks have a lending rate not exceeding 20% (the upper limit of the corresponding interval).

In the process of grouping data when constructing variation series, unequal intervals are sometimes used. This applies to those cases when the attribute values obey the rule of arithmetic or geometric progression, or when the application of the Sturgess formula leads to the appearance of "empty" interval groups that do not contain a single observation unit. Then the boundaries of the intervals are set arbitrarily by the researcher himself, based on common sense and the objectives of the survey, or by formulas. So, for data that changes in arithmetic progression, the size of the intervals is calculated as follows.

When constructing an interval distribution series, three questions are resolved:

1. How many intervals should you take?
2. How long are the intervals?
3. What is the order of inclusion of population units in the boundaries of intervals?
1. Number of intervals can be determined by the Sturjess formula:

2. Interval length, or interval step, usually determined by the formula

where R - the range of variation.

3. The order of inclusion of units of the population in the boundaries of the interval

may be different, but when constructing an interval series, the distribution is necessarily strictly defined.

For example, this: [), in which the aggregate units are included in the lower bounds, and are not included in the upper bounds, but are carried over to the next interval. The exception to this rule is the last interval, the upper bound of which includes the last number of the ranked series.

The boundaries of the intervals are:

closed - with two extreme values of the attribute;
open - with one extreme value of the feature (before such and such a number or over of such and such a number).

In order to assimilate the theoretical material, we introduce background information for solutions cross-cutting task.

There are conditional data on the average number of sales managers, the number of single-quality goods sold by them, the individual market price for this product, as well as the sales volume of 30 firms in one of the regions of the Russian Federation in the first quarter of the reporting year (Table 2.1).

Table 2.1

Background information for the cross-cutting challenge

Number managers,	Price, thousand rubles	Sales volume, RUB mln

Number managers,	Number of goods sold, pcs.	Price, thousand rubles	Sales volume, RUB mln

Based on the initial information, as well as additional information, we will make the setting of individual tasks. Then we will present the methodology for solving them and the solutions themselves.

Cross-cutting task. Assignment 2.1

Using the initial data table. 2.1 is required build a discrete series of distribution of firms by the amount of goods sold (Table 2.2).

Solution:

Table 2.2

Discrete series of distribution of firms by the amount of goods sold in one of the regions of the Russian Federation in the first quarter of the reporting year

Cross-cutting task. Assignment 2.2

required build a ranked row of 30 firms based on the average headcount of managers.

Solution:

15; 17; 18; 20; 20; 20; 22; 22; 24; 25; 25; 25; 27; 27; 27; 28; 29; 30; 32; 32; 33; 33; 33; 34; 35; 35; 38; 39; 39; 45.

Cross-cutting task. Assignment 2.3

Using the initial data table. 2.1, required:

1. Construct an interval series of distribution of firms by the number of managers.
2. Calculate the frequency of the distribution series of firms.
3. Draw conclusions.

Solution:

We calculate by the Sturgess formula (2.5) number of intervals:

Thus, we take 6 intervals (groups).

Interval length, or interval step, we will calculate by the formula

Note. The order of inclusion of population units in the boundaries of the interval is as follows: I), in which the population units are included in the lower boundaries, and not included in the upper ones, but are transferred to the next interval. The exception to this rule is the last interval I], the upper bound of which includes the last number of the ranked series.

We build an interval series (Table 2.3).

Interval series of distribution of firms but the average number of managers in one of the regions of the Russian Federation in the first quarter of the reporting year

Output. The most numerous group of firms is a group with an average number of managers of 25-30 people, which includes 8 firms (27%); the smallest group with an average number of managers of 40-45 people includes only one firm (3%).

Using the initial data table. 2.1, as well as the interval series of the distribution of firms by the number of managers (Table 2.3), required to construct an analytical grouping of the relationship between the number of managers and the volume of sales of firms and, on the basis of it, draw a conclusion about the presence (or absence) of a connection between the indicated characteristics.

Solution:

The analytical grouping is based on the factorial attribute. In our task, the factorial attribute (x) is the number of managers, and the resultant attribute (y) is the volume of sales (Table 2.4).

Let's build now analytical group(Table 2.5).

Output. Based on the data of the constructed analytical group, it can be said that with an increase in the number of sales managers, the average sales volume of the firm in the group also increases, which indicates the presence of a direct connection between the indicated characteristics.

Table 2.4

Auxiliary table for constructing an analytical grouping

Number of managers, people,	Company number	Sales volume, RUB mln,











		"= 59 f = 9.97








		I- ™ 4 - Yu.22







		74'25 1PY1 U4 = 7 = 10,61










			at = ’ =10,31 30

Table 2.5

Dependence of sales volumes on the number of managers of firms in one of the regions of the Russian Federation in the first quarter of the reporting year

CONTROL QUESTIONS

1. What is the essence of statistical observation?
2. What are the stages of statistical observation.
3. What are organizational forms statistical observation?
4. Name the types of statistical observation.
5. What is a statistical summary?
6. What are the types of statistical summaries.
7. What is a statistical grouping?
8. Name the types of statistical groupings.
9. What is a distribution series?
10. Name structural elements distribution series.
11. What is the order of constructing a distribution series?

What is the grouping of statistical data, and how it is related to distribution series, was discussed in this lecture, there you can also learn about what a discrete and variation distribution series is.

Distribution series is one of the varieties of statistical series (apart from them, dynamics series are used in statistics), they are used to analyze data on the phenomena of social life. The construction of the series of variations is quite a feasible task for everyone. However, there are rules to remember.

How to plot a discrete variational distribution series

Example 1. There are data on the number of children in 20 families surveyed. Construct a discrete variation series distribution of families by the number of children.

0 1 2 3 1
2 1 2 1 0
4 3 2 1 1
1 0 1 0 2

Solution:

We'll start with a layout for the table, into which we will then fill in the data. Since the distribution rows have two elements, the table will consist of two columns. The first column is always an option - what we are studying - we take its name from the task (the end of the sentence with the task in the conditions) - by the number of children- so our option is the number of children.

The second column is the frequency - how often does our variant occur in the studied phenomenon - we also take the name of the column from the task - distribution of families - so our frequency is the number of families with the corresponding number of children.

Now, from the initial data, select those values that occur at least once. In our case it is

And we will arrange this data in the first column of our table in a logical order, in this case increasing from 0 to 4. We get

And in conclusion, let's count how many times each value of the options occurs.

0 1 2 3 1

2 1 2 1 0

4 3 2 1 1

1 0 1 0 2

As a result, we get a complete table or the required series of distribution of families by the number of children.

Exercise . There are data on the wage categories of 30 workers of the enterprise. Construct a discrete variation series for the distribution of workers by wage category. 2 3 2 4 4 5 5 4 6 3

1 4 4 5 5 6 4 3 2 3

4 5 4 5 5 6 6 3 3 4

How to plot an interval variation series of a distribution

Let's construct an interval distribution series, and see how its construction differs from a discrete series.

Example 2. There are data on the amount of profit received by 16 enterprises, mln. Rubles. - 23 48 57 12 118 9 16 22 27 48 56 87 45 98 88 63. Construct an interval variation series of distribution of enterprises in terms of profit, highlighting 3 groups at equal intervals.

The general principle of constructing the series, of course, will remain the same, the same two columns, the same options and frequency, but here the option will be located in the interval and the frequencies will be counted differently.

Solution:

Let's start in a similar way to the previous task by building a table layout, into which we will then enter the data. Since the distribution rows have two elements, the table will consist of two columns. The first column is always the option - what we are studying - we take its name from the task (the end of the sentence with the task in the conditions) - by the volume of profit - this means that our option is the amount of profit received.

The second column is the frequency - as our variant often occurs in the studied phenomenon - we also take the name of the column from the task - distribution of enterprises - so our frequency is the number of enterprises with the corresponding profit, in this case falling into the interval.

As a result, the layout of our table will look like this:

where i is the value or length of the interval,

Xmax and Xmin - the maximum and minimum value of the feature,

n is the required number of groups according to the problem statement.

Let's calculate the size of the interval for our example. To do this, among the initial data, we find the largest and smallest

23 48 57 12 118 9 16 22 27 48 56 87 45 98 88 63 - the maximum value is 118 million rubles, and the minimum is 9 million rubles. Let's calculate using the formula.

In the calculation, we received the number 36, (3) three in the period, in such situations the value of the interval must be rounded up to a larger one so that after the calculations the maximum data is not lost, which is why in the calculation the value of the interval is 36.4 million rubles.

Now let's build the intervals - our options in this problem. The first interval begins to build from the minimum value, the value of the interval is added to it and the upper limit of the first interval is obtained. Then the upper limit of the first interval becomes the lower limit of the second interval, the value of the interval is added to it and the second interval is obtained. And so on as many times as required to plot intervals by condition.

Let's pay attention if we had not rounded the value of the interval to 36.4, but would have left it at 36.3, then the last value would have turned out to be 117.9. In order to avoid data loss, it is necessary to round the interval to a larger value.

Let's calculate the number of enterprises that fell into each specific interval. When processing data, it should be remembered that the upper value of the interval in this interval is not taken into account (not included in this interval), but is taken into account in the next interval (the lower limit of the interval is included in given interval, and the upper one is not included), except for the last interval.

When processing data, it is best to designate the selected data conventional icons or color for easier processing.

23 48 57 12 118 9 16 22

27 48 56 87 45 98 88 63

The first interval is denoted by yellow- and determine how much data falls in the interval from 9 to 45.4, while this 45.4 will be taken into account in the second interval (provided that it is in the data) - as a result, we get 7 enterprises in the first interval. And so on at all intervals.

(additional action) Let's calculate the total profit received by enterprises for each interval and in general. To do this, add the data marked different colors and get the total value of the profit.

By the first interval - 23 + 12 + 9 + 16 + 22 + 27 + 45 = 154 million rubles.

For the second interval - 48 + 57 + 48 + 56 + 63 = 272 million rubles.

For the third interval - 118 + 87 + 98 + 88 = 391 million rubles.

Exercise . There is data on the size of the deposit in the bank of 30 depositors, thousand rubles. 150, 120, 300, 650, 1500, 900, 450, 500, 380, 440,

600, 80, 150, 180, 250, 350, 90, 470, 1100, 800,

500, 520, 480, 630, 650, 670, 220, 140, 680, 320

Build interval variation series distribution of depositors, according to the size of the contribution, highlighting 4 groups at equal intervals. For each group, count overall size deposits.

Grouping- This is the division of the population into groups that are homogeneous in some way.

Service purpose... Using the online calculator, you can:

build a variation series, build a histogram and a polygon;
find indicators of variation (mean, mode (including and graphically), median, range of variation, quartiles, deciles, quartile coefficient of differentiation, coefficient of variation and other indicators);

Instruction. To group the series, you must select the type of the resulting variation series (discrete or interval) and indicate the amount of data (number of lines). The resulting solution is saved in a Word file (see an example of grouping statistics).

If the grouping has already been carried out and given discrete variation series or interval series, then you need to use the online calculator Variation indicators. Testing the hypothesis about the type of distribution is performed using the service Study of the form of distribution.

Types of statistical groupings

Variational series... In the case of observations of a discrete random variable the same value can be found several times. Such values x i of a random variable are recorded indicating n i the number of times it appears in n observations, this is the frequency of this value.
In the case of a continuous random variable, grouping is used in practice.

Typological grouping- This is the division of the studied qualitatively heterogeneous population into classes, socio-economic types, homogeneous groups of units. To build this grouping, use the Discrete variation series parameter.
A structural grouping is called, in which a homogeneous population is divided into groups that characterize its structure according to some varying feature. To build this grouping, use the Interval series parameter.
A grouping that identifies the relationship between the phenomena under study and their features is called analytical group(see analytic grouping of the series).

Principles of building statistical groupings

A series of observations, ordered in ascending order, is called a variation series. Grouping sign is called the attribute by which the population is divided into separate groups. It is called the base of the group. The grouping can be based on both quantitative and qualitative characteristics.
After determining the basis of the grouping, it is necessary to decide the question of the number of groups into which the studied population should be divided.

When using personal computers to process statistical data, the grouping of object units is performed using standard procedures.
One of these procedures is based on the use of the Sturgess formula to determine the optimal number of groups:

k = 1 + 3.322 * log (N)

Where k is the number of groups, N is the number of units in the population.

The length of the partial intervals is calculated as h = (x max -x min) / k

Then count the number of hits of observations in these intervals, which are taken as frequencies n i. Small frequencies, the values of which are less than 5 (n i< 5), следует объединить. в этом случае надо объединить и соответствующие интервалы.
The midpoints of the intervals x i = (c i-1 + c i) / 2 are taken as new values for the variant.

The most important stage in the study of socio-economic phenomena and processes is the systematization of primary data and obtaining, on this basis, a summary characteristics of the entire object using generalizing indicators, which is achieved by summarizing and grouping primary statistical material.

Statistical summary is a complex of sequential operations to generalize specific individual facts that form a set, to identify typical features and patterns inherent in the phenomenon under study as a whole. Conducting a statistical summary includes the following steps :

selection of a grouping attribute;
determination of the order of formation of groups;
development of a system of statistical indicators to characterize groups and the object as a whole;
development of layouts of statistical tables for presenting summary results.

Statistical grouping is called the division of units of the studied population into homogeneous groups according to certain essential features for them. Groupings are the most important statistical method for summarizing statistical data, the basis for the correct calculation of statistical indicators.

There are the following types of groupings: typological, structural, analytical. All these groupings are united by the fact that the units of the object are divided into groups according to some criterion.

Grouping sign is called the attribute by which the division of the units of the population into separate groups is carried out. From the right choice the grouping attribute depends on the conclusions of the statistical study. As a basis for grouping, it is necessary to use essential, theoretically grounded features (quantitative or qualitative).

Quantitative signs of grouping have a numerical expression (trading volume, person's age, family income, etc.), and qualitative signs of grouping reflect the state of a unit of the population (gender, marital status, industry affiliation of the enterprise, its form of ownership, etc.).

After the basis of the grouping has been determined, the question of the number of groups into which the studied population must be divided should be decided. The number of groups depends on the objectives of the study and the type of indicator underlying the grouping, the size of the population, the degree of variation of the feature.

For example, the grouping of enterprises by type of ownership takes into account the municipal, federal and property of the subjects of the federation. If the grouping is carried out according to a quantitative criterion, then it is necessary to reverse Special attention on the number of units of the object under study and the degree of variability of the grouping attribute.

When the number of groups has been determined, the grouping intervals should be determined. Interval - these are the values of a variable feature that lie within certain boundaries. Each interval has its own value, upper and lower boundaries, or at least one of them.

The lower boundary of the interval is called the smallest value of the feature in the interval, and upper bound - the largest value of the feature in the interval. The interval value is the difference between the upper and lower limits.

The grouping intervals, depending on their size, are: equal and unequal. If the variation of the trait manifests itself in relatively narrow boundaries and the distribution is uniform, then a grouping is built at equal intervals. The value of the equal interval is determined by the following formula :

where Xmax, Xmin are the maximum and minimum values of the attribute in the aggregate; n is the number of groups.

The simplest grouping, in which each selected group is characterized by one indicator, is a distribution series.

Statistical series distribution - This is an ordered distribution of units of the population into groups according to a certain characteristic. Depending on the feature underlying the formation of a distribution series, attributive and variation distribution series are distinguished.

Attributive call the distribution series, built according to qualitative characteristics, that is, characteristics that do not have a numerical expression (distribution by type of labor, by sex, by profession, etc.). Attributive distribution series characterize the composition of the population for one or another essential characteristics. Taken over several periods, this data allows one to investigate the change in structure.

Variational series are called distribution series, built on a quantitative basis. Any variation series consists of two elements: options and frequencies. Variants the individual values of the attribute, which it takes in the variation series, are called, that is, the specific value of the varying attribute.

Frequencies the number of individual variants or each group of the variation series is called, that is, these are numbers that show how often certain options are found in the distribution series. The sum of all frequencies determines the size of the entire population, its volume. Frequencies called frequencies, expressed in fractions of a unit or as a percentage of the total. Accordingly, the sum of the frequencies is 1 or 100%.

Depending on the nature of the variation of the trait, three forms of the variational series are distinguished: ranked series, discrete series and interval series.

Ranked variation series - This is the distribution of individual units of the population in ascending or descending order of the trait under study. Ranking allows you to easily divide quantitative data into groups, immediately find the smallest and largest values of a feature, highlight the values that are most often repeated.

Discrete variation series characterizes the distribution of population units over discrete feature that only accepts integer values. For example, tariff category, the number of children in the family, the number of employees at the enterprise, etc.

If a feature has continuous change, which within certain limits can take on any values ("from - to"), then for this feature you need to build interval variation series ... For example, the amount of income, work experience, the cost of fixed assets of the enterprise, etc.

Examples of solving problems on the topic "Statistical summary and grouping"

Problem 1 ... There is information about the number of books received by students by subscription for the past academic year.

Construct a ranked and discrete variational distribution series, designating the elements of the series.

Solution

This set represents many options for the number of books students receive. Let us count the number of such options and arrange them in the form of variational ranged and variational discrete series distribution.

Task 2 ... There are data on the cost of fixed assets for 50 enterprises, thousand rubles.

Construct a series of distributions, highlighting 5 groups of enterprises (at equal intervals).

Solution

To solve the problem, we will choose the largest and the smallest values of the value of fixed assets of enterprises. These are 30.0 and 10.2 thousand rubles.

Let's find the size of the interval: h = (30.0-10.2): 5 = 3.96 thousand rubles.

Then the first group will include enterprises with fixed assets ranging from 10.2 thousand rubles. up to 10.2 + 3.96 = 14.16 thousand rubles. There will be 9 such enterprises. The second group will include enterprises, the size of fixed assets of which will be from 14.16 thousand rubles. up to 14.16 + 3.96 = 18.12 thousand rubles. There will be 16 such enterprises. Similarly, we will find the number of enterprises included in the third, fourth and fifth groups.

The resulting distribution series is placed in the table.

Problem 3 ... For a number of light industry enterprises, the following data were obtained:

Group the enterprises according to the number of workers, forming 6 groups at equal intervals. Count for each group:

1.number of enterprises
2.number of workers
3.volume of products produced per year
4.the average actual output of one worker
5.volume of fixed assets
6.the average size of fixed assets of one enterprise
7.the average value of products produced by one enterprise

Fill out the calculation results in tables. Draw conclusions.

Solution

For the solution, we will choose the largest and the smallest values of the average number of workers in the enterprise. These are 43 and 256.

Find the size of the interval: h = (256-43): 6 = 35.5

Then the first group will include enterprises, the average number of workers in which is from 43 to 43 + 35.5 = 78.5 people. There will be 5 such enterprises. The second group will include enterprises, the average number of workers at which will be from 78.5 to 78.5 + 35.5 = 114 people. There will be 12 such enterprises. Similarly, we will find the number of enterprises included in the third, fourth, fifth and sixth groups.

We put the resulting distribution series in a table and calculate the necessary indicators for each group:

Output : As can be seen from the table, the second group of enterprises is the most numerous. It includes 12 enterprises. The smallest are the fifth and sixth groups (two enterprises). These are the largest enterprises (in terms of the number of workers).

Since the second group is the most numerous, the volume of products produced per year by the enterprises of this group and the volume of fixed assets are significantly higher than others. At the same time, the average actual output of one worker at the enterprises of this group is not the highest. Here, enterprises of the fourth group are in the lead. This group also accounts for a fairly large amount of fixed assets.

In conclusion, we note that the average size of fixed assets and the average value of the output of one enterprise are directly proportional to the size of the enterprise (by the number of workers).