Rates of growth− is the ratio of the levels of a series of one period to another.
Growth rates can be calculated as base rates when all levels of the series refer to the level of the same period, taken as the base:
T R = y i /y 0 − base growth rate
and as chain ones, this is the ratio of each level of the series to the level of the previous period:
T R = y i /y i-1− chain growth rate.
Growth rates can be expressed as a ratio or percentage.
Basic growth rates characterize a continuous line of development, and chain rates characterize the intensity of development in each individual period, and the product of chain rates is equal to the base rate. And the quotient of dividing the base rates is equal to the intermediate chain rate.
8.3 Growth and growth rate. Absolute value of 1% increase.
There is a distinction between the concepts of absolute and relative growth. The absolute increase is calculated as the difference between the levels of the series and expressed in units of measurement of the indicators of the series.
If the previous level is subtracted from the subsequent level, then we have a chain absolute increase:
If the same level, the base one, is subtracted from each level, then this is the base absolute increase:
The following relationship exists between chain and basic absolute increases: the sum of successive chain increases is equal to the corresponding basic increase, which characterizes the total increase for the entire relevant period of time.
Relative score the values of absolute growth compared to the initial level give indicators of the growth rate ( T ∆ i). It is defined in two ways:
As the ratio of absolute growth (chain) to the previous level:
This is a chain growth rate.
As the ratio of the base absolute increase to the base level:
This is the base growth rate.
2 As the difference between the growth rate and one, if the growth rate is expressed by a coefficient:
T ∆ = T R-1, or
T ∆ = T R- 100 if the growth rate is expressed as a percentage.
Rate of increase shows by what percentage the size of the phenomenon increased over the period under study. If the growth rate has a minus sign, then we talk about the rate of decline.
Absolute value of 1 percent increase equal to the ratio of absolute growth (chain) to the chain growth rate, expressed as a percentage:
A i= 0.01x U i;
8.4 Calculation of average dynamics indicators
The average level of the series is called the chronological average.
Average chronological− this average value from indicators that change over time.
In an interval series with equal intervals average level series is determined by the simple arithmetic mean formula.
The average level of a series in an interval dynamics series requires that it be indicated for what period of time it was calculated (monthly average, annual average, etc.).
Example 1
Calculate the average monthly turnover for the first quarter.
Because We are given an interval series with equal intervals; we apply the simple arithmetic mean formula:
If the interval series has different intervals, then it first needs to be reduced to a series with equal intervals, and then it will be possible to use the simple arithmetic average formula.
Example 2 The following data on trade turnover are available, monetary units:
Since the indicators of moment series do not have the property of totality, the average cannot be calculated using the simple arithmetic average formula, due to the fact that the balances changed continuously throughout the month, and the data are given for a specific day.
Therefore, we will use an approximate method based on the assumption that the phenomenon under study changed uniformly throughout each month. The shorter the series interval, the less error will be made when using this assumption.
We get the formula:
This formula is used to calculate average level in moment series with equal intervals.
Example 3 There is data on the balances of building materials at the beginning of the month, den. units:
Determine the average balance for the 1st quarter.
.
If the intervals in the moment series are not equal, then the average level of the series is calculated using the formula:
where is the average level in the intervals between dates,
t- time period (series interval)
Example 4 There is data on the balances of raw materials and supplies, den. units
Find the average monthly balances of raw materials and supplies for the first half of the year.
We apply the formula:
Average absolute increase calculated in two ways:
1 As the simple arithmetic average of annual (chain) increases, i.e.
2 As the quotient of base growth divided by the number of periods:
Calculation of the average absolute value of 1% increase over several years is produced using the simple arithmetic average formula:
When calculating the average annual growth rate You cannot use a simple arithmetic average, because the sum of the annual rates will not make sense. In this case, the geometric mean is used, i.e.:
Where Tr i− annual chain growth rates;
n− number of tempos.
Since the product of chain rates is equal to the base rate, then average tempo growth can be calculated as follows:
Error: Reference source not found
When calculating using this formula, it is not necessary to know the annual growth rate. The average tempo will depend on the ratio of the initial and final levels of the series.
Example 5 The nominal wages of workers in the national economy of the Republic of Belarus are characterized by the data presented in Table 1.
Table 1 – Nominal wages of workers in the national economy of the Republic of Belarus
To analyze dynamics wages define:
average annual salary for 8 years;
annual and basic absolute increases, growth rates and wage increases;
absolute value of 1% increase;
average annual absolute growth;
average annual growth rate and average annual growth rate;
average 1% increase.
Present the results in a table and draw conclusions.
Solution
1 We determine the average annual salary using the simple arithmetic average formula
2 Annual (chain) absolute growth () is determined by the formula
where , is the value of the indicator, respectively, in the th period and the period preceding it.
For example, for 2005, thousand rubles, i.e. wages in 2005 compared to 2004 increased by 64.1 thousand rubles; for 2006 thousand R. etc.
The basic absolute increase () is determined by the formula
where , is the value of the indicator in the th and base (2004) periods, respectively.
For example, for 2005, thousand rubles; for 2006 thousand rubles, i.e. wages in 2006 compared to 2004 increased by 130.3 thousand rubles. etc.
The chain growth rate is determined by the formula
For example, for 2005, i.e. wages in 2001 compared to 2004 increased by 108.8%; for 2006, etc.
The base growth rate is determined by the formula
For example, for 2001; for 2002, i.e. wages in 2002 compared to 2000 increased by 221.2%, etc.
We find the growth rate using the formula
So, the chain growth rate
for 2005: ;
for 2006: .
Base growth rate
for 2005: ;
for 2006: .
3 The absolute value of 1% growth () is found using the formula
This indicator can also be calculated as one hundredth of the previous level:
For example, for 2005, thousand rubles; for 2006 thousand R.
Calculations of indicators for points 1, 2, 3 will be presented in Table 2
Table 2 - Indicators of wage dynamics for 2004-2011.
|
wages, |
Absolute increase, thousand rubles |
Growth rate, % |
Growth rate, % |
Absolute value of 1% increase, thousand rubles |
|||||||
|
basic |
basic |
basic | |||||||||
As the growth rate as a percentage and the corresponding growth rate. At the same time, everything is usually clear with the first, but the second often raises various questions regarding both the interpretation of the obtained value and the calculation formula itself. The time has come to figure out how these quantities differ from each other and how they need to be correctly determined.
Growth rate
This indicator is calculated in order to find out what percentage one value of a series is from another. In the role of the latter, the previous value or the basic one is most often used, that is, the one that stands at the beginning of the series under study. If the result is more than 100%, this means that there is an increase in the studied indicator, and vice versa. The calculation is very simple: just find the ratio of the value for to the value of the previous or base period of time.
Unlike the previous one, this indicator allows you to find out not by how much, but by how much the value being studied has changed. A positive value of the calculation results means that it is observed, and a negative value means the rate of decline of the value being studied in comparison with the previous or base period. How to calculate the growth rate? First, they find the ratio of the indicator under study to the base or previous one, and then subtract one from the result obtained, after which, as a rule, they multiply the total by 100 to get it as a percentage. This method is used most often, but it happens that instead actual value of the analyzed indicator, only the value of the absolute increase is known. How to calculate the growth rate in this case? Here you need to use an alternative formula. The second calculation option is to find the percentage ratio to the level in comparison with which it was calculated.
Practice
Suppose we learned that in 2010 Joint-Stock Company“Svetly Put” received a profit of 120,000 rubles, in 2011 - 110,400 rubles, and in 2012 the amount of income increased compared to 2011 by 25,000 rubles. Let's see how to calculate the growth rate and growth rate based on the available data and what can be concluded from it.
Growth rate = 110,400 / 120,000 = 0.92 or 92%.
Conclusion: In 2011, the enterprise’s profit compared to previous year amounted to 92%.
Growth rate = 110,400 / 120,000 - 1 = -0.08, or -8%.
This means that in 2011 the income of JSC “Svetly Put” decreased by 8% compared to 2010.
2. Calculation of indicators for 2012.
Growth rate = (120,000 + 25,000) / 120,000 ≈ 1.2083 or 120.83%.
This means that the profit of our company in 2012 compared to the previous year, 2011, amounted to 120.83%.
Growth rate = 25,000 / 120,000 - 1 ≈ 0.2083 or 20.83%.
Conclusion: financial results of the analyzed enterprise in 2012 turned out to be 20.83% higher than the corresponding figure for 2011.
Conclusion
After we have figured out how to calculate the growth rate and growth rate, we note that on the basis of just one indicator it is impossible to give an unambiguously correct assessment of the phenomenon under study. For example, it may well turn out that the magnitude of the absolute increase in profit increases, and the development of the enterprise slows down. Therefore, any signs of dynamics must be analyzed jointly, that is, comprehensively.
Instructions
Growth rates are expressed as percentages. If we calculate the average annual growth rate, the analyzed period under consideration is from January 1 to December 31. It coincides not only with the calendar, but also with the usually taken into account financial year. It is most convenient to take the value of the base indicator for which the growth rate will be determined as 100%. Its meaning in in absolute terms must be known by January 1st.
Determine the absolute values of the indicators at the end of each month of the year (APi). Calculate the absolute values of the increase in indicators (Pi) as the difference between two compared, one of which will be the base value of the indicators as of January 1 (To), the second - the values of the indicators at the end of each month (Pi):
APi = Po – Pi,
You should have twelve such absolute values of monthly growth, according to the number of months.
Add up all the absolute values of the increase for each month and divide the resulting amount by twelve - the number of months in a year. You will receive the average annual growth rate in absolute units (P):
P = (AP1 + AP2 + AP3 +…+ AP11 + AP12) / 12.
Determine the average annual base growth rate of KB:
Kb = P / Po, where
By - the value of the base period indicator.
Express the average annual base growth rate as a percentage and you will get the average annual growth rate (ARg):
TRsg = Kb * 100%.
Using indicators of average annual growth rates over several years, you can track the intensity of their changes over the long-term period under consideration and use the obtained values to analyze and forecast the development of the situation, industry, financial sector.
In analytical calculations, both coefficients and growth rates are equally often used. They have identical essence, but are expressed in different units of measurement.
Sources:
- business growth rate
- Let's calculate the average annual growth rate
To determine the intensity of changes in any indicators over a certain period of time, a set of characteristics is used, which are obtained by comparing several levels of indicators measured at different points on the time scale. Depending on how the measured indicators are compared with each other, the resulting characteristics are called growth coefficient, growth rate, growth rate, absolute growth or the absolute value of 1% growth.
Instructions
Determine which indicators and how should be compared with each other in order to obtain the desired value of absolute growth. Proceed from the fact that this should show the absolute rate of change of the thing under study and be calculated as the difference between the current level and the level taken as .
Subtract from the current value of the indicator under study its value measured at that point on the time scale that is taken as the base. For example, let's say that the number of workers employed in production at the beginning of the current month is 1549 people, and at the beginning of the year, which is considered the base period, it was equal to 1200 workers. In this case, for the period from the beginning of the year to the beginning of the current month it was 349 units, since 1549-1200=349.
If you need not only this indicator for one last period, but also to determine the average value of absolute growth over several periods, then you need to calculate this value for each time mark in relation to the previous one, then add the resulting values and divide them by the number of periods. For example, let’s say that we need to calculate the average value of the absolute increase in the number of people employed in production by current year. In this case, subtract the corresponding value for the beginning of January from the indicator value as of the beginning of February, then perform similar operations for the pairs March/, /March, etc. Having finished with this, add up the resulting values and divide the result by the serial number of the last month of the current year participating in the calculation.
The term " pace growth» used in industry, economics, and finance. This is a statistical quantity that allows you to analyze the dynamics of ongoing processes, the speed and intensity of the development of a particular phenomenon. For determining pace ov growth it is necessary to compare values obtained at certain intervals.
Instructions
Determine the period of time for which you need an average pace growth. Usually this period is taken calendar year or its multiple. This allows us to eliminate the influence of such factors as seasonality, caused by changing climatic conditions. In the case when the period under study is equal to a year, we speak of average annual pace Oh growth.
Analysis of the intensity of change over time is carried out using indicators obtained as a result of comparison of levels. These indicators include: absolute growth, growth rate, growth rate, absolute value of one percent. Dynamics analysis indicators can be calculated on constant and variable comparison bases. In this case, it is customary to call the level being compared the reporting level, and the level with which the comparison is being made, the base level. To calculate dynamics analysis indicators on a constant basis, each level of the series is compared with the same basic level. Either the initial level in the dynamics series, or the level from which a certain level begins, is selected as the base level. new stage development of the phenomenon. The indicators calculated in this case are called basic. To calculate dynamics analysis indicators on a variable basis, each subsequent level of the series is compared with the previous one. The dynamics analysis indicators calculated in this way are called chain The most important statistical indicator of dynamics analysis is the absolute increase (decrease), i.e. absolute change, characterizing an increase or decrease in the level of a series over a certain period of time. Absolute growth with a variable base is called growth rate.
Absolute increase:
Chain and basic absolute increases are interconnected: the sum of successive chain absolute increases is equal to the basic one, i.e. overall growth for the entire period of time
To assess the intensity, i.e. relative change in level time series for any period of time, calculated growth rate (decrease). The intensity of level changes is assessed by the ratio of the reporting level to the base level. The indicator of the intensity of change in the level of a series, expressed in fractions of a unit, is called the growth coefficient, and in percentage - the growth rate. These intensity indicators differ only in units of measurement. Growth (decrease) coefficient shows how many times the level being compared is greater than the level with which the comparison is being made (if this coefficient is greater than one) or what part (share) of the level with which the comparison is being made is the level being compared (if it is less than one). Growth rate is always a positive number.
Growth rate:
Growth rate:
Thus,
There is a relationship between chain and basic growth coefficients (if the basic coefficients are calculated in relation to the initial level of the dynamics series): the product of successive chain growth coefficients is equal to the basic growth coefficient for the entire period:
and the quotient of dividing the subsequent base growth rate by the previous one is equal to the corresponding chain growth rate.
A relative assessment of the rate of measurement of the level of a series per unit time is given by indicators of the rate of growth (decrease).Growth rate (decrease)shows by what percentage the level being compared is greater or less than the level taken as the base of comparison and is calculated as the ratio of the absolute increase to the absolute level taken as the base of comparison. The growth rate can be positive, negative or equal to zero, it is expressed as a percentage or as a fraction of a unit (growth rates).
Rate of increase:
The rate of growth (decrease) can be obtained by subtracting 100% from the growth rate expressed as a percentage:
The growth rate is obtained by subtracting one from the growth rate:
When analyzing the dynamics of development, you should also know what absolute values are hidden behind the rates of growth and gain. In order to correctly assess the value of the resulting growth rate, it is considered in comparison with the absolute growth rate. The result is expressed by an indicator called the absolute value (content) of one percent of growth and is calculated as the ratio of absolute growth to the growth rate over this period of time, %:
An example of calculating indicators of dynamics series using the basic and chain method:
- Absolute growth;
- Growth rate;
- Growth rate;
- The value is 1% increase.
Basic scheme provides for comparison of the analyzed indicator ( dynamics series level) with a similar one, relating to the same period (year). At chain analysis method Each subsequent level of the series is compared (matched) with the previous one.
|
Year |
Conditional convoy |
Production volume million rubles |
Absolute increase |
Growth rate |
Rate of increase |
Meaning 1% increase |
|||
|
bases |
chain |
bases |
chain |
bases |
chain |
P=A i/T i P=0.01Y i-1 |
|||
|
Y i -Y 0 |
Y i -Y i-1 |
Y i/Y 0 |
Y i/Y i-1 |
T=T p -100 |
|||||
|
2000 |
Y 0 |
17,6 |
|||||||
|
2001 |
Y 1 |
18,0 |
0,17 |
||||||
|
2002 |
Y 2 |
18,9 |
0,18 |
||||||
|
2003 |
Y 3 |
22,7 |
0,19 |
||||||
|
2004 |
Y 4 |
25,0 |
0,23 |
||||||
|
2005 |
Y 5 |
30,0 |
12,4 |
0,25 |
|||||
|
2006 |
Y 6 |
37,0 |
19,4 |
0,30 |
|||||
|
169,2 |
19,4 |
||||||||
Determination of annual averages using calculation formulas for the average (simple arithmetic mean, simple geometric mean).
1) Def. average annual absolute growth:
2) Def. average annual growth rate:
Either by geometric mean simple:
3) Def. average annual growth rate:
See also
Analytical indicators of changes in series levels
|
Indicator name |
Basic |
|||
|
Absolute increase |
|
|
||
|
Growth rate, % |
|
|
|
|
|
Growth rate, % |
|
|
|
|
|
Absolute value 1% increase |
|
|||
;
;
To illustrate the calculations of statistical indicators presented in Table 1.10.3, let us consider the time series of cement production in the economic region for 1991 – 2002. (Table 1.10.4.).
Absolute increase(
)
-
this is the difference between the next level of the series and the previous (or basic). If the difference between the subsequent and the previous one is chain absolute increase:
(1.10.1)
if between the subsequent and the basic, then basic:
Substituting the values of cement production from column 1 (Table 1.10.4) into formula (1.10.1), we obtain absolute chain increments (Column 2 of Table 1.10.4), into formula (1.10.2) - basic increments (Column 3 of Table .1.10.4).
Average absolute increase calculated in two ways:
1) as the simple arithmetic average of annual chain increments:
Substituting into formula (1.10.3) the values from column 2 (Table 1.10.4) into the numerator and n=11 (the number of years being compared or the number of periods) into the denominator, we get:
2) as the ratio of base growth to the number of periods:
Chain growth rate- this is the ratio of the next level to the previous one, multiplied by 100%, if the calculation is as a percentage, as in our case:
Substituting the corresponding data in column 1 of table into formula (1.10.5). 1.10.4, we obtain the values of the chain growth rate, see column 4 of the table. 1.10.4.
Baseline growth rate is the ratio of each subsequent level to one level taken as the basis of comparison:
(1.10.6)
Substituting the same data into formula (1.10.6) as in the previous one, we obtain the values of the basic growth rate, see column 5 of Table 1.10.4.
It should be noted that there is a relationship between chain and base growth rates. Knowing the basic rates, you can calculate the chain rates by dividing each subsequent basic rate by the previous one.
Average growth rate is calculated using the formula for the geometric mean of chain growth coefficients:
(1.10.7)
To do this, we convert the indicators of column 4, expressed as percentages, into coefficients, substituting them into formula (1.10.7), we get:
Average growth rate can be counted the second way, based on the final and initial levels according to the formula:
From this calculation we can conclude that the average annual growth rate for 1991-2002 was 100.75%.
Along with the growth rate, you can calculate the indicator growth rate, characterizing the relative rate of change in the level of the series per unit time. The growth rate shows by what fraction (or percentage) the level of a given period or point in time is greater (or less) than the base level.
The growth rate is the ratio of absolute growth to the level of the series taken as the base. The growth rate is a positive value if the compared level is greater than the base level, and vice versa.
Defined as the difference between the growth rate and 100%, if the growth rate is expressed as a percentage:
chain - 
(1.10.8)
basic - 
(1.10.9)
For determining chain growth rate we take the difference between the chain growth rate (column 4 of Table 1.10.4) and one hundred percent, for the base one - between the base growth rate (column 5 of Table 1.10.4) and one hundred percent.
Substituting all the relevant data into formulas (1.10.8 and 1.10.9), we obtain the values of the growth rates of chain (column 6 of table 1.10.4) and basic (column 7 of table 1.10.4).
Average annual growth rate is calculated similar to the growth rate using the formula:
Thus, cement production over the years under study increased by an average of 0.75% per year.
In statistical practice, instead of calculating and analyzing growth rates and increments, they often consider absolute value of one percent increase. It represents one hundredth of the base level and at the same time the ratio of absolute growth to the corresponding growth rate:
Substituting the data in column 1 for the previous year, divided by 100% (1942:100=19.4) into formula (1.10.10), we obtain the absolute value of 1% growth (see column 8 of table 1.10.4).
Average level series of dynamics ( 
) is calculated using the chronological average. Middle chronological is called the average, calculated from values that change over time. Such averages summarize chronological variation. The chronological average reflects the totality of the conditions in which the phenomenon under study developed in a given period of time.
Methods for calculating the average level of interval and moment time series are different. For interval equally spaced rows, the average level is found using the simple arithmetic average formula and for unequally spaced rows using the weighted arithmetic average:
(1.10.11)
(1.10.11)
Where 
- level of the dynamics series;
n - number of levels;
Thus, Table 1.10.4 shows an interval series of dynamics with equally spaced levels. Using these data, it is possible to calculate the average annual level of cement production for 1991-2002. It will be equal to:
The average level of the moment series of dynamics cannot be calculated in this way, since individual levels contain elements of repeated calculation.
The average level of a momentary equidistant series of dynamics is found using the average chronological formula:
(1.10.12)
The average level of moment series of dynamics with unequally spaced levels is determined by the average chronological weighted formula:
Where , 
- levels of dynamics series;
Duration of the time interval between levels.
Methods for aligning time series
An important task of statistics when analyzing time series is to determine the main development trend inherent in a particular time series. For example, fluctuations in the yield of an agricultural crop in individual years may not directly indicate a trend toward growth (decrease) in yield, and therefore must be identified by statistical methods.
Methods for analyzing the main trend in time series are divided into two main groups:
1) smoothing or mechanical alignment of individual members of the dynamics series using the actual values of adjacent levels;
2) alignment using a curve drawn between specific levels in such a way that it reflects the tendency inherent in the series and at the same time frees it from minor fluctuations.
Let's look at the methods of each group.
Interval enlargement method. If we consider the levels of economic indicators over short periods of time, then due to the influence of various factors acting in different directions, a decrease and increase in these levels is observed in the dynamics series. This makes it difficult to see the main trend in the development of the phenomenon being studied. In this case, to visually represent the trend, the method of enlarging intervals is used, which is based on enlarging the time periods to which the series levels relate. For example, a series of daily production output is replaced by a series of monthly production output, etc.
Simple moving average method. Smoothing a dynamic series using a moving average consists of calculating the average level from a certain number of the first in order levels of the series, then the average level from the same number of levels, starting from the second, then starting from the third, etc. Thus, when calculating the average level, they “slide” along the series of dynamics from its beginning to the end, each time discarding one level at the beginning and adding the next one. Hence the name - moving average.
The smoothed yield series for three years is shorter than the actual one by one member of the series at the beginning and at the end, for five years - by two at the beginning and at the end of the series. It is less susceptible to fluctuations due to random reasons than the actual one, and more clearly expresses the main trend of productivity growth over the period under study, associated with the action of long-term causes and conditions of development
The disadvantage of the simple moving average method is that the smoothed time series is reduced due to the impossibility of obtaining smoothed levels for the beginning and end of the series. This drawback is eliminated by using the analytical alignment method to analyze the underlying trend.
Analytical alignment involves representing the levels of a given series of dynamics as a function of time - y 
=f(t).
To display the main trend in the development of phenomena over time, various functions are used: polynomials of degree, exponentials, logistic curves and other types. The polynomials have the following form:
first degree polynomial:
polynomial of the second degree:
third degree polynomial:
polynomial of nth degree: Abstract >> Marketing
... STATISTICAL STUDYING SPEAKERS SOCIAL-ECONOMIC PHENOMENA CONCEPT AND CLASSIFICATION OF DYNAMICS SERIES Process of development, movement socially-economic phenomena... - number of elements statistical totality, variation which is free (unlimited...
Statistical studying relationships socially-economic phenomena
Coursework >> Economics... "Statistics" on the topic: " Statistical studying relationships socially-economic phenomena" Introduction The essence of studying the relationships between characteristics... () - shows which part variations result is due variation factor under study. (73%) Coefficient...
Statistical studying relationships socially-economic phenomena and processes
Textbook >> Economic and mathematical modelingAnd management" A.V. Chernova I.A. Krasnobokaya STATISTICAL STUDYING RELATIONSHIPS SOCIAL-ECONOMIC PHENOMENA AND PROCESSES Guidelines by execution... shows what part of the total variations the effective characteristic (y) is explained by the influence...
Statistical data about socially-economic phenomena and processes
Test >> SociologyEssence socially-economic phenomena and certain statistical patterns. Statistical summary... 1) selection socially-economic types phenomena; 2) studying structures phenomena and structural... by nature variations meanings of the studied...
Regression analysis in statistical studying relationship between indicators
Abstract >> MarketingTyumen, 2010 CONTENTS Introduction 3 1. Statistical studying relationships socially-economic phenomena and processes 5 2. Characteristics of regression... α and the number of degrees of freedom variations. IN socially-economic In studies, the significance level α is usually...