Statistical distribution series. Statistical discrete distribution

Let out population a sample is extracted, and X 1 observed P 1 time, X 2 - P 2 times, x k - p to times and is the sample size. Observed values X 1 are called variants, and the sequence of variants is written in ascending order - variation series .

The number of observation variants is called frequency, and its relation to the sample size is called relative frequency.

Definition. Statistical (empirical) law of sample distribution, or simply statistical distribution of the sample name the sequence of options and their corresponding frequencies n i or relative frequencies.

Statistical distribution It is convenient to present samples in the form of a table of frequency distributions, called statistical discrete series distributions:

(the sum of all relative frequencies is equal to one).

Example 1. When measuring in homogeneous groups of subjects, the following samples were obtained: 71, 72, 74, 70, 70, 72, 71, 74, 71, 72, 71, 73, 72, 72, 72, 74, 72, 73, 72.74 ( heart rate). Based on these results, compile a statistical series of frequency distributions and relative frequencies.

Solution. 1) Statistical series of frequency distribution:

Control: 0.1 + 0.2 + 0.4 + 0.1 + 0.2 = 1.

Frequency polygon called a broken line, segments that connect points To construct a frequency polygon, options are laid out on the abscissa axis X 2, and on the ordinate - the corresponding frequencies p i . The points are connected by segments and a frequency polygon is obtained.

Polygon of relative frequencies called a broken line, segments that connect points. To construct a polygon of relative frequencies, options are plotted on the abscissa axis X i , and on the ordinate axis the corresponding frequencies w i. The points are connected by segments and a polygon of relative frequencies is obtained

Example 2. Construct a frequency polygon and a relative frequency polygon based on the data in Example 1.

Solution: Using the discrete statistical distribution series compiled in example 1, we will construct a frequency polygon and a relative frequency polygon:

2. Statistical interval distribution series. bar chart.

A statistical discrete series (or empirical distribution function) is usually used when great friend from each other there are not too many options in the sample, or when discreteness for one reason or another is significant for the researcher. If the characteristic of the general population X that interests us is distributed continuously or its discreteness is impractical (or impossible) to take into account, then the options are grouped into intervals.

The statistical distribution can also be specified as a sequence of intervals and the frequencies corresponding to them (the sum of frequencies falling within this interval is taken as the frequency corresponding to the interval).

1. R(span) = X max -X ​​min

2. k- number of groups

3. (Sturges formula)

4. a = x min, b = x max

It is convenient to present the resulting grouping in the form of a frequency table, which is called statistical interval distribution series:

Intervals factions			...
Frequencies			...

An analogous table can be formed by replacing frequencies n i relative frequencies.

Sample obtained during experimental research, is an unordered set of numbers written in the sequence in which the measurements were made. Typically, the sample is drawn up in the form of a table, the first row (or column) of which contains the experiment number i, and in the second (second) - the fixed value random variable sign. In this form, the sample represents the primary form of recording statistical material that can be processed different ways. As an example, consider the results shown at athletics competitions by shot putters and shown in Table 1. The first line of this table contains the numbers of measurements, and the second - their numerical values ​​in meters.

Table 1

Shot put competition results

№
x i	16,36	14,91	15,31	14,26	14,77	13,88	14,97	14,01	14,07	14,48

№
x i	14,44	14,81	13,81	15,15	15,23	15,69	14,29	14,15	14,57	13,92

№
x i	13,62	14,92	15,73	13,22	14,65	14,8	13,04	15,1	13,3

As can be seen from Table 1, a simple statistical aggregate ceases to be a convenient form of presenting statistical material even with a relatively small sample size: it is quite cumbersome and not very visual. It is very difficult to analyze the experimental data obtained, much less draw any conclusions based on them. Based on this, the resulting statistical material must be processed for further research. The simplest way to process a sample is ranking. Ranking is the arrangement of options in ascending or descending order of their values. Table 2 below shows a ranked sample, the elements of which are arranged in ascending order.

table 2

Ranked competition results in shot put

№
x i	13,04	13,22	13,3	13,62	13,81	13,88	13,92	14,01	14,07	14,15

№
x i	14,26	14,29	14,44	14,48	14,57	14,65	14,77	14,8	14,81	14,91

№
x i	14,92	14,97	15,1	15,15	15,23	15,31	15,69	15,73	16,36

But even in this form, the experimental data obtained are poorly observable and are of little use for direct analysis. That is why, in order to make the statistical material more compact and clear, it must be subjected to further processing - a so-called statistical series is constructed. The construction of a statistical series begins with grouping.

Grouping is the process of organizing and systematizing data obtained during an experiment, aimed at extracting the information contained in them. In the process of grouping, the sample is distributed into groups or grouping intervals, each of which contains a certain range of values ​​of the characteristic being studied. The grouping process begins by dividing the entire range of variation of a characteristic into grouping intervals.

For each specific purpose statistical research, the volume of the sample under consideration and the degree of variation of the characteristic in it, there is an optimal value for the number of intervals and the width of each of them. Approximate value optimal number of intervals k can be determined based on sample size P either using the data given in Table 3, or using the Sturgess formula:

k = 1 + 3.322 lg n.

Table 3

Determining the number of grouping intervals

The value obtained from the formula k almost always turns out to be a fractional value that must be rounded to a whole number, since the number of intervals cannot be fractional. Practice shows that, as a rule, it is better to round down, because the formula gives good results at large values n, and when small - somewhat overestimated.

Consider grouping the sample option into specific example. To do this, let's look at the example of shot putters (see tables 1, 2). We will determine the number of grouping intervals based on the data given in Table 3. With a sample size n=29 it is advisable to choose the number of intervals equal to k=5 (Sturgess formula gives the value k =5,9).

Let us agree to use intervals of equal width in the example under consideration. In this case, after the number of grouping intervals is determined, the width of each of them should be calculated using the relationship:

Here h- the width of the intervals, and X max and X min - respectively the maximum and minimum value of the attribute in the sample. Quantities X max and X min are determined directly from the source data table (see Table 2). In this case:

(m).

Here it is necessary to dwell on the accuracy of determining the width of the interval. Two situations are possible: the accuracy of the calculated value h matches the accuracy of the experiment or exceeds it. IN the latter case It is possible to use two approaches to determine the boundaries of the intervals. WITH theoretical point view it is most correct to use the obtained value h to construct intervals. This approach will not introduce additional distortions associated with the processing of experimental data. However, for practical purposes in statistical studies related to physical culture and sports, it is customary to round the resulting value h to the accuracy of data measurement. This is due to the fact that for a visual presentation of the results obtained, it is convenient for the boundaries of the intervals to be the possible values ​​of the attribute. Thus, the resulting value of the interval width should be rounded taking into account the accuracy of the experiment. We especially note that rounding must be done not in the generally accepted mathematical sense, but upward, i.e. in excess, so as not to reduce the overall range of variation of the characteristic - the sum of the width of all intervals should not be less than the difference between the maximum and minimum values ​​of the characteristic. In the example under consideration, the experimental data are determined to the nearest hundredth (0.01 m), therefore the value of the interval width obtained above should be rounded up to the nearest hundredth. As a result we get:

h= 0.67 (m).

After determining the width of the grouping intervals, their boundaries must be determined. It is advisable to take the lower limit of the first interval equal to the minimum value of the attribute in the sample x min:

x H1 = x min.

In the example under consideration x H1 = 13.04 (m).

To obtain the upper limit of the first interval ( x B1) you should add the value of the interval width to the value of the lower boundary of the first interval:

x B1 = X H1 + h.

Note that the upper limit of each interval (here, the first) will simultaneously be the lower limit of the next one (in in this case second) interval: x H2 = x IN 1 .

The values ​​of the lower and upper boundaries of all remaining intervals are determined in a similar way:

x B i = x N i +1 = x N i + h.

In this example:

x B1 = x H2 = x H1 + h=13.04+0.67=13.71 (m),

x B2 = x H3 = x H2+ h=13.71+0.67=14.38 (m),

x B3 = x H4 = x H3+ h=14.38+0.67=15.05 (m),

x B4 = x H5 = x H4 + h=15.05+0.67=15.72 (m),

x B5 = x H5+ h=15.72+0.67=16.39 (m).

Before grouping the option, we introduce the concept median value of the interval x i, equal to the value feature equidistant from the ends of this interval. Considering that it is spaced from the lower boundary by an amount equal to half the width of the interval, to determine it it is convenient to use the relation:

x i=x N i+ h/2,

Where x N i - lower limit i-ro interval, and h- its width. The median values ​​of the intervals will be used later when processing grouped data.

After determining the boundaries of all intervals, the sample options should be distributed across these intervals. But first you need to decide to which interval to include a value located exactly on the border of two intervals, that is, when the value of the options coincides with the upper limit of one and the lower limit of the interval adjacent to it. In this case, the option can be assigned to any of the two adjacent intervals and, to eliminate ambiguity in grouping, we agree in such cases to assign the options to the upper interval. The following argument can be made in favor of this approach. Since the minimum value of the attribute coincides with the lower limit of the first interval and is included in this interval, then the option that falls on the boundary of two intervals should be classified as one of them, the value of the lower limit of which is equal to the option under consideration.

Let's move on to consider the statistical table - see table 4, which consists of seven columns.

Table 4

Table view shot put results

The first three columns of the statistical table contain, respectively, the numbers of grouping intervals i, their boundaries x N i - x IN i and median values ​​of intervals x i .

The fourth column contains the frequencies of the intervals. Frequency interval is a number showing how many options there are, i.e. measurement results fell within this interval. To denote this quantity it is customary to use the symbol n i. The sum of all frequencies of all intervals is always equal to the sample size P, which can be used to check the correctness of the grouping.

The fifth column of table 4 is intended for entering into it accumulated frequency interval - a number obtained by summing the frequency of the current interval with the frequencies of all previous intervals. The accumulated frequency is usually denoted Latin letter N i. The accumulated frequency shows how many options have values ​​no greater than the upper limit of the interval.

The sixth column of the table contains frequency. Frequency is called a frequency presented in relative terms, i.e. ratio of frequency to sample size. The sum of all frequencies is always equal to 1. The symbol is used to indicate frequency f i:

f i=n i /n.

The frequency of an interval is related to the probability of a random variable falling into this interval. According to Bernoulli's theorem, with an unlimited increase in the number of experiments, the frequency of an event converges in probability to its probability. If we understand by an event that the value of the studied quantity falls into a certain interval, then it becomes clear that when large number experiments, the frequency of the interval approaches the probability of the measured random variable falling into this interval.

Both frequency and frequency describe the repeatability of results in a sample. Comparing them statistical significance, it should be noted that the information content of frequency is significantly higher than that of frequency. Indeed, if, as, for example, in Table 4, the frequency of the second interval is 8 and, therefore, 8 results fell into this interval, then it is difficult to understand whether this is a little or a lot; if there are 1000 variants in the sample, then this frequency is small, and if there are 20, then it is large. In this case, for objective assessment it is necessary to compare the frequency value with the sample size. If you use frequency, you can immediately tell what proportion of the results fell within the interval under consideration (approximately 28% in the example given). Therefore, frequency gives more visual representation about the repeatability of a characteristic in a sample. Another important advantage of frequency should be especially noted. Its use makes it possible to compare samples of different sizes. Frequency is not applicable for such purposes.

The seventh column of the table contains the accumulated frequency. Cumulative frequency is the ratio of the accumulated frequency to the sample size. The accumulated frequency is indicated by the letter F i:

The accumulated frequency shows what proportion of the sample variant has values ​​that do not exceed the value of the upper limit of the interval.

The last row of the statistical table is used to control the grouping.

After filling out the table, let's return to defining the statistical series. As a rule, a statistical series is presented in the form of a table, the first line of which lists the intervals, and the second line lists the frequencies or frequencies corresponding to them. Thus, statistically close called double number series, establishing a connection between the numerical value of the characteristic under study and its frequency in the sample. A significant advantage of statistical series is that they, unlike statistical aggregates, give a clear idea of characteristic features variation of signs.

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The simplest way to summarize statistical material is to construct series. The summary result of a statistical study can be distribution series.

After determining the grouping characteristic, the number of groups and grouping intervals, the summary and grouping data are presented in the form of distribution series and presented in the form of statistical tables.

A distribution series is one of the types of groupings.

Near distribution in statistics, an ordered distribution of population units into groups according to any one characteristic is called: qualitative or quantitative.

1. Types of distribution series

Depending on the characteristic underlying the formation of the distribution series, attributive and variation series distributions:

distribution series constructed according to qualitative characteristics are called attributive;

Variational series are distribution series constructed in ascending or descending order of the values ​​of a quantitative characteristic.

The variation series of the distribution consists of two columns. The first column provides quantitative values ​​of the varying characteristic, which are called variants and are designated. Discrete option - expressed as an integer. The interval option ranges from and to. Depending on the type of options, you can construct a discrete or interval variation series. The second column contains the number of specific options, expressed in terms of frequencies or frequencies:

frequencies are absolute numbers, showing the number of times a given attribute value occurs in total; the sum of all frequencies must be equal to the number of units in the entire population;

frequencies are frequencies expressed as a percentage of the total; the sum of all frequencies expressed as percentages must be equal to 100% in fractions of one.

Variation series characterized by two elements: variant (X) and frequency (f). A variant is a separate value of a characteristic of an individual unit or group of a population. A number showing how many times a particular value of a characteristic occurs is called frequency. If frequency is expressed as a relative number, then it is called frequency.

The variation series can be:

interval, when the boundaries “from” and “to” are defined, interval distribution series can be represented graphically in the form of a histogram;

discrete when the characteristic being studied is characterized by a certain number.

1. Graphic representation of distribution series

The distribution series are visually presented using graphical images.

The distribution series are depicted as:

landfill;

histograms;

cumulates;

When building testing ground on the horizontal axis (abscissa axis) the values ​​of the varying characteristic are plotted, and on vertical axis(y-axis) - frequencies or frequencies.

For building histograms The values ​​of the boundaries of the intervals are indicated along the abscissa axis and rectangles are constructed on their basis, the height of which is proportional to the frequencies (or frequencies).

The distribution of a characteristic in a variation series over accumulated frequencies (frequencies) is depicted using a cumulate.

Cumulates or a cumulative curve, unlike a polygon, is constructed from accumulated frequencies or frequencies. In this case, the values ​​of the characteristic are placed on the abscissa axis, and accumulated frequencies or frequencies are placed on the ordinate axis.

Ogiva is constructed similarly to the cumulate with the only difference being that the accumulated frequencies are placed on the abscissa axis, and the characteristic values ​​are placed on the ordinate axis.

A type of cumulate is a concentration curve or Lorentz plot. To construct a concentration curve, a scale scale in percentages from 0 to 100 is plotted on both axes of the rectangular coordinate system. At the same time, the accumulated frequencies are indicated on the abscissa axis, and the accumulated values ​​of the share (in percent) by volume of the characteristic are indicated on the ordinate axis.

Let us assume that as a result of measurements of the parameters of the objects under study, there is a statistical population representing a set of values ​​of SV X obtained as a result of measurements (observations).

The histogram is constructed in the following order.

1. The entire measurement range of SV () is divided into intervals and the number of values ​​falling on each interval is calculated. This number is divided by total measurements (products) and the frequency corresponding to this interval is determined.

The sum of the frequencies of all digits must obviously be equal to one.

2. Table 1.1 is constructed, which shows the intervals in the order of their location along the abscissa axis and the corresponding frequencies. This table is called statistically close.

Table 1.1

Statistical series of SV values

Interval,			…		…
Number of values			…		…
Frequency,			…		…

Here is the designation of the i-th interval; - its boundaries; k is the number of intervals.

When grouping observed SW values ​​into intervals, a situation may arise in which the value falls on the boundary of the interval. In this case, the question arises as to which category this value should be assigned to. It is recommended to count given value belonging to equally both intervals and add 0.5 to the numbers of both intervals.

3. Determination of the number of intervals.

The number of intervals into which a statistical series should be grouped should not be too large, since in this case the distribution series becomes inexpressive, and the frequencies in it exhibit irregular fluctuations. On the other hand, it should not be too small, since with a small number of intervals the distribution properties are described by the statistical series too roughly.

Practice shows that in most cases it is rational to choose the number of intervals within 10¸20. The larger and more homogeneous the statistical material, the more large quantity intervals can be selected when compiling a statistical series.

You can also use empirical formulas, offered by various authors. In this work, it is proposed to use as such formulas the following expressions

These expressions are obtained for the most common distributions in practice with kurtosis ranging from 1.8 to 6, that is, from uniform to Laplace distribution.

The lengths of the intervals can be either the same or different. Obviously, it’s easier to take them the same. However, when preparing data on CBs that are distributed too unevenly, it is sometimes convenient to select in the area highest density distribution intervals are narrower than in the low-density region.

4. Design of the histogram graphically.

The statistical series is presented graphically in the form of the so-called histograms(Fig. 1.1). It is constructed as follows. Intervals are plotted along the abscissa axis, and on each of the intervals a rectangle is constructed as a base, the area of ​​which is equal to the frequency of the given interval. To construct a histogram, you need to divide the frequency of each interval by its length and take the resulting number as the height of the rectangle. In the case of intervals of equal length, the heights of the rectangles are proportional to the corresponding frequencies. From the method of constructing the histogram it follows that total area its equal to one.

Obviously, as the number of experiments increases, smaller and smaller intervals can be chosen, and at the same time, the top of the histogram will become increasingly closer to the curve limiting the area, equal to one. This curve is a graph probability density function f(x) ( differential function distribution for continuous CB ).

5. Statistical distribution function .

Using the data of a statistical series, it is possible to construct statistical (empirical) distribution function SV X. To do this, points x i of the boundaries of the intervals and the corresponding sums of frequencies pi corresponding to the histogram rectangles lying to the left of these points are taken from the series. These frequencies and their sums are denoted as F(x i). Then we obtain a system of expressions that determine the points of the statistical distribution function. By connecting them with a broken line or a smooth curve, we obtain an approximate graph of the statistical distribution function ( cumulative distribution function for continuous CB ) F(x) (Fig. 1.2).

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

Statistical summary - it's a complex sequential operations by generalizing specific individual facts that form a set in order to identify typical features and patterns inherent in the phenomenon being studied as a whole. Conducting a statistical summary includes the following steps :

• selection of grouping characteristics;
• determining the order of group formation;
• system development statistical indicators to characterize groups and the object as a whole;
• development of statistical table layouts to present summary results.

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

Distinguish the following types 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 characteristic.

Grouping feature is a characteristic by which the units of a population are divided into separate groups. From the right choice The grouping characteristic determines the conclusions of the statistical study. As a basis for grouping, it is necessary to use significant, theoretically based characteristics (quantitative or qualitative).

Quantitative characteristics of grouping have a numerical expression (trading volume, person’s age, family income, etc.), and qualitative signs of grouping reflect the state of a population unit (gender, Family 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 population under study should be divided must be decided. The number of groups depends on the objectives of the study and the type of indicator underlying the grouping, the volume of the population, and the degree of variation of the characteristic.

For example, the grouping of enterprises by type of ownership takes into account municipal, federal and federal subject property. If grouping is done by quantitative characteristic, 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 characteristic.

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

Lower limit of the interval is called the smallest value of the characteristic in the interval, and upper limit - the highest value of the characteristic in the interval. The value of the interval is the difference between the upper and lower boundaries.

Grouping intervals, depending on their size, are: equal and unequal. If the variation of a characteristic manifests itself within relatively narrow boundaries and the distribution is uniform, then a group is built at equal intervals. Magnitude equal interval determined by the following formula :

where Xmax, Xmin - maximum and minimum value characteristics in the aggregate; n - number of groups.

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

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

Attributive are called distribution series constructed according to qualitative features, that is, signs that do not have numerical expression(distribution by type of labor, by gender, by profession, etc.). Attribute series distributions characterize the composition of the population according to certain essential characteristics. Taken over several periods, these data make it possible to study changes in structure.

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

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

Depending on the nature of the variation of a characteristic, three forms of variation 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 characteristic being studied. Ranking allows you to easily divide quantitative data into groups, immediately detect the smallest and highest value characteristic, highlight the values ​​that are most often repeated.

Discrete variation series characterizes the distribution of population units according to a discrete characteristic that takes only integer values. For example, tariff category, number of children in the family, number of employees in the enterprise, etc.

If a characteristic has a continuous change, which within certain limits can take any values ​​(“from - to”), then for this characteristic it is necessary to build interval variation series . For example, the amount of income, length of service, 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 students received through subscriptions over the past academic year.

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

Solution

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

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

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

Solution

To solve, we choose the largest and smallest value 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 whose fixed assets amount 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 whose fixed assets amount from 14.16 thousand rubles. up to 14.16+3.96=18.12 thousand rubles. There will be 16 such enterprises. Similarly let's find the number enterprises included in the third, fourth and fifth groups.

We place the resulting distribution series in the table.

Problem 3 . The following data was obtained for a number of light industry enterprises:

Group the enterprises by the number of workers, forming 6 groups at equal intervals. Calculate for each group:

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

Present the calculation results in tables. Draw conclusions.

Solution

To solve, we will select the largest and smallest values ​​of the average number of workers at the enterprise. These are 43 and 256.

Let's find the size of the interval: h = (256-43):6 = 35.5

Then the first group will include enterprises whose average number of workers is from 43 to 43 + 35.5 = 78.5 people. There will be 5 such enterprises. The second group will include enterprises whose average number of workers 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 place the resulting distribution series in a table and calculate the necessary indicators for each group:

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

Since the second group is the largest, the volume of products produced per year by enterprises of this group and the volume of fixed assets are significantly higher than others. At the same time, the average actual output per worker at enterprises in this group is not the greatest. Enterprises of the fourth group are leading here. This group also accounts for a fairly large volume of fixed assets.

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