Unorganized data collected in the process statistical observation, form the primary data series. With a sufficiently large volume of the population, the primary data series becomes difficult to see and its direct consideration cannot give an idea of ​​the distribution of population units according to the size of the attribute.

The first step in ordering the primary row is its ranking, i.e. the arrangement of all variants of the series (characteristic values) in ascending or descending order. Data ranking allows you to:

• immediately see the maximum and minimum value a sign in the aggregate and outline the difference between them (X max - X min);
• determine the number of repetitions of individual variants of the series (frequency).

As a result, the primary unordered data series is converted into an ordered series, which will reflect the number of repetitions of each option:

This row is called statistical series distribution. It characterizes the composition and structure of the phenomenon under study, makes it possible to judge the degree of homogeneity of the studied population, the patterns and limits of variation of the analyzed feature.

The elements of the statistical distribution series are variants X, and frequencies / (absolute value of the number repetitions d-th options).

To characterize the structure of the population, an indicator is used, which is called the frequency (4) and is determined by the formula

The following equalities follow from the definition of frequency and frequency: where N - the volume of the population.

A distribution series can be obtained as a result of grouping. Distribution series can be attribute and variation.

An attribute series is a statistical series of distribution, which is built on an attribute basis. As an example of such a series, one can consider, in particular, the distribution of workers in an enterprise's workshop by profession (Table 3.2).

Distribution of shop workers by profession

variation series is a statistical distribution series, which is built according to quantitative attribute. The variation series can be considered discrete series, if the sign on which) "it is built, respectively, is discrete. The variation distribution series can also be interval if the sign on which it is built is continuous. An example of such a series is the distribution of workers in a workshop or enterprise by skill level (Table 3.3).

Table 3.3

Distribution of shop workers by skill level

As an example of an interval distribution series, we can give an example of the distribution of enterprises by production volume (see paragraph 3.3). interval distribution in this case, it is performed in the process of constructing the corresponding analytical grouping presented in Table. 3.4.

The interval distribution series, along with the discrete distribution series, allows one to reveal and study the structure of the phenomenon under study (observation object).

Table 3.4

Distribution of enterprises by production volume

Groups of enterprises by production volume, million rubles	Quantity enterprises (frequency)	Share in the total number of enterprises		Accumulated
			/about
Xi	BUT	BUT	d,

The statistical series of distribution can be considered as an obligatory result of any statistical grouping. When constructing distribution series, the number of groups and the length of the interval are determined according to the rules used when performing statistical groupings(see paragraph 3.2).

For clarity and better understanding statistical distribution series can be presented not in tabular, but in graphical form.

Most often graphic view distribution series is used to display variational statistical distribution series.

To display a discrete series, line charts are used, which are called distribution polygons. When constructing a distribution polygon in a rectangular coordinate system, variants (values) of the analyzed feature are plotted on the x-axis. On the y-axis, the frequency of distribution of variants or trait values ​​is plotted. The expediency of displaying frequencies on the ordinate axis is explained by the following:

• this is the most convenient method for a large volume of the studied statistical population;
• this makes it possible, within the framework of one graph, to depict the statistical series of the distribution of two or more features with different number aggregate units.

The intersection of points along the abscissa axis and the ordinate axis forms broken line, which is the distribution polygon (Fig. 3.1 - based on the data in Table 3.3).

For graphical display of the interval series, as a rule, bar charts are used, which are accepted in this case called histograms.

You can build a histogram of the interval series of the distribution of enterprises in terms of production volume (see Table 3.4). The abscissa axis in this case is the segments, equal to intervals of the distribution series (in the accepted scale). Further, rectangles are built on these segments, which, in height, plotted along the y-axis, are equal to the frequency or frequency of each interval (Fig. 3.2).

Rice. 3.1.

Rice. 3.2.

To solve such problems as determining structural averages, monitoring the process of concentration of the phenomenon under study, etc., it is customary to transform the distribution series into cumulative series, which line up depending on the accumulated frequencies or frequencies. The rule for calculating the accumulation of frequencies (frequencies) of each interval of the distribution series is quite simple. The accumulation of frequencies (frequencies) is calculated as the sum of the frequency (frequency) of a given interval and the frequencies (frequencies) of all intervals preceding this interval.

As an example of constructing a cumulative series, we take the data in Table. 3.4 from the last column (see cumulative frequency s,) and build the corresponding diagram (Fig. 3.3).

When constructing cumulative series in a rectangular coordinate system, the upper boundaries of the intervals of the distribution series are plotted on the abscissa axis, and the accumulated frequencies (frequencies) that correspond to these intervals are plotted on the ordinate axis.

Rice. 3.3.

With the use of cumulates, the concentration process can be illustrated, where, along with the accumulation of frequencies (frequencies), there are distributions and sums of accumulated grouping (or other important) features of the phenomenon under study in the statistical series. Such curves, which reflect the process of concentration, are called Lorenz curves.

So, if we refer to the data in Table. 3.4 and fig. 3.3, it can be noted that the accumulated frequency of the second interval indicates that seven enterprises out of 25 produce about 19% of the total output, while each of the seven enterprises has a production volume of no more than 8.2 million rubles. and these seven enterprises account for 28% total considered enterprises.

by the most important requirement of all that can be presented to the construction of statistical distribution series, is the requirement of comparability in time and space of data on intervals. At the same time, it is quite clear that in the rows with at equal intervals this requirement is met automatically. In those distribution series, the intervals of which are not equal, it is customary to calculate the distribution density as a quotient of dividing the frequency of the interval by its length. In the graphical display of distribution series with unequal intervals, it is customary to plot the ns of the frequency (frequency) on the y-axis, and the values ​​of the distribution density.

To facilitate the construction of groupings and graphical displays of statistical series, editors can be used. spreadsheets(e.g. Excel).

• See: Makarova N. V., Trofimets V. S. Statistics in Excel. M.: Finance and statistics, 2009; and other similar publications.

They are presented in the form of distribution series and are formatted as .

A distribution series is one type of grouping.

Distribution range- represents an ordered distribution of units of the studied population into groups according to a certain varying attribute.

Depending on the trait underlying the formation of a distribution series, there are attributive and variational distribution ranks:

• attributive- call the distribution series built on qualitative grounds.
• Distribution series constructed in ascending or descending order of values ​​of a quantitative attribute are called variational.

The variation series of the distribution consists of two columns:

The first column contains quantitative values variable trait, which are called options and are marked. Discrete variant - expressed as an integer. The interval option is in the range from and to. Depending on the type of variants, it is possible to construct a discrete or interval variational series.
The second column contains number of specific option, expressed in terms of frequencies or frequencies:

Frequencies- these are absolute numbers showing how many times in the aggregate it occurs given value signs that represent . The sum of all frequencies should be equal to the number of units of the entire population.

Frequencies() are the frequencies expressed as a percentage of the total. The sum of all frequencies expressed as a percentage must be equal to 100% in fractions of one.

Graphical representation of distribution series

The distribution series are visualized using graphic images.

The distribution series are displayed as:

• Polygon
• Histograms
• Cumulates
• ogives

Polygon

When constructing a polygon on the horizontal axis (abscissa axis), the values ​​of the varying attribute are plotted, and on vertical axis(y-axis) - frequencies or frequencies.

The polygon in fig. 6.1 was built according to the micro-census of the population of Russia in 1994.

6.1. Distribution of households by size

Condition: Data are given on the distribution of 25 employees of one of the enterprises by tariff categories:
4; 2; 4; 6; 5; 6; 4; 1; 3; 1; 2; 5; 2; 6; 3; 1; 2; 3; 4; 5; 4; 6; 2; 3; 4
A task: Build a discrete variational series and depict it graphically as a distribution polygon.
Solution:
AT this example options is the wage category of the employee. To determine the frequencies, it is necessary to calculate the number of employees with the appropriate wage category.

The polygon is used for discrete variation series.

To build a distribution polygon (Fig. 1), along the abscissa (X), we plot the quantitative values ​​of the varying trait - variants, and along the ordinate - frequencies or frequencies.

If the characteristic values ​​are expressed as intervals, then such a series is called an interval series.
interval series distributions are shown graphically as a histogram, cumulate or ogive.

Statistical table

Condition: Data on the size of deposits 20 are given individuals in one bank (thousand rubles) 60; 25; 12; ten; 68; 35; 2; 17; 51; 9; 3; 130; 24; 85; 100; 152; 6; eighteen; 7; 42.
A task: Build an interval variation series with equal intervals.
Solution:

1. The initial population consists of 20 units (N = 20).
2. Using the Sturgess formula, we define required amount used groups: n=1+3.322*lg20=5
3. Let's calculate the value of the equal interval: i=(152 - 2) /5 = 30 thousand rubles
4. We divide the initial population into 5 groups with an interval of 30 thousand rubles.
5. The grouping results are presented in the table:

With such a recording of a continuous feature, when the same value occurs twice (as the upper limit of one interval and the lower limit of another interval), then this value belongs to the group where this value acts as the upper limit.

bar chart

To construct a histogram along the abscissa, indicate the values ​​of the boundaries of the intervals and, on their basis, construct rectangles whose height is proportional to the frequencies (or frequencies).

On fig. 6.2. the histogram of distribution of the population of Russia in 1997 by age groups is shown.

Rice. 6.2. Distribution of the population of Russia by age groups

Condition: The distribution of 30 employees of the company according to the size of the monthly salary is given

A task: Display the interval variation series graphically as a histogram and cumulate.
Solution:

1. The unknown border of the open (first) interval is determined by the value of the second interval: 7000 - 5000 = 2000 rubles. With the same value, we find the lower limit of the first interval: 5000 - 2000 = 3000 rubles.
2. To construct a histogram in a rectangular coordinate system, along the abscissa axis, we set aside segments whose values ​​correspond to the intervals of the variant row.
   These segments serve as the lower base, and the corresponding frequency (frequency) serves as the height of the rectangles formed.
3. Let's build a histogram:

To construct the cumulate, it is necessary to calculate the accumulated frequencies (frequencies). They are determined by successive summation of the frequencies (frequencies) of the previous intervals and are denoted by S. The accumulated frequencies show how many units of the population have a feature value no greater than the one under consideration.

Cumulate

The distribution of a trait in a variational series according to the accumulated frequencies (frequencies) is depicted using the cumulate.

Cumulate or the cumulative curve, in contrast to the polygon, is built on the accumulated frequencies or frequencies. At the same time, the values ​​of the feature are placed on the abscissa axis, and the accumulated frequencies or frequencies are placed on the ordinate axis (Fig. 6.3).

Rice. 6.3. Cumulative distribution of households by size

4. Calculate the accumulated frequencies:
The knee frequency of the first interval is calculated as follows: 0 + 4 = 4, for the second: 4 + 12 = 16; for the third: 4 + 12 + 8 = 24, etc.

When constructing the cumulate, the accumulated frequency (frequency) of the corresponding interval is assigned to its upper bound:

Ogiva

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

A variation of the cumulate is the concentration curve or Lorenz plot. To plot the concentration curve on both axes rectangular system coordinates, a scale scale is applied in percent from 0 to 100. In this case, the abscissa indicates the accumulated frequencies, and the ordinate indicates the accumulated values ​​of the share (in percent) by the volume of the feature.

The uniform distribution of the sign corresponds to the diagonal of the square on the graph (Fig. 6.4). With uneven distribution, the graph is a concave curve depending on the concentration level of the trait.

6.4. concentration curve

Theory of Statistics: Lecture Notes Burkhanova Inessa Viktorovna

1. Statistical distribution series

As a result of processing and systematization of the primary data of statistical observation, groupings are obtained, called distribution series.

Statistical distribution series represent an ordered arrangement of units of the studied population into groups according to a grouping attribute.

There are attributive and variation distribution series.

Attributive is a distribution series constructed according to qualitative features. It characterizes the composition of the population according to various essential features.

Built on a quantitative basis variation series of distribution. It consists of the frequency (number) of individual variants or each group of the variation series. These numbers show how common various options(feature values) in the distribution series. The sum of all frequencies determines the size of the entire population.

Group sizes are expressed in absolute and relative values Oh. AT absolute values expressed as the number of population units in each selected group, and in relative terms - in the form of shares, specific gravity presented as a percentage of the total.

Depending on the nature of the variation of the trait, discrete and interval variation distribution series are distinguished. In a discrete variational distribution series, the groups are composed according to a feature that varies discretely and takes only integer values.

In the interval variation series of the distribution, the grouping attribute, which forms the basis of the grouping, can take any values ​​in a certain interval.

Variation series consist of two elements: frequencies and variants.

Variant name a separate value of a variable attribute, which it takes in a distribution series.

Frequency- this is the number of individual variants or each group of the variation series. If the frequencies are expressed in fractions of a unit or as a percentage of the total, then they are called frequencies.

The rules and principles for constructing interval distribution series are built according to similar rules and principles for constructing statistical groupings. If the interval variation series of the distribution is built with equal intervals, the frequencies make it possible to judge the degree of filling the interval with population units. For comparative analysis the occupancy of the intervals determines the indicator that will characterize the distribution density.

Distribution density is the ratio of the number of population units to the width of the interval.

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The results of the summary and grouping of statistical observation materials are drawn up in the form of statistical distribution series. Statistical distribution series represent an ordered distribution of units of the studied population into groups according to a grouping (variable) attribute. They characterize the composition (structure) of the phenomenon under study, make it possible to judge the homogeneity of the population, the boundaries of its change, and the patterns of development of the observed object. Depending on the feature, the statistical distribution series are divided into:

Attributive (qualitative);

Variational (quantitative)

a) discrete;

b) interval.

Attribute distribution series

Attribute series are formed according to qualitative characteristics, which can be the position held by trade workers, profession, gender, education, etc.

Table 1 - Distribution of employees of the enterprise by education.

In this example, the grouping feature is the education of employees of the enterprise (higher, secondary). These distribution series are attributive, since the variable feature is represented not by quantitative, but by qualitative indicators. Largest number workers with secondary education (about 40%); the rest of the employees are divided into groups according to this qualitative feature: medium special education- 25%; with incomplete higher - 20%; with the highest - 15%.

Variational distribution series

Variation series are built on the basis of a quantitative grouping attribute. Variation series consist of two elements: variant and frequencies.

Option- this is a separate value of a variable attribute, which it takes in a distribution series. They can be positive or negative, absolute or relative. Frequency- this is the number of individual variants or each group of the variation series. Frequencies expressed as fractions of a unit or as a percentage of the total are called frequencies. The sum of frequencies is called the volume of the population and determines the number of elements of the entire population.

Frequencies are frequencies expressed as relative values ​​(fractions of units or percentages). The sum of the frequencies is equal to one or 100%. The replacement of frequencies by frequencies makes it possible to compare variational series with different numbers of observations.

Variation series, depending on the nature of the variation, are divided into: discrete (discontinuous) and interval (continuous). Discrete distribution series are based on discrete (discontinuous) features that have only integer values ​​(for example, the wage category of workers, the number of children in a family).

Interval distribution series are based on a continuously changing value of a feature that takes any (including fractional) quantitative expressions, i.e. the value of features in such rows is given as an interval.

Sufficient if available a large number Variants of the values ​​of the attribute, the primary series is difficult to see, and direct consideration of it does not give an idea of ​​the distribution of units according to the value of the attribute in the aggregate. Therefore, the first step in ordering the primary series is its ranking - the arrangement of all options in ascending (descending) order.

To construct a discrete series with a small number of options, all occurring variants of the attribute values ​​are written out X i, and then the frequency of repetition of the variant is calculated f i. It is customary to arrange a distribution series in the form of a table consisting of two columns (or rows), one of which presents options, and the other - frequencies.

To build a distribution series of continuously changing features, or discrete ones, presented as intervals, it is necessary to establish the optimal number of groups (intervals) into which all units of the studied population should be divided.

The results of the summary and grouping, the materials of statistical observation are drawn up in the form of distribution series and statistical tables.

Statistical distribution series is an ordered arrangement of units of the studied population into groups according to a grouping criterion. They characterize the composition, make it possible to judge the homogeneity of the population, the boundaries of its change, and the patterns of development of the observed object.

Depending on the trait underlying the distribution series, attribute and variation series are distinguished.

Options are called individual values features that he takes in the variation series, i.e. specific value of the variable attribute.

Frequencies - they call the number of individual options or each group of the variation series, i.e. this is a number that shows how often certain options occur in the distribution series.

The sum of all frequencies determines the size of the entire population or its volume. Variational series consist of two elements: variants and frequencies. Frequencies are expressed in fractions of units or as a percentage of the total (called frequencies). Accordingly, the sum of the frequencies is equal to 1 or 100%.

Depending on the nature of the variation of the trait, discrete and interval series are distinguished.

Discrete series characterize the distribution of population units according to a discrete attribute that takes only a fixed value, most often an integer.

Interval variation series are series in which variant values ​​are given as intervals.

Graphically, discrete series are represented as a distribution polygon. Interval series - in the form of a distribution histogram.

Statistical tables

The results of the summary and grouping of observation materials, as a rule, are presented in the form of statistical tables. This is the most rational way to present summary results. The value of statistical tables is that they allow you to cover the materials of the statistical summary as a whole.

By appearance statistical tables are a series of intersecting vertical and horizontal lines. Vertically - rows, horizontally - columns.

A compiled but not completed table is called a table layout. The statistical table consists of two elements: the subject and the predicate. The subject - the object of study - units of the population, which are characterized by numerical indicators. The predicate is a list of numerical indicators that characterize the object of study, i.e. subject of the table.

The name of the units or groups that form the subject is given on the left side of the table in the row headings, and the name of the indicators they characterize, i.e. predicate, at the top of the table in the headings of the column.

Depending on the construction, the subject of the statistical table is divided into three types:

1. Simple

2. Group

3. Combination

1) Simple - in the subject of which there are no groupings. By the nature of the material presented, simple tables are:

list;

· territorial;

chronological.

2) Group - in which the object under study is divided in the subject into groups according to one or another attribute.

3) Combination - tables, in the subject of which the grouping of units of the population is given according to two or more characteristics taken in combination.

When there are several indicators in the predicate, the development of the predicate can be simple and complex. A simple development of the predicate provides parallel arrangement indicators, but a complex combined one.

Statistical Graphs

Developed statistical material, located in tables, often needs to be visualized using the construction of statistical graphs.

A graph in statistics is a visual representation of statistical data using geometric lines and figures or geographic charts (cartogram).

Each graph has the following elements:

1. Graphic image - the basis of the graph - geometric signs, a set of points, lines, figures, with the help of which statistical information is displayed.

2. Printing is the place where the graphic image is located.

3. Spatial landmarks - compiled using a coordinate system.

4. Scale guidelines - depend on the scale and scale of the graph.

5. Operation of the graph is the name and corresponding fields of its individual parts.

Depending on the use of geometric symbols, graphs are divided into dot, line, strip, square and circular. Graphs are in the form of non-geometric figures, they are called curly.

Statistical graphs according to the method of construction and tasks are divided into:

1. Charts:

a) comparisons;

b) dynamics;

c) structural.

2. Statistical maps:

a) cartograms;

b) charts.

Diagram - the most common way of graphic images, used to visually compare different quantities from each other.

A chart is a graph of quantitative relationships.

Statistical maps are plots of quantitative distribution over a surface. In their main purpose, they are close to diagrams, but differ in that they are conditional representations of statistical data on a contour map.

Statistical maps show the spatial distribution or spatial distribution of statistical data.

1. Statistical maps include cartograms - this is a schematic map or plan of the area on which individual territories, depending on the value of the displayed indicator, are indicated using graphic symbols.

2. Cartograms - a combination of a cartogram with a diagram.

In special cases, when you need to depict some statistic, which is obtained by multiplying two other quantities, and they must be shown on the graph, use special graphic signs, they are called Warzal signs.

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