Variation indicators. When studying a varying characteristic among units of a population, one cannot limit oneself to only calculating the average value from individual variants, since the same average may not apply to populations of the same composition.
Variation of a characteristic is the difference in individual values of a characteristic within the population being studied.
The term “variation” comes from the Latin variatio – change, fluctuation, difference. However, not all differences are usually called variation.
In statistics, variation is understood as such quantitative changes in the value of the characteristic under study within a homogeneous population, which are caused by the intersecting influence of various factors. The variability of individual values is characterized by variation indicators. The greater the variation, the further apart the individual values are on average.
Variation of a trait is distinguished in absolute and relative values.
Absolute indicators include: range of variation, average linear deviation, standard deviation, dispersion. All absolute indicators have the same dimension as the quantities being studied.
Relative indicators include oscillation coefficients, linear deviation and variations.
The indicators are absolute. Let's calculate the absolute indicators characterizing the variation of the trait.
The range of variation is the difference between the maximum and minimum values of a characteristic.
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R = Xmax – Xmin. |
The range of variation indicator is not always applicable, since it takes into account only the extreme values of a characteristic, which can be very different from all other units.
It is possible to more accurately determine the variation in a series using indicators that take into account the deviations of all options from the arithmetic mean.
There are two such indicators in statistics: linear average and standard deviation.
Average linear deviation (L) represents the arithmetic mean of the absolute values of deviations of individual options from the average.
The practical use of the average linear deviation is as follows: with the help of this indicator, the composition of workers, the rhythm of production, and the uniformity of supplies of materials are analyzed.
The disadvantage of this indicator is that it complicates calculations of the probable type and complicates the use of mathematical statistics methods.
Standard deviation () is the most common and accepted measure of variation. It is slightly larger than the average linear deviation. For moderately asymmetric distributions, the following relationship between them is established
To calculate it, each deviation from the average is squared, all squares are summed up (taking into account the weight), after which the sum of squares is divided by the number of terms of the series and the square root is extracted from the quotient.
All these actions are expressed by the following formula
those. The standard deviation is the square root of the arithmetic mean of the squares of the deviations from the mean.
The standard deviation is a measure of the reliability of the mean. The smaller σ, the better the arithmetic mean reflects the entire represented population.
The arithmetic mean of the squared deviations of variant values of a characteristic from the average value is called dispersion (), which is calculated using the formulas
A distinctive feature of this indicator is that when squaring (), the proportion of small deviations decreases, and large ones increase in the total amount of deviations.
The variance has a number of properties, some of which make it easier to calculate:
1. The variance of a constant value is 0.
If , then and .
Then 
.
2. If all variants of attribute (x) values are reduced by the same number, then the variance will not decrease.
Let , but then in accordance with the properties of the arithmetic mean and .
The variance in the new series will be equal to
Those. the variance in the series is equal to the variance of the original series.
3. If all variants of attribute values are reduced by the same number of times (k times), then the variance will decrease by k2 times.
Let , then and .
The variance of the new series will be equal to
4. The variance calculated in relation to the arithmetic mean is minimal. The average square of deviations calculated with respect to an arbitrary number is greater than the variance calculated with respect to the arithmetic mean by the square of the difference between the arithmetic mean and the number, i.e. 
. The variance from the average has the property of minimality, i.e. it is always less than the variances calculated from any other quantities. In this case, when we equate to 0 and, therefore, do not calculate deviations, the formula takes the following form:
The calculation of variation indicators for quantitative characteristics was discussed above, but in economic calculations the task may be set to assess the variation of qualitative characteristics . For example, when studying the quality of manufactured products, products can be divided into high-quality and defective.
In this case we're talking about about alternative signs.
Alternative characteristics are those that some units of the population possess and others do not. For example, the presence of industrial experience among applicants, academic degree from university teachers, etc. The presence of a characteristic in population units is conventionally denoted by 1, and the absence by 0. Then, if the proportion of units possessing the characteristic (in the total number of population units) is denoted by p, and the proportion of units not possessing the characteristic by q, the variance of the alternative characteristic can be calculate by general rule. In this case, p + q = 1 and, therefore, q = 1– p.
First, we calculate the average value of the alternative attribute:
Let's calculate the average value of the alternative characteristic
,
those. the average value of an alternative characteristic is equal to the proportion of units possessing this characteristic.
The variance of the alternative characteristic will be equal to:
Thus, the variance of an alternative characteristic is equal to the product of the proportion of units possessing this characteristic by the proportion of units not possessing this characteristic.
And the standard deviation will be equal to =.
The indicators are relative. For the purpose of comparing the variability of different characteristics in the same population or when comparing the variability of the same characteristic in several populations, variation indicators expressed in relative values are of interest. The basis for comparison is the arithmetic mean. These indicators are calculated as the ratio of the range of variation, average linear deviation or standard deviation to the arithmetic mean or median.
Most often they are expressed as a percentage and determine not only a comparative assessment of variation, but also characterize the homogeneity of the population. The population is considered homogeneous if the coefficient of variation does not exceed 33%. The following relative indicators of variation are distinguished:
1. The oscillation coefficient reflects the relative fluctuation of the extreme values of a characteristic around the average.
3. The coefficient of variation evaluates the typicality of average values.
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.The smaller , the more homogeneous the population is in terms of the characteristic being studied and the more typical the average. If ≤33%, then the distribution is close to normal, and the population is considered homogeneous. From the above example, the second population is homogeneous.
Types of variances and the rule for adding variances. Along with studying the variation of a characteristic throughout the population as a whole, it is often necessary to trace quantitative changes in the characteristic in the groups into which the population is divided, as well as between groups. This study of variation is achieved through calculation and analysis various types variances.
In this case, it is possible to determine three indicators of the variability of a sign in the aggregate:
1. The general variation of an aggregate which results from the action of all causes. This variation can be measured by the total variance (), which characterizes the deviations of individual values of a population characteristic from the overall average
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.2. Variation of group averages, expressing deviations of group averages from the general average and reflecting the influence of the factor by which the grouping was made. This variation can be measured by the so-called between-group variance (δ2)
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,where are group averages, a is the overall average for the entire population, and is the number of individual groups.
3. Residual (or intragroup) variation, which is expressed in the deviation of individual values of the attribute in each group from their group average and, therefore, reflects the influence of all other factors except the one underlying the grouping. Since the variation in each group is reflected by the group variance
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,then for the entire population the residual variation will be reflected by the average of the group variances. This variance is called the average of the intragroup variances () and is calculated using the formula
This equality, which has a strictly mathematical proof, is known as the rule of adding variances.
The rule for adding variances allows you to find total variance according to its components, when the individual values of a characteristic are unknown, and only group indicators are available.
Determination coefficient. The variance addition rule allows you to identify the dependence of results on certain factors using the coefficient of determination.
It characterizes the influence of the characteristic that forms the basis of the group on the variation of the resulting characteristic. The correlation ratio varies from 0 to 1. If , then the grouping characteristic does not affect the resultant one. If , then the resulting characteristic changes only depending on the characteristic underlying the grouping, and the influence of other factorial characteristics is zero.
Indicators of asymmetry and kurtosis. In the field of economic phenomena, strictly symmetrical series are extremely rare; more often one has to deal with asymmetrical series.
In statistics, several indicators are used to characterize asymmetry. If we take into account that in a symmetric series the arithmetic mean coincides in value with the mode and median, then the simplest indicator of asymmetry () will be the difference between the arithmetic mean and the mode, i.e.
The value of kurtosis is calculated using the formula
If >0, then the kurtosis is considered positive (the distribution is peaked), if<0, то эксцесс считается отрицательным (распределение низковершинно).
The coefficient of variation, VAR or CV, is a key indicator in assessing the risk of projects and the profitability of securities. It allows you to analyze in advance two indicators that have values that change over time. If the indicator is less than 0.1, the investment direction is characterized by a low level of risk. If the indicator is above 0.3, the risk level is unreasonably high. For calculations, it is most convenient to use the STANDARDEVAL and AVERAGE functions of the Excel spreadsheet editor.
In order to form a high-quality investment portfolio, investors sometimes have to resort to evaluating the assets included in it, which have different levels of risk and return. For this purpose, an indicator widely known in investment analysis and econometrics is used.
The coefficient of variation(Coefficient of variation - CV, VAR) is a relative financial indicator that demonstrates a comparison of the dispersion of the values of two random indicators that have different units of measurement relative to the expected value.
Reference! Since the coefficient of variation allows one to obtain comparable results, its use is optimal within the framework of portfolio analysis. In it, it allows you to effectively combine the risk and return values and output the resulting value.
Coefficient of variation is an indicator from among the relative statistical methods, which, like NPV and IRR, is used as part of investment analysis. It is measured as a percentage and can be used to compare variations in two unrelated criteria. It is most often used by financial and investment analysts.
Reference! Based on the coefficient of variation, the so-called “unitized risk” is estimated, since it evaluates the relative spread of two indicators in relation to the predicted value.
What is VAR used for?
- for the purpose of comparing two different indicators;
- to determine the degree of stability of forecast models (mainly for investments and portfolio investment);
- to perform XYZ analysis.
Reference! XYZ analysis is an analytical tool within which a company’s products are assessed according to two parameters: stability of consumption and sales.
Formula for calculating the coefficient of variation
The essence of calculating the coefficient of variation is that for a set of values, first calculate the standard deviation, and then the arithmetic mean, and then find their ratio.
In general, the formula for calculating VAR is as follows:
CV = σ / t avg, where:
CV - coefficient of variation;
σ - standard deviation;
t is the arithmetic mean for the random variable.
The formula for calculating the VAR indicator can take on a wide variety of interpretations depending on the object being assessed.
Important point! It is obvious that applying the above formulas manually, especially when there is a wide range of values, is very difficult. That is why the Excel spreadsheet editor is used for calculations.
VAR values in investment analysis
There is no standard value for this indicator. However, there are some reference criteria that help in its analysis and interpretation.
Important point! The CV coefficient has several disadvantages - it does not take into account the size of the initial investment, assumes the symmetry of scattered values with respect to the average, and also cannot be used for options whose profitability may be less than 0. Therefore, if in doubt, it is worth additionally using the IRR and NPV indicators.
Examples of VAR calculation in Excel
Calculating the coefficient of variation manually is a complex and time-consuming procedure. If the sample is large, then manually calculating the standard deviation from it is extremely fraught with errors and inaccuracies.
A convenient way to determine VAR is offered by the Excel spreadsheet editor. On its basis you can calculate:
- standard deviation (STANDEVAL function);
- arithmetic mean (AVERAGE function).
In order to understand the intricacies of using CV, it makes sense to give an example of its calculation.
Calculation example: evaluation of two projects with different profits
There are two businesses that have shown different financial results over the course of 5 years. In order to make a choice between them, an investor should calculate the coefficient of variation.
First, let's calculate the standard deviation using the Excel statistical function STANDARDEV.V.
Similarly, based on the statistical function AVERAGE, the arithmetic mean is calculated for both projects
After this, it remains to divide the standard deviation by the arithmetic mean and get the result - the value of the coefficient of variation.
Conclusion! For project A, the risk level turned out to be 40%. In this situation, it seems risky and unstable. For Project B, the risk level is acceptable - only 11.64%. It is appropriate for an investor to invest in a more reliable project B, although in certain periods project A brings greater profits.
A detailed algorithm for calculating the indicator is presented in a sample based on the Excel spreadsheet editor.
The detailed process for calculating the variation index is presented in the video.
CALCULATION OF VARIATION INDICATORS
PRACTICAL WORK 3
Goal of the work: obtaining practical skills in calculating various indicators (measures) of variation depending on the objectives set by the study.
Work order:
1. Determine the type and form (simple or weighted) of variation indicators.
3. Formulate conclusions.
1. Determination of the type and form of variation indicators.
Variation indicators are divided into two groups: absolute and relative. The absolute ones include: range of variation, quartile deviation, average linear deviation, dispersion and standard deviation. Relative indicators are coefficients of oscillation, variation, relative linear deviation, relative quartile variation, etc.
Range of variation (R) is the simplest measure of variation of a trait and is determined by the following formula:
where is the highest value of the varying characteristic;
– the smallest value of the varying characteristic.
Quartile deviation (Q)– used to characterize the variation of a characteristic in the aggregate. Can be used instead of range of variation to avoid the disadvantages associated with using extreme values.
where and are the first and third quartiles of the distribution, respectively.
Quartiles– these are the values of the characteristic in the ranked series of the distribution, selected in such a way that 25% of the population units will be less in value; 25% of the units will be contained between and ; 25% of the units will be contained between and , and the remaining 25% exceed .
Quartiles 1 and 3 are determined by the formulas:
,
Where is the lower limit of the interval in which the first quartile is located;
– the sum of the accumulated frequencies of intervals preceding the interval in which the first quartile is located;
– frequency of the interval in which the first quartile is located.
where Me is the median of the series;
,
The symbols are the same as for quantities.
In symmetric or moderately asymmetric distributions Q»2/3s. Since the quartile deviation is not affected by the deviations of all values of the attribute, its use should be limited to cases where determining the standard deviation is difficult or impossible.
Average linear deviation () represents the average value of the absolute deviations of the attribute variants from their average. It can be calculated using the arithmetic mean formula, both unweighted and weighted, depending on the absence or presence of frequencies in the distribution series.
Unweighted average linear deviation,
- weighted average linear deviation.
variance()– the average square of deviations of individual values of a characteristic from their average value. The variance is calculated using the simple unweighted and weighted formulas.
- unweighted,
- weighted.
Standard deviation (s)– the most common indicator of variation, is the square root of the variance value.
The range of variation, quartile deviation, average linear and square deviations are named quantities and have the dimension of the characteristic being averaged. Dispersion has no unit of measurement.
For the purpose of comparing the variability of different characteristics in the same population or when comparing the variability of the same characteristic in several populations, relative indicators of variation are calculated. The basis for comparison is the arithmetic mean. Most often, relative indicators are expressed as percentages and characterize not only a comparative assessment of variation, but also characterize the homogeneity of the population.
Oscillation coefficient(relative range of variation) is calculated by the formula:
,
Linear coefficient of variation(relative linear deviation):
Relative quartile variation index:
or 
The coefficient of variation:
,
The most commonly used indicator of relative variability in statistics is the coefficient of variation. It is used not only for a comparative assessment of variation, but also as a characteristic of the homogeneity of the population. The greater the coefficient of variation, the greater the spread of attribute values around the average, the greater the heterogeneity of the population. There is a scale for determining the degree of homogeneity of a population depending on the values of the coefficient of variation (17; P.61).
To obtain an approximate idea of the shape of the distribution, distribution graphs (polygon and histogram) are constructed.
In the practice of statistical research one encounters a wide variety of distributions. When studying homogeneous populations, we usually deal with single-vertex distributions. Multivertex indicates the heterogeneity of the population being studied; the appearance of two or more vertices indicates the need to regroup the data in order to identify more homogeneous groups. Determining the general nature of the distribution involves assessing the degree of its homogeneity, as well as calculating indicators of asymmetry and kurtosis. Symmetrical is a distribution in which the frequencies of any two options, equally spaced on both sides of the distribution center, are equal to each other. For symmetric distributions, the arithmetic mean, mode and median are equal. In this regard, the simplest indicator asymmetry is based on the ratio of indicators of the distribution center: the greater the difference between the means, the greater the asymmetry of the series.
To characterize the asymmetry in the central part of the distribution, that is, the bulk of units, or for a comparative analysis of the degree of asymmetry of several distributions, the relative asymmetry index of K. Pearson is calculated:
The value of the As indicator can be positive and negative. A positive value of the indicator indicates the presence of right-sided asymmetry (the right branch relative to the maximum ordinate is more elongated than the left). With right-sided asymmetry, there is a relationship between the indicators of the distribution center: . A negative sign of the asymmetry index indicates the presence of left-sided asymmetry (Fig. 1). In this case, there is a relationship between the indicators of the distribution center: .
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Rice. 1. Distribution:
1 – with left-sided asymmetry; 2 – with right-sided asymmetry.
Another indicator, proposed by the Swedish mathematician Lindbergh, is calculated using the formula:
where P is the percentage of those characteristic values that exceed the arithmetic mean in value.
The most accurate and widespread indicator is based on the determination of the third-order central moment (in a symmetric distribution its value is zero):
where is the third-order central moment:
σ – standard deviation.
The use of this indicator makes it possible not only to determine the magnitude of asymmetry, but also to answer the question about the presence or absence of asymmetry in the distribution of a characteristic in the general population. An assessment of the degree of significance of this indicator is given using the mean square error, which depends on the volume of observations n and is calculated by the formula:
.
If the ratio is , the asymmetry is significant and the distribution of the trait in the population is not symmetrical. If the ratio , asymmetry is insignificant, its presence can be explained by the influence of various random circumstances.
For symmetric distributions, the indicator is calculated excess(sharpness). Lindbergh proposed the following indicator for assessing kurtosis:
,
where P is the proportion (%) of the number of options lying in the interval equal to half the standard deviation in one direction or another from the arithmetic mean.
The most accurate indicator is using the fourth order central moment:
where is the central moment of the fourth moment;
- for ungrouped data;
- for grouped data.
Figure 2 shows two distributions: one is peaked (the kurtosis value is positive), the second is flat-topped (the kurtosis value is negative). Kurtosis is the extent of the top of the empirical distribution moving up or down from the top of the normal distribution curve. In a normal distribution the ratio is .
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Rice. 2. Distribution:
1.4 – normal; 2 – pointed; 3 – flat top
The mean square error of kurtosis is calculated using the formula:
,
where n is the number of observations.
If , then the kurtosis is significant, if , then it is not significant.
Assessing the significance of the asymmetry and kurtosis indicators allows us to conclude whether this empirical study can be classified as a type of normal distribution curve.
2. Let's consider the methodology for calculating variation indices.
Relative indicators of variation - section Economy, Data on the activities of banks in one of the regions of the Russian Federation 1. Coefficient of Variation (Vσ) – Relative So far...
A population is considered qualitatively homogeneous if the coefficient of variation does not exceed 0.33 (or 33%).
Table 5.1.3.
Scale for assessing population homogeneity
In this case, the average value of the studied characteristic can be considered a typical, reliable characteristic of a statistical population.
If the coefficient of variation more than 0.33 (or 33%) then, therefore, the variation of the trait under study great, and the found average poorly represents the entire statistical population, is not its typical, reliable characteristic, and the population itself is heterogeneous in terms of the characteristics under consideration.
Similarly to the coefficient of variation, calculate other relative measures of variation, which are used less frequently in statistical practice:
2. Oscillation indicator: ; (5.1.12.)
3. Linear coefficient of variation: . (5.1.13)
Let's calculate the variation indicators for the end-to-end problem:
Table 5.1.4.
Calculation table for finding the characteristics of the distribution series
| Groups of banks by volume of loan investments, million rubles. X | Middle of the interval | Number of banks | Product of variants by frequencies | |||
| gr.4= gr.2*gr.3 | gr.6= gr.5*gr.5 | gr.7= gr.6*gr.3 | ||||
| 375,00 - 459,00 | =417 | 417*4= | 417-585= -168 | = | 28224*4= | |
| 459,00 - 543,00 | ? | ? | ? | ? | ||
| 543,00 - 627,00 | ? | ? | ? | ? | ||
| 627,00 - 711,00 | ? | ? | ? | ? | ||
| 711,00 - 795,00 | ? | ? | ? | ? | ||
| Total | ? | X | X | ? |
Calculation of the arithmetic weighted average:
Variance calculation:
σ2=
Calculation of standard deviation:
Calculation of coefficient of variation:
Conclusion. Analysis of the obtained values of indicators and σ suggests that the average volume of bank credit investments is _______? million. rub., the deviation from the average volume in one direction or another is on average _________? million. rub. (or ______?%), the most typical values of the volume of credit investments are in the range from ______________? million. rub. up to _______________? million rub. (range).(see Table 3.2.5 -_____? banks or ______?% are included in this interval).
Value V σ = ______?% _____? exceeds 33%, therefore, the variation of credit investments in the studied set of banks is insignificant and the set is qualitatively homogeneous on this basis. The discrepancy between the values of , Mo and Me is insignificant (=585 million rubles, Mo=593.40 million rubles, Me=588.818 million rubles), which confirms the conclusion about the homogeneity of the population of banks. Thus, the found average value of the volume of bank credit investments (585 million rubles) ______? is a typical, reliable characteristic of the population of banks under study.
End of work -
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Data on the activities of banks in one of the regions of the Russian Federation
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Subject, method and tasks of statistics
1.1. Subject, methods, tasks of statistics The term “statistics” comes from the Latin “status”, which came into use in Germany in the mid-18th century. Statistics was taught for the first time
Individual objects or phenomena that form a statistical population are called units of the population
For example, when conducting a census of retail equipment, the unit of observation is the retail establishment, and the unit of population is their equipment (counters, refrigeration units, etc.).
A sign is a characteristic property of the phenomenon being studied that distinguishes it from other phenomena
Different branches of statistics study different characteristics. So, for example, the object of study is an enterprise, and its characteristics are the type of product, volume of output, number of employees, etc. Or volume
The concept of stat. observations. Requirements for the information collected
Statistical observation is the initial stage of economic and statistical observation. It is a scientific and organizational work on collecting materials
Main types, forms and methods of observation
Specially organized statistical observation is the collection of information through censuses, one-time records and surveys. An example of a specially organized statistical
Observation accuracy and surveillance data control
Any statistical observation poses the task of obtaining data that would more accurately reflect reality. Deviations, or differences between calculated indicators and actual ones (true
Absolute and relative values
To characterize mass phenomena, statistics uses statistical quantities (indicators). They are divided into absolute, relative and
Each selected group is characterized by the AVERAGE value(s) of the resulting characteristic
Table 3.2.3. Analytical grouping of the relationship between credit investments and bank profits Group number Groups of banks by size of credit investments
By volume of credit investments
To construct an interval variation series characterizing the distribution of banks by volume of credit investments, it is necessary to calculate the value and boundaries of the intervals of the series.
A statistical distribution series is an ordered distribution of population units into groups according to the characteristic being studied.
Depending on the type of attribute considered as a grouping, series can be variational (quantitative) and attributive (qualitative).
Tabular and graphical presentation of statistical data
Statistical tables are a kind of statistical sentence that consists of a statistical subject and a statistical predicate. Statistical tables - uh
Or 15 16 17
4. the lack of data can be due to various reasons and this should be reflected in the tables in different ways: a) if this item cannot be filled out at all, then
Graphical representation of statistical data
The use of graphs in statistics goes back more than two centuries. The founder of the graphical method in business statistics is considered to be the English economist W. Playfair
Frequency distribution polygon
Based on the data in table. 3.4.3. Let's build a frequency polygon Table 3.4.3. Distribution of shoe sizes among male survey respondents Size No. Number
Histograms
A histogram is used to display an interval distribution series. When constructing it, the values of the intervals (
Cumulates
To depict distribution series, a cumulative curve (sum curve) is used. When constructing the cumulates of an interval variation series, variants of the series are plotted along the abscissa axis (
The essence of averages. Two forms of averages
The average value is an indicator that provides a general characteristic of a varying characteristic of a homogeneous population. Properties of the average size: 1. Average characterizes the entire scoop
Harmonic mean
Harmonic – similarity, consonance, harmonic mean is close to the arithmetic mean. Harmonic mean is used in cases where statistical information
The concept of variation. Basic indicators of variation
Variation is the difference in the individual values of a characteristic among units of the population being studied. The need to study variation is due to the fact that
Other unaccounted factors
This indicator is calculated using the formula, (5.2.1.) where yi
Volume of credit investments (our factor sign - x)
The indicator is calculated using the formula
Other unaccounted factors
(5.2.9.) Average of within-group variances (
The curve is bell shaped
2. Since the normal distribution function is even, that is, f(-t)=f(t), then the normal distribution curve is symmetrical about the maximum ordinate, equal to
Therefore, the asymmetry is left-sided
The most accurate skewness coefficient is the coefficient calculated using the third-order central moment of the distribution.
Concept of sample observation and sampling errors
Selective observation is such a non-continuous observation in which characteristics are recorded in individual units of the studied statistical population, selected using
Average and maximum sampling errors
The use of a sample observation method is always associated with establishing the degree of reliability of estimates of indicators of the general population obtained on the basis of the values so far
Determination of the sampling error for the average volume of bank credit investments and the boundaries within which the general average will be located
According to the conditions of the cross-cutting problem, the sample population includes 30 banks, the sample is 20% mechanical, therefore, the general population includes (______?)=________? banks.
The proportion of units in the sample population that have one or another given property is expressed by the formula
, (6.3.4.) where m is the number of units in the population that have
Determination of the required sample size with a given value of the permissible maximum sampling error equal to 10 million rubles
For purely random and mechanical sampling with a non-repetitive selection method, the required sample size for the average quantitative characteristic is calculated by the formula:
The concept of correlation. Types and forms of correlations
Among the many forms of connections that are quantitative in nature and studied by quantitative methods, a special place is occupied by factor connections, for the study of which cortical methods are used.
Functional connections
The relationship of the resultant characteristic Y with the factor characteristic X is called functional if each possible value xi of the characteristic X
If the model takes into account the dependence of trait Y on a number of factors, then the model has the form
(7.1.5.) A characteristic feature of stochastic connections is
Visually, one can assume the existence of a correlation
3. The correlation table is a combination of two distribution series. The rows of the table correspond to the grouping of population units according to factor characteristics
Analytical grouping method
When using the method of analytical grouping, an interval series of distribution of population units is constructed according to the factor characteristic X and for each j-th group of the series the average group is determined
Regression method for relationship analysis
The line that smoothes the empirical polyline is called the theoretical regression line of Y on X, or simply the regression line. This line is from
Method of expressing series levels
Table 8.1.2 Number of apartments built by enterprises and organizations of all forms of ownership and their average size in the Russian Federation Indicators
Average indicators in dynamics series
In table 8.2.1. data are presented that characterize the dynamics of changes in the levels of the series for individual periods of time. For a general assessment of changes in series levels for the entire period of time under consideration
Forecasting product sales volumes using the average growth rate
Forecasting the level of a series of dynamics using the average growth rate (coefficient) is carried out using the following formula:
Methods for detecting seasonal variations
In a number of cases, differences in series levels depending on the time of year are naturally repeated. The challenge is to measure such differences so that they are not accidental.
Methods for analyzing the main trend in time series
Trend is the main fairly stable tendency for the development of a phenomenon in the dynamics series, in other words, a smooth and stable change in levels (y) over time. On t
Grain production in the Russian Federation, million tons
Years t production, million tons y Average for 3 years Moving amount for 5 years, Moving average for 5 years, calculated
Individual and general indices. Problems of measuring indexed quantities in aggregate indices
Individual index - characterizes the dynamics of the level of the phenomenon being studied over time for two compared periods or expresses the ratio of individual elements of the population.
The Paasche formula is preferred when the price index is considered in a system with a turnover index and a physical volume index
Example 9.2.2. Table 9.2.3. Data on product sales in the Zvezdochka store Product Unit. change Base period O
Indices average of individual
The average index is an index calculated as the average of the individual indices. These indices are used in cases where there is no data in the source information
The trade turnover index is the product of the price index (according to Paasche) and physical volume
, let's check this:
Indices of constant and variable composition. Fixed Structure Indexes
When studying qualitative indicators, it is often necessary to consider the change in time (or space) of the AVERAGE value of the index
Index of structural changes
All indices discussed above were calculated for several goods sold in one place. Let us now consider the case when ONE product is sold in several places. Example 9.5.1.
Practical lesson
Task 01 Calculate analytical and average indicators of annual changes in the levels of the series, draw appropriate conclusions. Table 1. Sales volume by product
Average growth rate -
Years (t) Sales volume, thousand tons. Absolute growth, thousand tons Growth rate, % Growth rate, % Absolute value
Many people are faced with the variability of the characteristic being studied in individual units of the population, its fluctuation relative to a certain value, that is, its variation. This is something that should be taken into account in order to obtain the most reliable information about the progress of a particular scientific research.
Most researchers, when determining the interval of change in the value of a particular parameter, most often resort to absolute ones. Among the latter, the coefficient of variation is most widely used, which, if the value under study is characterized by a normal distribution, is a criterion for the homogeneity of the population. This indicator allows you to determine what degree of scattering the values of the parameter under study will have, regardless of the scale and unit of measurement.
The coefficient of variation can be calculated by dividing by the arithmetic mean of the variable, expressed as a percentage. The result of this calculation can fall in the range from zero to infinity, increasing as the variation of the trait increases. If the obtained value is less than 33.3%, the variation of the trait is weak. If more - strong. In the latter case, the data set under study is heterogeneous, it is considered atypical, and therefore cannot be a generalizing indicator. Therefore, for this population it is worth using other indicators.
It is worth noting that the coefficient of variation not only characterizes the homogeneity of a certain population, but is also used as a comparative assessment of it. For example, it is used if fluctuations of a particular characteristic are necessary in populations for which the calculated average value is different. In this case, the scattering of the data obtained does not allow an objective assessment of the acquired meaning. The coefficient of variation characterizes the relative variability of a variable, and therefore can be a relative measure of fluctuations in the value of the parameter being studied.
However, there are some limitations here. In particular, it is possible to assess the degree of fluctuation in parameter values only for a specific characteristic and if the population has a certain composition. Moreover, the equality of these indicators may indicate both strong and weak variation. This is the case if the signs are different or the studies are conducted on different populations. This result is formed under the influence of very objective reasons, and this should be taken into account when processing the obtained experimental data.
The coefficient of variation is widely used in various fields of science and technology. In particular, it is actively used when assessing fluctuations in parameters in economics and sociology. At the same time, the use of the coefficient becomes impossible if it is necessary to assess the variability of variables that can change their sign to the opposite one. After all, then, as a result of the calculations, incorrect values of this indicator will be obtained: either it will be very small or will have a negative sign. In the latter case, it is worth checking the correctness of the calculations performed.
Thus, we can say that the coefficient of variation is a parameter that will allow you to evaluate the degree of dispersion and relative variability of the average value. The use of this indicator allows us to identify the most significant factors, focusing on which will allow us to achieve our goals and solve the necessary problems.