How to determine the base growth rate. Average indicators in dynamics series
Growth rate - relative rate of change in the level of a time series per unit of time.
Growth rate is the ratio of one level of a time series to another, taken as a basis for comparison; expressed as a percentage or growth rates.
Absolute increase - the difference between two levels of a time series, one of which (the one under study) is considered as current, the other (with which it is compared) as the base one. If each current level (yt or y(t)) is compared with the immediately preceding one (yt-1) or y(t-1)), then chain absolute increases are obtained. If the level yt is compared with the initial level of the series (y0) or another level taken as the basis of comparison (yt), then basic absolute increases are obtained. Increases are expressed either in absolute values or as a percentage, in units.
Rate of increase
TP growth rate is defined as the ratio of the absolute increase of a given level to the previous or basic one.
Rate of increase - the ratio of the growth of the indicator under study to the corresponding level of the time series taken as the basis of comparison.
Averages
Absolute value of one percent increase in Ai serves as an indirect measure of the baseline level. It represents one hundredth of the base level, but at the same time it also represents the ratio of absolute growth to the corresponding growth rate.
To characterize the dynamics of the phenomenon being studied over a long period, a group of average dynamics indicators is calculated. We can distinguish two categories of indicators in this group: a) average levels of the series; b) average indicators of changes in the levels of the series.
Average series levels are calculated depending on the type of time series.
For an interval series of dynamics of absolute indicators average level series is calculated using the simple arithmetic average formula.
Average level of moment series with unequal intervals is calculated using the weighted arithmetic mean formula, where the duration of time intervals between time points of changes in the levels of the dynamic series is taken as weights.
Average absolute increase (average growth rate) is defined as the arithmetic average of the growth rate indicators for individual periods of time.
Average growth rate calculated using the geometric mean formula from the growth coefficients for individual periods.
Average growth rate expressed as a percentage:
Average growth rate , for the calculation of which the average growth rate is initially determined, which is then reduced by 100%. It can also be determined by decreasing the average growth rate by one.
Section 7 Indexes in Statistics
7.1. The concept of statistical indices and their role in economics
Individual indices
Statistical science has in its arsenal a method that allows one to compare the indicators of a phenomenon in time and space and compare actual data with any standard, which can be a plan, forecast or some kind of standard. This is an index method that operates with relative indicators, called indices in statistics.
In the practice of statistics, indices, along with average values, are the most common statistical indicators. With their help, the development of the national economy as a whole and its individual sectors is characterized, the role of individual factors in the formation of the most important economic indicators is studied, indices are also used in international comparisons of economic indicators, determining the standard of living, monitoring business activity in the economy, etc.
Index (Latin index) is a relative value showing how many times the level of the phenomenon being studied under given conditions differs from the level of the same phenomenon under other conditions. Differences in conditions can manifest themselves in time (dynamic indices), in space (territorial indices) and in the choice of any conditional level as a basis for comparison.
According to the coverage of the elements of the population (its objects, units and their characteristics), indices are distinguished individual e (elementary) and summary (complex), which, in turn, are divided into general and group.
In statistics, an index is understood as a relative indicator that expresses the ratio of the magnitudes of a phenomenon in time, space, or a comparison of actual data with any standard.
The following tasks are solved using indexes:
measuring the dynamics of a socio-economic phenomenon over two periods of time or more;
measuring the dynamics of the average economic indicator;
measuring the ratio of indicators in different regions;
determining the degree of influence of changes in the values of some indicators on the dynamics of others.
In international practice, indices are usually denoted by the symbols i and I ( initial Latin word index). The letter “i” denotes individual (private) indices, the letter “I” denotes general indices.
In addition, certain symbols are used to denote index structure indicators:
q - quantity (volume) of any product in physical terms;
p - unit price of goods;
z is the cost per unit of production;
t is the time required to produce a unit of product;
w - production output in value terms per worker or per unit of time;
v - production output in physical terms per worker or per unit of time;
T - total time expenditure (tq) or number of workers;
pq - cost of production or turnover;
zq - production costs.
The sign at the bottom right of the symbol means the period: 0 - base; 1 - reporting.
All indices can be classified according to the following criteria:
extent of coverage of the phenomenon;
comparison base;
type of scales (co-meter);
form of construction;
object of study
composition of the phenomenon;
calculation period.
According to the degree of coverage of the phenomenon, indices are individual And summary (are common).
Individual indices serve to characterize changes in individual elements of a complex phenomenon. For example, a change in the volume of production of certain types of products (TVs, electricity, etc.), as well as the price of shares of an enterprise.
Summary (complex) indexes serve to measure a complex phenomenon, the component parts of which are directly incommensurable. For example, changes in the physical volume of products, including goods of different names, the price index of shares of regional enterprises, etc.
According to the comparison base, indices are dynamic And territorial.
Dynamic Indexes serve to characterize changes in a phenomenon over time. For example, the product price index in 1996 compared to the previous year. When calculating dynamic indices, the value of the indicator in the reporting period is compared with the value of the same indicator for the previous period, which is called the base period. Dynamic indexes can be basic or chain.
Territorial indices serve for interregional comparisons. They are usually used in international statistics.
According to the type of scales, indices come with permanent And variable scales.
According to the form of construction they distinguish aggregate And average indices . The aggregate form is the most common. Average indices are derived from aggregate indices.
Depending on the nature of the object of study, indices can be labor productivity, cost, physical volume of production, etc.
According to the composition of the phenomenon, indices are permanent (fixed) composition and variable composition.
According to the calculation period, indices are annual, quarterly, monthly, weekly.
Depending on the economic purpose, individual indices are: physical volume of production, cost, prices, labor intensity, etc.
individual index of physical volume of production shows how many times the output of any one product increased (decreased) in the reporting period compared to the base one or what percentage is the increase (decrease) in the output of the product; if you subtract 100% from the index value expressed as a percentage, the resulting value will show how much production output has increased (decreased);
individual price index characterizes the change in the price of one specific product in the current period compared to the base period;
an individual unit cost index shows the change in the cost of one specific type of product in the current period compared to the base period;
labor productivity can be measured by the quantity of products produced per unit of time (v), or the cost of working time to produce a unit of output (t); therefore, it is possible to construct an index of the quantity of products produced per unit of time;
labor productivity index based on labor costs;
the individual product cost (turnover) index reflects how many times the cost of any product has changed in the current period compared to the base period or what percentage is the increase (decrease) in the cost of the product.
The growth rate is one of the dynamic, that is, changing indicators economic system. To calculate dynamics indicators, you need to set a base level - that is, the one with which all further indicators will be compared.
In economics, the variable base principle is often used. This means that each subsequent indicator is compared with the previous one. To understand how to calculate the growth rate, you need to be able to calculate basic indicators.
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Absolute increase
First of all, we need such a concept as absolute growth. Calculating the absolute growth is quite simple: to do this, calculate the difference between the last economic indicators and previous ones.
For example, if the selected indicator in the reporting period amounted to X rubles, and in the previous reporting period Y rubles, then the absolute increase will be X-Y rubles.
Absolute growth can be positive or negative. Using this indicator, you can immediately see the increase or decrease of the selected indicator for the selected period.
Rate of increase
The growth rate indicates relative growth. This is a relative value and is calculated as a percentage or fraction, as a growth factor. In order to calculate the growth rate for a selected indicator, you need to divide the absolute growth for the selected period by the indicator for the initial period. We multiply the resulting value by 100 to obtain a percentage.
Let's look at the example already given:
- Behind reporting period revenue - X rubles, and for the previous one - Y rubles.
- The absolute increase is X-Y.
- The growth rate can now be calculated from the available data: (X-Y)/Y *100. This indicator can also be positive or negative.
To calculate the growth rate for the entire period, you need to select an initial, base level (for example, the year the company was founded). Then the absolute increase is calculated as the difference between the indicators last year and first year. By dividing this difference by the indicator for the first year, you can calculate the growth rate for the entire period.
Dynamic indicators of the economic system show its viability and profitability. One of these indicators is the growth rate, which shows the percentage of growth in indicators.
When analyzing the development of phenomena, there is often a need to give a generalized description of the intensity of development over a long period. What are average dynamics used for:
1. Average absolute increase is found by the formula:
Where n- number of periods (levels), including the base one.
2. Average growth rate is calculated using the formula for the geometric mean simple of chain growth coefficients:
, 
.
When it is necessary to calculate average growth rates for periods of different lengths (unequally spaced levels), then a geometric mean weighted by the duration of the periods is used. The weighted geometric mean formula will look like:
where t is the time interval during which this growth rate is maintained.
3. Average growth rate cannot be determined directly from successive growth rates or average absolute growth rates. To calculate it, you must first find the average growth rate and then reduce it by 100%:
Example 7.1. There is data on increases in sales volumes by month (as a percentage of the previous month): January – +4.5, February – +5.2, March – +2.4, April – -2.1.
Determine the growth and gain rates for 4 months and the monthly averages.
Solution: we have data on chain growth rates. Let's convert them into chain growth rates using the formula: T r = T r + 100%.
We get the following values: 104.5; 105.2; 102.4; 97.9
For calculations only growth factors are used: 1.045; 1.052; 1.024; 0.979.
The product of chain growth coefficients gives the base growth rate.
K = 1.045 1.052 1.024 0.979 = 1.1021
Growth rate for 4 months T r= 1.1021·100= 110.21%
Growth rate for 4 months T pr= 110,21 – 100 = +10,21%
The average growth rate is found using the simple geometric mean formula:
Average growth rate for 4 months = 1.0246·100= 102.46%
Average growth rate for 4 months = 102.46 – 100 = +2.46%
4. Average level of interval series is found by the simple arithmetic average formula if the intervals are equal, or by the weighted arithmetic average if the intervals are not equal:
, 
.
where t is the duration of the time interval.
5. Average level of the moment series of dynamics it is impossible to calculate this way, since individual levels contain elements of repeated counting.
a) Average torque level equidistant row dynamics is found using the average chronological formula:
.
Where at 1 And y n- level values at the beginning and end of the period (quarter, year).
b) Average level moment series speakers with unequally spaced levels determined by the chronological weighted average formula:
Where t- duration of the period between adjacent levels.
Example 7.2. The following data are available on production volumes for the first quarter (thousand units) - January - 67, February - 35, March - 59. Determine the average monthly production volume for the 1st quarter.
Solution: according to the conditions of the problem, we have an interval series of dynamics with equal periods. The average monthly production volume is found using the simple arithmetic average formula:
thousand pieces
Example 7.3. The following data are available on production volumes for the first half of the year (thousand tons) - the average monthly volume for the 1st quarter is 42, April - 35, May - 59, June - 61. Determine the average monthly production volume for the six months.
Solution: according to the conditions of the problem, we have an interval series of dynamics with unequal periods. The average monthly production volume is found using the weighted arithmetic average formula:
Example 7.4. The following data is available on the balance of goods in the warehouse, million rubles: 1.01 – 17; on 1.02 – 35; on 1.03 – 59; at 1.04 – 61.
Determine the average monthly balance of raw materials and materials in the enterprise's warehouse for the first quarter.
Solution: According to the conditions of the problem, we have a moment series of dynamics with equally spaced levels, therefore the average level of the series will be calculated using the average chronological formula:
million rubles
Example 7.5. The following data is available on the balance of goods in the warehouse, million rubles: 1.01.11 – 17; at 1.05 – 35; on 1.08 – 59; on 1.10 – 61, on 1.01.12 – 22.
Determine the average monthly balance of raw materials and materials in the enterprise's warehouse for the year.
Solution: According to the conditions of the problem, we have a moment series of dynamics with unequally spaced levels, therefore the average level of the series will be calculated using the chronological weighted average formula.
Topic 5. Methods for studying the dynamics of socio-economic phenomena
The concept of dynamic series, their type and main elements.
System of dynamic series characteristics.
Average levels of a series and methods of calculating them.
The concept of dynamics series, their type and main elements
To characterize and analyze socio-economic phenomena over a certain period, indicators and methods are used that characterize these processes over time (dynamics).
The process of development, movement of socio-economic phenomena over time is called dynamics.
Dynamics series are series of sequentially located statistical indicators characterizing the state and change of phenomena over time.
Any The dynamics series consists of two elements:
1) row level, which refers to the value of a statistical indicator relating to a specific point or period of time;
2) periodtime- these are moments or periods of time to which the numerical values of indicators refer (year, quarter, month, etc.).
Each series of dynamics can be presented in tabular form - in the form of pairs of values And ; and in graphical form - in the form of a line diagram.
When processing statistical data Dynamic series are used that differ in the following characteristics: time, form of presentation of levels, distance between dates or intervals.
By time differentiate moment and interval time series.
In moment series, levels express the state of a phenomenon at a critical point in time– the beginning of the month, quarter, year, etc.
For example, population size, number of employees, etc. In such series, each subsequent level fully or partially contains the value of the previous level, so it is impossible to sum the levels, since this leads to repeated counting.
In interval ones, levels reflect the state of a phenomenon over a certain period of time– day, month, year, etc. These are series of indicators of production volume, sales volume by month of the year, number of man-days worked, etc.
By form of representation of levels differentiate series of absolute, relative and average values.
By distance between dates or intervals the dynamics series are divided into series with equidistant and unequally spaced levels.
In series with equally spaced levels, the distance between dates or periods is the same, in series with equally spaced levels it is different.
Using series of dynamics in statistics, the following are solved:tasks :
Obtaining characteristics of the intensity of changes in a phenomenon over time and characteristics of individual levels;
Identification and quantitative assessment of the main long-term trend in the development of the phenomenon;
Study of periodic and seasonal fluctuations of the phenomenon;
Extrapolation and forecasting.
Processing of dynamics series is carried out in 3 stages:
1. Definition of a system for characterizing a time series;
2. Decomposition of the series into individual components;
3. Forecasting based on extrapolation.
Time series characteristics system
Time series characteristics system includes :
individual (private) characteristics;
summary (generalizing) characteristics.
Individual indicators of the intensity of changes in the phenomenon include:
- absolute increaseΔ ;
- growth rate (growth coefficient);
- growth rate;
- absolute value of one percent increase.
The first three of these characteristics can be calculated in two ways, depending on the basis of comparison used. The comparison base can be constant or variable. Accordingly, one can calculate basic or chain characteristics of a time series.
Absolute gain (Δ)– characterizes the size of the increase (decrease) in the level of the series compared to the selected base:
- chain absolute increase shows how much the value of a given level has changed compared to the previous one, that is, the increment of the level compared to the previous one:
-base absolute increase shows how much the value of a given level has changed compared to the original (initial) level:
There is a relationship between basic and chain absolute increases: the sum of all chain absolute increases is equal to the basic increase of the final level.
Growth rate (relative growth)characterizes the intensity of changes in the levels of a series (the rate of change in levels). He shows, how many times is the level of a given period higher or lower than the base level. This indicator as a relative value expressed in fractions of a unit is called growth coefficient (index); expressed as a percentage is called growth rate.
Chain growth rate shows how many times the current level is higher or lower than the previous one:
Basic growth rate shows how many times the current level is higher or lower than the initial level:
There is a relationship between the basic and chain growth rates (coefficients): the product of successive chain growth coefficients is equal to the basic growth coefficient for the entire period of time.
Growth rate there is always a positive value, the range of its permissible values is (0 - + ∞).
Rate of increasecharacterizes the relative rate of change in the level of the series per unit time. Shows by what percentage the level of a given period or point in time is above or below the base level.
Chain growth rate calculated by the formula:
It shows by what percentage the level of the current period is higher or lower than the previous level.
Base growth rate is equal to:
Base growth rate shows by what percentage the level of the current period is higher or lower than the initial level of the series.
Absolute value of one percent increaseis used to estimate the value of the resulting growth rate. It shows what absolute value corresponds to one percent of growth. The indicator is calculated according to chain characteristics:
Average levels of a series and methods of calculating them
The second part of the system of characteristics of a dynamic series consists of general characteristics, which include its average indicators:
- average row level;
- average absolute increase ;
- average growth rate (growth rate);
- average growth rate;
The calculation of the average level of a dynamics series is determined by the type of series and the size of the interval corresponding to each level. Average levelcharacterizes the most typical value of levels, the center of the series.
In interval series with equally spaced intervals the average level of the series is determined by simple arithmetic mean formula:
where is the average level of the dynamics series;
n – number of levels
In interval series with unequally spaced levels formula used arithmetic average weighted:
where is the duration of the time interval between levels.
Average level of moment series dynamics cannot be calculated this way, since individual levels contain elements of repeated counting. For moment series with equally spaced levels average level is found according to the formula of the average chronological:
Average level of moment series of dynamics with unequally spaced levels determined by the formula average chronological weighted:
Average absolute increase is a general indicator of changes in a phenomenon over time. He shows how much the level of the series changes on average per unit of time and is calculated as a simple arithmetic average of absolute chain increment indicators:
Average absolute increase can also be calculated in a basic way according to the formula :
Average growth rate (average relative increase)shows how many times the level of the dynamic series has changed on average per unit of time. This characteristic is important in identifying and describing the main long-term development trend and is used as a general indicator of the intensity of the development of a phenomenon over a long period of time.
Average growth rate by chain method calculated by the formula simple geometric mean:
where m is the number of growth coefficients,
- growth coefficients calculated by the chain method.
Basic method for calculating the average growth rate carried out according to the formula :
Average growth rate calculated by multiplying the growth factor by 100%.
Average growth rateshows by what percentage on average the level of the series changes per unit of time. It is determined based on the average growth rate.
By investing money in business development, buying shares, real estate or bonds, an entrepreneur expects to increase investments, that is, to receive growth. To figure out how to calculate growth, you will need to understand what it is. Gain is an increase in the value of fixed capital, ensuring the receipt of more funds (profit) upon its sale. Until the asset is sold, it is considered that no income has been received.
To calculate, you will need the values of the current price and the previous one. The calculation results are used to manage financial and economic activity, as well as for maintaining statistics. The growth value allows you to determine whether income, the number of clients, or any other indicator has increased or decreased during the period under review.
Types of growth
- Realized– it is received if the investment objects were sold and a profit was made on them.
- Unrealized– occurs when there are investments that are not realized, but can bring profit after sale.
Management
To calculate, you will need to set a time interval and decide on the starting (base) point. It could be the beginning of a year, a month, or another time period.
The increase can be absolute. Its value is equal to the difference between the indicators of the current and base (or previous) periods. For example, the cost of producing a unit of output at the beginning of the year was 150 rubles, and at the end - 175 rubles. The absolute increase in value was 175-150 = 25 rubles.
Growth is often considered in relative terms (growth rate). To do this, the value of the current indicator is divided by the base or previous value. For example, 175/150=1.16. This suggests that the cost of production has increased by 1.16 times. To get the percentage value, you need to multiply the result by 100%. In the example under consideration this will be 16%.
To analyze the effectiveness of activities or investments, it is necessary to determine the growth rate. To do this, determine the indicators corresponding to the starting and ending points. For example, the price of shares at the beginning of 2014 was 250 thousand rubles, and by the end of the year - 420 thousand rubles. Then the initial value is subtracted from the final indicator value (420000-250000=170000). The result must be divided by the initial value and multiplied by 100%. (170000/420000*100=40%). In the example considered, the rate of increase in the value of shares over the year was 40%.
To generalize the results over a long period (for example, several years), calculate average absolute growth. To do this, find the difference between the final and initial indicators, then it must be divided by the number of periods.
The increase may turn out to be negative. For example, if the value of shares by the end of the year was 210 thousand rubles, then the increase will be equal to:
(210000-250000)/210000*100=-19%.
Depending on the purpose of calculating absolute growth, the basic or chain methods are used. The basis of the basic method is the comparison of indicators of any period with the base one. In the chain method, current indicators are compared with previous ones.
Question: How to calculate profit growth?
Answer: Absolute indicator is the difference between the current and basic (or previous) indicators. Relative – the result of dividing the current indicator by the base (or previous) one.
Question: How to get the average monthly increase if you take into account several different periods?
Answer: For this purpose, indicators for each month are calculated separately. Then they need to be added up and divided by their number.
Question: When calculating, I received a negative value. What does it mean?
Answer: This means that the investment did not bring profit, but became unprofitable.