Growth rate reduction formula between negative indicator. Task: Determine absolute growth using basic and chain methods

Task

The following data is available:

Determine using the basic and chain methods:

• Absolute increase;
• Growth rate (%);
• Growth rate (%);
• Average annual growth rate.

Provide calculations of all indicators, summarize the calculation results in a table. Draw conclusions by describing each indicator in the table in comparison with the previous and baseline indicators. The result of this work is a detailed conclusion.

Computations

1. Absolute increase (decrease) (A pr)

• Absolute increase (decrease) in a “chain” way.

If we determine the absolute increase (decrease) in the presence of flower beds in the city of Arkhangelsk each time compared to the previous year, then it will be:

In 1991: 17159 - 16226 = 933 units.

In 1992: 15833 - 17159 = - 1326 units.

In 1993: 11455 - 15833 = - 4378 units.

In 1994: 12668 - 11455 = 1213 units.

In 1995: 13126 - 12668 = 458 units.

In 1996: 14553 - 13126 = 1427 units.

In 1997: 14120 - 14553 = - 433 units.

In 1998: 15663 - 14120 = 1543 units.

In 1999: 17290 - 15663 = 1627 units.

In 2000: 18115 - 17290 = 825 units

In 2001: 19220 - 18115 = 1105 units.

• Absolute increase (decrease) in the “basic” way.

If we take 1990 as the base of comparison, then in relation to it the absolute increase (decrease) in the presence of flower beds in the city of Arkhangelsk in subsequent years will be:

In 1991: 17159-16226 = 933 units.

In 1992: 15833 - 16226 = - 393 units.

In 1993: 11455 - 16226 = - 4771 units.

In 1994: 12668 - 16226 = 3558 units.

In 1995: 13126 - 16226 = - 3100 units.

In 1996: 14553 - 16226 = - 1673 units.

In 1997: 14120 - 16226 = - 2106 units.

In 1998: 15663 - 16226 = - 563 units.

In 1999: 17290 - 16226 = 1064 units.

In 2000: 18115 - 16226 = 1889 units

In 2001: 19220 - 16226 = 2994 units.

1. Growth (decrease) rate (T r)

• The rate of growth (decrease) in a “chain” way.

If we determine the rate of growth (decrease) in the presence of flower beds in the city of Arkhangelsk each time compared to the previous year, then it will be:

In 1992: 15833 / 17159 * 100% = 92.3 (%)

In 1993: 11455 / 15833 * 100% = 72.3 (%)

In 1994: 12668 / 11455 * 100% = 110.6 (%)

In 1995: 13126 / 12668 * 100% = 103.6 (%)

In 1996: 14553 / 13126 * 100% = 110.8 (%)

In 1997: 14120 / 14553 * 100% = 97.0 (%)

In 1998: 15663 / 14120 * 100% = 110.9 (%)

In 1999: 17290 / 15663 * 100% = 110.4 (%)

In 2000: 18115 / 17290 * 100% = 104.8 (%)

In 2001: 19220 / 18115 * 100% = 106.1 (%)

• Growth (decrease) rate in a “basic” way.

If we take 1990 as the base of comparison, then in relation to it the rate of growth (decrease) in the presence of flower beds in the city of Arkhangelsk in subsequent years will be:

In 1991: 17159 / 16226 * 100% = 105.7(%)

In 1992: 15833 / 16226 * 100% = 97.6 (%)

In 1993: 11455 / 16226 * 100% = 70.6 (%)

In 1994: 12668 / 16226 * 100% = 78.0 (%)

In 1995: 13126 / 16226 * 100% = 80.9 (%)

In 1996: 14553 / 16226 * 100% = 89.7 (%)

In 1997: 14120 / 16226 * 100% = 87.0 (%)

In 1998: 15663 / 16226 * 100% = 96.5 (%)

In 1999: 17290 / 16226 * 100% = 106.5 (%)

In 2000: 18115 / 16226 * 100% = 111.6 (%)

In 2001: 19220 / 16226 * 100% = 118.5 (%)

1. Rate of increase (decrease) (T pr)

• The rate of increase (decrease) in a “chain” way.

If we determine the rate of increase (decrease) in the presence of flower beds in the city of Arkhangelsk each time compared to the previous year, then it will be:

In 1992: (15833 - 17159) / 17159 * 100% = - 7.7(%)

In 1993: (11455 - 15833) / 15833 * 100% = - 27.7(%)

In 1994: (12668 - 11455) / 11455 * 100% = 10.6(%)

In 1995: (13126 - 12668) / 12668 * 100% = 3.6(%)

In 1996: (14553 - 13126) / 13126 * 100% = 10.9(%)

In 1997: (14120-14553) / 14553 * 100% = -3.0(%)

In 1998: (15663 - 14120) / 14120 * 100% = 10.9(%)

In 1999: (17290 - 15663) / 15663 * 100% = 10.4(%)

In 2000: (18115 - 17290) / 17290 * 100% = 4.8(%)

In 2001: (19220 - 18115) / 18115 * 100% = 6.1(%)

• Rate of growth (decrease) in a “basic” way.

If we take 1990 as the base of comparison, then in relation to it the rate of increase (decrease) in the presence of flower beds in the city of Arkhangelsk in subsequent years will be:

In 1991: (17159 - 16226) / 16226 * 100% = 5.8(%)

In 1992: (15833 - 16226) / 16226 * 100% = - 2.4(%)

In 1993: (11455 - 16226) / 16226 * 100% = - 29.4(%)

In 1994: (12668 - 16226) / 16226 * 100% = - 21.9(%)

In 1995: (13126 - 16226) / 16226 * 100% = - 19.1(%)

In 1996: (14553 - 16226) / 16226 * 100% = - 10.3(%)

In 1997: (14120-16226) / 16226 * 100% = - 13.0(%)

In 1998: (15663 - 16226) / 16226 * 100% = - 3.5(%)

In 1999: (17290 - 16226) / 16226 * 100% = 6.6(%)

In 2000: (18115 - 16226) / 16226 * 100% = 11.6(%)

In 2001: (19220 - 16226) / 16226 * 100% = 18.5(%)

Average annual growth rate (T r)

• The average annual growth rate determined by the “chain” method will be:

1,057*0,923*0,723*1,106*1,036*1,108*0,970*1,109*1,104*1,048*1,061 = 1,183

• The average annual growth rate determined by the “basic” method is:

1,057*0,976*0,706*0,780*0,809*0,897*0,870*0,965*1,065*1,116*1,185 = 0,487

Dynamics of indicators of absolute increase (decrease), growth rate (decrease), rate of increase (decrease) in the presence of flower beds in the city of Arkhangelsk in the period from 1990 to 2001, calculated by the “chain” and “basic” methods

conclusions

In 1990, the presence of flower beds in the city of Arkhangelsk amounted to 16,226.

In 1991, the presence of flower beds in the city of Arkhangelsk amounted to 17,159 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk compared to 1990 was 933 units. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1991 compared to 1990 was 105.7 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1991 compared to 1990 was 5.8 percent.

In 1992, the presence of flower beds in the city of Arkhangelsk amounted to 15,833 units. The absolute decrease in the presence of flower beds in the city of Arkhangelsk in 1992 compared to 1991 was 1,326 units. The absolute decrease in the presence of flower beds in the city of Arkhangelsk in 1992 compared to 1990 was 393 units. The rate of decline in the presence of flower beds in the city of Arkhangelsk in 1992 compared to 1991 was 92.3 percent. The rate of decline in the presence of flower beds in the city of Arkhangelsk in 1992 compared to 1990 was 97.6 percent. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1992 compared to 1991 was 7.7 percent. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1992 compared to 1990 was 2.4 percent.

In 1993, the presence of flower beds in the city of Arkhangelsk amounted to 11,455 units. The absolute decrease in the presence of flower beds in the city of Arkhangelsk in 1993 compared to 1992 amounted to 4,378 units. The absolute decrease in the presence of flower beds in the city of Arkhangelsk in 1993 compared to 1990 was 4,771 units. The rate of decline in the availability of flower beds in the city of Arkhangelsk in 1993 compared to 1992 was 72.3 percent. The rate of decline in the presence of flower beds in the city of Arkhangelsk in 1993 compared to 1990 was 70.6 percent. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1993 compared to 1992 was 27.7 percent. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1993 compared to 1990 was 29.4 percent.

In 1994, the presence of flower beds in the city of Arkhangelsk amounted to 12,668 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 1994 compared to 1993 was 1213 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 1994 compared to 1990 was 3,558 units. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1994 compared to 1993 was 110.6 percent. The rate of decline in the presence of flower beds in the city of Arkhangelsk in 1994 compared to 1990 was 78.0 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1994 compared to 1993 was 10.6 percent. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1994 compared to 1990 was 21.9 percent.

In 1995, the presence of flower beds in the city of Arkhangelsk amounted to 13,126 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 1995 compared to 1994 was 458 units. The absolute decrease in the presence of flower beds in the city of Arkhangelsk in 1995 compared to 1990 was 3,100 units. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1995 compared to 1994 was 103.6 percent. The rate of decline in the presence of flower beds in the city of Arkhangelsk in 1995 compared to 1990 was 80.9 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1995 compared to 1994 was 3.6 percent. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1995 compared to 1990 was 19.1 percent.

In 1996, the presence of flower beds in the city of Arkhangelsk amounted to 14,553 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 1996 compared to 1995 was 1,427 units. The absolute decrease in the presence of flower beds in the city of Arkhangelsk in 1996 compared to 1990 amounted to 1,673 units. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1996 compared to 1995 was 110.8 percent. The rate of decline in the presence of flower beds in the city of Arkhangelsk in 1996 compared to 1990 was 89.7 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1996 compared to 1995 was 10.9 percent. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1996 compared to 1990 was 10.3 percent.

In 1997, the presence of flower beds in the city of Arkhangelsk amounted to 14,120 units. The absolute decrease in the presence of flower beds in the city of Arkhangelsk in 1997 compared to 1996 was 433 units. The absolute decrease in the presence of flower beds in the city of Arkhangelsk in 1997 compared to 1990 was 2,106 units. The rate of decline in the presence of flower beds in the city of Arkhangelsk in 1997 compared to 1996 was 97.0 percent. The rate of decline in the presence of flower beds in the city of Arkhangelsk in 1997 compared to 1990 was 87.0 percent. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1997 compared to 1996 was 3.0 percent. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1997 compared to 1990 was 13.0 percent.

In 1998, the presence of flower beds in the city of Arkhangelsk amounted to 15,663 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 1998 compared to 1997 was 1,543 units. The absolute decrease in the presence of flower beds in the city of Arkhangelsk in 1998 compared to 1990 was 563 units. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1998 compared to 1997 was 110.9 percent. The rate of decline in the presence of flower beds in the city of Arkhangelsk in 1998 compared to 1990 was 96.5 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1998 compared to 1997 was 10.9 percent. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1998 compared to 1990 was 3.5 percent.

In 1999, the presence of flower beds in the city of Arkhangelsk amounted to 17,290 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 1999 compared to 1998 was 1,627 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 1999 compared to 1990 was 1064 units. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1999 compared to 1998 was 110.4 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1999 compared to 1990 was 106.5 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1999 compared to 1998 was 10.4 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 1999 compared to 1990 was 6.6 percent.

In 2000, the presence of flower beds in the city of Arkhangelsk amounted to 18,115 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 2000 compared to 1999 was 825 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 2000 compared to 1990 was 1889 units. The growth rate of the presence of flower beds in the city of Arkhangelsk in 2000 compared to 1999 was 104.8 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 2000 compared to 1990 was 111.6 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 2000 compared to 1999 was 4.8 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 2000 compared to 1990 was 11.6 percent.

In 2001, the presence of flower beds in the city of Arkhangelsk amounted to 19,220 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 2001 compared to 2000 was 1,105 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 2001 compared to 1990 was 2994 units. The growth rate of the presence of flower beds in the city of Arkhangelsk in 2001 compared to 2000 was 106.1 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 2001 compared to 1990 was 118.5 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 2001 compared to 2000 was 6.1 percent. The growth rate of the presence of flower beds in the city of Arkhangelsk in 2001 compared to 1990 was 18.5 percent.

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If you have ever dealt with the analysis of time series, then you have probably heard a lot about such statistical indicators as growth rate and growth rate. But if the growth rate is a fairly simple concept, then the growth rate often raises many questions, including the formula for calculating it. This article will be useful both for those for whom these concepts are not new, but slightly forgotten, and for those who hear these terms for the first time. Next, we will explain the concepts of growth rate and gain for you and tell you how to find the growth rate.

Growth rate and growth rate: what's the difference?

The growth rate is an indicator that is necessary to determine how much one value of a series occupies in another. As the latter, as a rule, they use the previous value, or the basic one, that is, the one that is at the beginning of the series under study. If the result of calculating the growth rate is more than one hundred percent, then this indicates that there is an increase in the indicator being studied. Conversely, if the result is less than one hundred percent, this means that the indicator under study is decreasing. Calculating the growth rate is quite simple: you need to find the ratio of the value for the reporting period to the value of the base or previous time period.

Unlike the growth rate, the growth rate allows us to calculate how much the value we are studying has changed. During calculations, the resulting positive value may indicate the presence of a growth rate, while at the same time, a negative value indicates that there is a rate of decline in the value relative to the previous or base period.

How is the growth rate calculated? To make this calculation, you must first find the ratio of the indicator to the previous one, and then subtract one from the result obtained and multiply the resulting amount by one hundred. By multiplying the number by one hundred you can get the total as a percentage.

This method of calculation is used more often than others, but it also happens that only the value of the absolute increase is known, and we do not know the actual value of the indicator that we are analyzing. Is it possible to calculate the growth rate in this case? It is possible, but the standard formula will no longer help us with this; we need to apply an alternative formula. Its essence is to find the percentage of absolute growth to a certain level in comparison with which it was calculated.

It is important that absolute growth can be both positive and negative. Having learned this information, you can determine whether the selected indicator increases or decreases over a certain period.

How to calculate growth rate

Since the growth rate is a relative value, it is calculated in shares or percentages, and acts as a growth coefficient. If we are faced with the question of how to determine the growth rate, we need to divide the absolute growth for the selected period by the indicator for the initial period and multiply the total by one hundred to obtain a percentage figure.

For clarity, consider an example. Let's say we have the following conditions:

• Revenue for the reporting period is Z rubles;
• Revenue for the previous period is R rubles.

We can already calculate that the absolute increase will be equal to Z-R under such conditions. Next, we calculate the growth rate for the entire selected period. To do this, it is necessary to determine the initial level (let's say this will be the year the enterprise was founded). In this case, the absolute increase is calculated as the difference between the indicators of the last and first year. Then we calculate the growth rate for the entire period by dividing this difference by the indicator for the first year.

Calculating the growth rate on a calculator

Of course, the growth rate formula is not at all complicated, but even with such calculations difficulties can sometimes arise. With the latest technologies, of course, we can find ways that will make our lives easier and help us with calculations even of such complexity. Nowadays you can find special calculators on the Internet designed to calculate analytical indicators of statistical time series. Now, knowledge of complex formulas is not at all necessary in order to find out the rate of growth or increase; it is enough to enter the available data into the appropriate fields of the calculator and it will do all the calculations itself.

After we have dotted all the i’s and found out what formulas can be used to find out the rate of growth and increase, it is important to note that in order to give the only correct assessment of the phenomenon under study, it is not enough to have information about only one indicator. For example, a case may arise when at an enterprise the absolute increase in profit gradually increases, but at the same time development slows down. This suggests that any signs of dynamics require a comprehensive analysis.

The growth rate is an important analytical indicator that allows you to answer the question: how did this or that indicator increase/decrease and how many times did it change over the analyzed period of time.

Correct calculation

Calculation using an example

Objective: the volume of Russian grain exports in 2013 amounted to 90 million tons. In 2014, this figure was 180 million tons. Calculate the growth rate as a percentage.

Solution: (180/90)*100%= 200% That is: the final indicator is divided by the initial indicator and multiplied by 100%.

Answer: the growth rate of grain exports was 200%.

Rate of increase

The growth rate shows how much a particular indicator has changed. It is very often confused with the growth rate, making annoying mistakes that can be easily avoided by understanding the difference between the indicators.

Calculation using an example

Problem: in 2010, the store sold 2,000 packs of washing powder, in 2014 - 5,000 packs. Calculate the growth rate.

Solution: (5000-2000)/2000= 1.5. Now 1.5*100%=150%. The base year is subtracted from the reporting period, the resulting value is divided by the base year indicator, then the result is multiplied by 100%.

Answer: the growth rate was 150%.

You might also be interested in learning about

Dynamics series- these are a series of statistical indicators characterizing the development of natural and social phenomena over time. Statistical collections published by the State Statistics Committee of Russia contain a large number of dynamics series in tabular form. Dynamic series make it possible to identify patterns of development of the phenomena being studied.

Dynamics series contain two types of indicators. Time indicators(years, quarters, months, etc.) or points in time (at the beginning of the year, at the beginning of each month, etc.). Row level indicators. Indicators of the levels of dynamics series can be expressed in absolute values ​​(product production in tons or rubles), relative values ​​(share of the urban population in %) and average values ​​(average wages of industry workers by year, etc.). A dynamics row contains two columns or two rows.

1. all indicators of a series of dynamics must be scientifically based and reliable;
2. indicators of a series of dynamics must be comparable over time, i.e. must be calculated for the same periods of time or on the same dates;
3. indicators of a number of dynamics must be comparable across the territory;
4. indicators of a series of dynamics must be comparable in content, i.e. calculated according to a single methodology, in the same way;
5. indicators of a number of dynamics should be comparable across the range of farms taken into account. All indicators of a series of dynamics must be given in the same units of measurement.

Statistical indicators can characterize either the results of the process being studied over a period of time, or the state of the phenomenon being studied at a certain point in time, i.e. indicators can be interval (periodic) and momentary. Accordingly, initially the dynamics series can be either interval or moment. Moment dynamics series, in turn, can be with equal or unequal time intervals.

The original dynamics series can be transformed into a series of average values ​​and a series of relative values ​​(chain and basic). Such time series are called derived time series.

The methodology for calculating the average level in the dynamics series is different, depending on the type of the dynamics series. Using examples, we will consider the types of dynamics series and formulas for calculating the average level.

Interval time series

The levels of the interval series characterize the result of the process being studied over a period of time: production or sales of products (for a year, quarter, month, etc.), the number of people hired, the number of births, etc. The levels of an interval series can be summed up. At the same time, we get the same indicator over longer time intervals.

Average level in interval dynamics series() is calculated using the simple formula:

• y— series levels ( y 1 , y 2 ,...,y n),
• n— number of periods (number of levels of the series).

Let's consider the methodology for calculating the average level of an interval dynamics series using data on the sale of sugar in Russia as an example.

This is the average annual volume of sugar sales to the Russian population for 1994-1996. In just three years, 8137 thousand tons of sugar were sold.

Moment dynamics series

The levels of moment series of dynamics characterize the state of the phenomenon being studied at certain points in time. Each subsequent level includes, in whole or in part, the previous indicator. For example, the number of employees on April 1, 1999 fully or partially includes the number of employees on March 1.

If we add up these indicators, we get a repeat count of those workers who worked throughout the month. The resulting amount has no economic content; it is a calculated figure.

In moment series of dynamics with equal time intervals, the average level of the series calculated by the formula:

• y-moment series levels;
• n-number of moments (series levels);
• n - 1— number of time periods (years, quarters, months).

Let's consider the methodology for such calculation using the following data on the payroll number of employees of the enterprise for the 1st quarter.

It is necessary to calculate the average level of a series of dynamics, in this example - an enterprise:

The calculation was made using the average chronological formula. The average number of employees of the enterprise for the 1st quarter was 155 people. The denominator is 3 months in a quarter, and the numerator (465) is a calculated number that has no economic content. In the vast majority of economic calculations, months, regardless of the number of calendar days, are considered equal.

In moment series of dynamics with unequal time intervals, the average level of the series is calculated using the weighted arithmetic mean formula. The length of time (t-days, months) is taken as the average weight. Let's perform the calculation using this formula.

The list of employees of the enterprise for October is as follows: on October 1 - 200 people, on October 7, 15 people were hired, on October 12, 1 person was fired, on October 21, 10 people were hired, and until the end of the month there were no hiring or dismissal of workers. This information can be presented as follows:

When determining the average level of a series, it is necessary to take into account the duration of the periods between dates, i.e. apply:

In this formula, the numerator () has economic content. In the example given, the numerator (6665 person-days) is the company’s employees in October. The denominator (31 days) is the calendar number of days in the month.

In cases where we have a moment series of dynamics with unequal time intervals, and the specific dates of change in the indicator are unknown to the researcher, then first we need to calculate the average value () for each time interval using the simple arithmetic average formula, and then calculate the average level for the entire series of dynamics, by weighing the calculated average values ​​over the duration of the corresponding time interval. The formulas are as follows:

The dynamics series discussed above consist of absolute indicators obtained as a result of statistical observations. The initially constructed series of dynamics of absolute indicators can be transformed into derivative series: series of average values ​​and series of relative values. Series of relative values ​​can be chain (in % of the previous period) and basic (in % of the initial period taken as the basis of comparison - 100%). The calculation of the average level in the derivative time series is performed using other formulas.

A series of averages

First, we transform the above moment series of dynamics with equal time intervals into a series of average values. To do this, we calculate the average number of employees of the enterprise for each month, as the average of the indicators at the beginning and end of the month (): for January (150+145): 2 = 147.5; for February (145+162): 2 = 153.5; for March (162+166): 2 = 164.

Let's present this in tabular form.

Average level in derivative series average values ​​are calculated by the formula:

Note that the average payroll number of employees of the enterprise for the 1st quarter, calculated using the chronological average formula based on the database on the 1st day of each month and the arithmetic average - according to the derived series - are equal to each other, i.e. 155 people. A comparison of the calculations allows us to understand why in the average chronological formula the initial and final levels of the series are taken in half size, and all intermediate levels are taken in full size.

Series of average values ​​derived from moment or interval series of dynamics should not be confused with series of dynamics in which levels are expressed by an average value. For example, the average wheat yield by year, the average salary, etc.

Series of relative quantities

In economic practice, series are widely used. Almost any initial series of dynamics can be converted into a series of relative values. In essence, transformation means replacing the absolute indicators of a series with relative values ​​of dynamics.

The average level of the series in relative dynamics series is called the average annual growth rate. Methods for its calculation and analysis are discussed below.

Analysis of time series

For a reasonable assessment of the development of phenomena over time, it is necessary to calculate analytical indicators: absolute growth, growth coefficient, growth rate, growth rate, absolute value of one percent of growth.

The table shows a numerical example, and below are calculation formulas and economic interpretation of the indicators.

Absolute increases (Δy) show how many units the subsequent level of the series has changed compared to the previous one (gr. 3. - chain absolute increases) or compared to the initial level (gr. 4. - basic absolute increases). The calculation formulas can be written as follows:

When the absolute values ​​of the series decrease, there will be a “decrease” or “decrease”, respectively.

Indicators of absolute growth indicate that, for example, in 1998, the production of product “A” increased by 4 thousand tons compared to 1997, and by 34 thousand tons compared to 1994; for other years, see table. 11.5 gr. 3 and 4.

Growth rate shows how many times the level of the series has changed compared to the previous one (gr. 5 - chain coefficients of growth or decline) or compared to the initial level (gr. 6 - basic coefficients of growth or decline). The calculation formulas can be written as follows:

Rates of growth show what percentage the next level of the series is compared to the previous one (gr. 7 - chain growth rates) or compared to the initial level (gr. 8 - basic growth rates). The calculation formulas can be written as follows:

So, for example, in 1997, the production volume of product “A” compared to 1996 was 105.5% (

Growth rate show by what percentage the level of the reporting period increased compared to the previous one (column 9 - chain growth rates) or compared to the initial level (column 10 - basic growth rates). The calculation formulas can be written as follows:

T pr = T r - 100% or T pr = absolute growth / level of the previous period * 100%

So, for example, in 1996, compared to 1995, product “A” was produced by 3.8% (103.8% - 100%) or (8:210)x100% more, and compared to 1994 - by 9% (109% - 100%).

If the absolute levels in the series decrease, then the rate will be less than 100% and, accordingly, there will be a rate of decline (the rate of increase with a minus sign).

Absolute value of 1% increase(column 11) shows how many units must be produced in a given period so that the level of the previous period increases by 1%. In our example, in 1995 it was necessary to produce 2.0 thousand tons, and in 1998 - 2.3 thousand tons, i.e. much bigger.

The absolute value of 1% growth can be determined in two ways:

• divide the level of the previous period by 100;
• chain absolute increases are divided by the corresponding chain growth rates.

Absolute value of 1% increase =

In dynamics, especially over a long period, a joint analysis of the growth rate with the content of each percentage increase or decrease is important.

Note that the considered methodology for analyzing time series is applicable both for time series, the levels of which are expressed in absolute values ​​(t, thousand rubles, number of employees, etc.), and for time series, the levels of which are expressed in relative indicators (% of defects , % ash content of coal, etc.) or average values ​​(average yield in c/ha, average wage, etc.).

Along with the considered analytical indicators, calculated for each year in comparison with the previous or initial level, when analyzing dynamics series, it is necessary to calculate the average analytical indicators for the period: the average level of the series, the average annual absolute increase (decrease) and the average annual growth rate and growth rate.

Methods for calculating the average level of a series of dynamics were discussed above. In the interval dynamics series we are considering, the average level of the series is calculated using a simple formula:

Average annual production volume of the product for 1994-1998. amounted to 218.4 thousand tons.

The average annual absolute growth is also calculated using the simple arithmetic average formula:

Annual absolute increases varied over the years from 4 to 12 thousand tons (see column 3), and the average annual increase in production for the period 1995 - 1998. amounted to 8.5 thousand tons.

Methods for calculating the average growth rate and average growth rate require more detailed consideration. Let us consider them using the example of the annual series level indicators given in the table.

Average annual growth rate and average annual growth rate

First of all, we note that the growth rates shown in the table (columns 7 and 8) are series of dynamics of relative values ​​- derivatives of the interval series of dynamics (column 2). Annual growth rates (column 7) vary from year to year (105%; 103.8%; 105.5%; 101.7%). How to calculate the average from annual growth rates? This value is called the average annual growth rate.

The average annual growth rate is calculated in the following sequence:

The average annual growth rate ( is determined by subtracting 100% from the growth rate.

The average annual growth (decrease) coefficient using geometric mean formulas can be calculated in two ways:

1) based on the absolute indicators of the dynamics series according to the formula:

• n— number of levels;
• n - 1- number of years in the period;

2) based on annual growth rates according to the formula

• m— number of coefficients.

The calculation results using the formulas are equal, since in both formulas the exponent is the number of years in the period during which the change occurred. And the radical expression is the growth rate of the indicator for the entire period of time (see Table 11.5, column 6, line for 1998).

The average annual growth rate is

The average annual growth rate is determined by subtracting 100% from the average annual growth rate. In our example, the average annual growth rate is

Consequently, for the period 1995 - 1998. The production volume of product "A" increased by 4.0% on average per year. Annual growth rates ranged from 1.7% in 1998 to 5.5% in 1997 (for each year’s growth rates, see Table 11.5, group 9).

The average annual growth rate (growth) allows you to compare the dynamics of development of interrelated phenomena over a long period of time (for example, the average annual growth rate of the number of workers in sectors of the economy, the volume of production, etc.), to compare the dynamics of a phenomenon in different countries, to study the dynamics of some or phenomena according to periods of historical development of the country.

Seasonal analysis

The study of seasonal fluctuations is carried out in order to identify regularly recurring differences in the level of time series depending on the time of year. For example, the sale of sugar to the population in the summer increases significantly due to the canning of fruits and berries. The need for labor in agricultural production varies depending on the time of year. The task of statistics is to measure seasonal differences in the level of indicators, and in order for the identified seasonal differences to be natural (and not random), it is necessary to build an analysis on the basis of data for several years, at least for at least three years. In table 11.6 shows the initial data and methodology for analyzing seasonal fluctuations using the simple arithmetic average method.

The average value for each month is calculated using the simple arithmetic average formula. For example, for January 2202 = (2106 +2252 +2249):3.

Seasonality index(Table 11.5, column 7.) is calculated by dividing the average values ​​for each month by the total average monthly value, taken as 100%. The average monthly for the entire period can be calculated by dividing the total fuel consumption for three years by 36 months (1188082 tons: 36 = 3280 tons) or by dividing the average monthly sum by 12, i.e. total total for gr. 6 (2022 + 2157 + 2464, etc. + 2870) : 12.

For clarity, a seasonal wave graph is constructed based on seasonality indices (Fig. 11.1). Months are located on the abscissa axis, and seasonality indices in percentage are located on the ordinate axis (Table 11.6, group 7). The overall average monthly for all years is located at the 100% level, and the average monthly seasonality indices in the form of points are plotted on the graph field in accordance with the accepted scale along the ordinate axis.

The points are connected by a smooth broken line.

In the example given, the annual fuel consumption differs slightly. If, in the dynamics series, along with seasonal fluctuations, there is a pronounced tendency of growth (decrease), i.e. levels in each subsequent year systematically significantly increase (decrease) compared to the levels of the previous year, then we obtain more reliable data on the extent of seasonality as follows:

1. for each year we calculate the average monthly value;
2. Let's calculate the seasonality indices for each year by dividing the data for each month by the average monthly value for that year and multiplying by 100%;
3. for the entire period, we calculate the average seasonality indices using the simple arithmetic average formula from the monthly seasonality indices calculated for each year. So, for example, for January we will obtain the average seasonality index if we add up the January values ​​of seasonality indices for all years (let’s say for three years) and divide by the number of years, i.e. on three. Similarly, we calculate the average seasonality indices for each month.

The transition for each year from absolute monthly values ​​of indicators to seasonality indices makes it possible to eliminate the tendency of growth (decrease) in the dynamics series and more accurately measure seasonal fluctuations.

In market conditions, when concluding contracts for the supply of various products (raw materials, materials, electricity, goods), it is necessary to have information about the seasonal needs for means of production, about the population’s demand for certain types of goods. The results of the study of seasonal fluctuations are important for the effective management of economic processes.

Reducing dynamics series to the same base

In economic practice, there is often a need to compare several series of dynamics (for example, indicators of the dynamics of electricity production, grain production, passenger car sales, etc.). To do this, you need to transform the absolute indicators of the compared time series into derived series of relative basic values, taking the indicators of any one year as one or 100%. Such a transformation of several time series is called bringing them to the same base. Theoretically, the absolute level of any year can be taken as the basis of comparison, but in economic research, for the basis of comparison it is necessary to choose a period that has a certain economic or historical significance in the development of phenomena. At present, it is advisable to take, for example, the 1990 level as a basis for comparison.

Methods for aligning time series

To study the pattern (trend) of development of the phenomenon under study, data over a long period of time is required. The development trend of a particular phenomenon is determined by the main factor. But along with the action of the main factor in the economy, the development of the phenomenon is directly or indirectly influenced by many other factors, random, one-time or periodically recurring (years favorable for agriculture, drought years, etc.). Almost all series of dynamics of economic indicators on the graph have the shape of a curve, a broken line with ups and downs. In many cases, it is difficult to determine even the general trend of development from actual data from a series of dynamics and from a graph. But statistics must not only determine the general trend in the development of a phenomenon (growth or decline), but also provide quantitative (digital) characteristics of development.

• Interval enlargement method
• Moving average method

In table Table 11.7 (column 2) shows actual data on grain production in Russia for 1981-1992. (in all categories of farms, in weight after modification) and calculations for leveling this series using three methods.

Method of enlarging time intervals (column 3).

Considering that the dynamics series is small, three-year intervals were taken and the averages were calculated for each interval. The average annual volume of grain production for three-year periods is calculated using the simple arithmetic average formula and referred to the average year of the corresponding period. So, for example, for the first three years (1981 - 1983), the average was recorded against 1982: (73.8 + 98.0 + 104.3): 3 = 92.0 (million tons). Over the next three-year period (1984 - 1986), the average (85.1 +98.6+ 107.5): 3 = 97.1 million tons was recorded against 1985.

For other periods, the calculation results in gr. 3.

Given in gr. 3 indicators of the average annual volume of grain production in Russia indicate a natural increase in grain production in Russia for the period 1981 - 1992.

Moving average method

Moving average method(see groups 4 and 5) is also based on the calculation of average values ​​for aggregated periods of time. The goal is the same - to abstract from the influence of random factors, to cancel out their influence in individual years. But the calculation method is different.

In the example given, five-tier (over five-year periods) moving averages are calculated and assigned to the middle year in the corresponding five-year period. Thus, for the first five years (1981-1985), using the simple arithmetic average formula, the average annual volume of grain production was calculated and recorded in table. 11.7 versus 1983 (73.8+ 98.0+ 104.3+ 85.1+ 98.6): 5= 92.0 million tons; for the second five-year period (1982 - 1986) the result was recorded against 1984 (98.0 + 104.3 +85.1 + 98.6 + 107.5): 5 = 493.5: 5 = 98.7 million tons

For subsequent five-year periods, the calculation is made in a similar way by eliminating the initial year and adding the year following the five-year period and dividing the resulting amount by five. With this method, the ends of the row are left empty.

How long should the time periods be? Three, five, ten years? The researcher decides the question. In principle, the longer the period, the more smoothing occurs. But we must take into account the length of the dynamics series; do not forget that the moving average method leaves cut ends of the aligned series; take into account the stages of development, for example, in our country for many years, socio-economic development was planned and accordingly analyzed according to five-year plans.

Analytical alignment method

Analytical alignment method(gr. 6 - 9) is based on calculating the values ​​of the aligned series using the corresponding mathematical formulas. In table 11.7 shows calculations using the equation of a straight line:

To determine the parameters, it is necessary to solve the system of equations:

The necessary quantities for solving the system of equations have been calculated and given in the table (see groups 6 - 8), let’s substitute them into the equation:

As a result of the calculations we get: α= 87.96; b = 1.555.

Let's substitute the values ​​of the parameters and get the equation of the straight line:

For each year we substitute the value t and get the levels of the aligned series (see column 9):

In the leveled series, there is a uniform increase in series levels on average per year by 1.555 million tons (the value of the “b” parameter). The method is based on abstracting the influence of all other factors except the main one.

Phenomena can develop in dynamics evenly (increase or decrease). In these cases, the straight line equation is most often suitable. If the development is uneven, for example, at first very slow growth, and from a certain moment a sharp increase, or, conversely, first a sharp decrease, and then a slowdown in the rate of decline, then the leveling must be performed using other formulas (equation of a parabola, hyperbola, etc.). If necessary, one should turn to textbooks on statistics or special monographs, where the issues of choosing a formula to adequately reflect the actual trend of the dynamics series being studied are described in more detail.

For clarity, we will plot the indicators of the levels of the actual dynamics series and the aligned series on a graph (Fig. 11.2). The actual data is represented by a broken black line, indicating increases and decreases in the volume of grain production. The remaining lines on the graph show that the use of the moving average method (line with cut ends) allows you to significantly align the levels of the dynamic series and, accordingly, make the broken curved line on the graph smoother and smoother. However, straight lines are still crooked lines. Constructed on the basis of theoretical values ​​of the series obtained using mathematical formulas, the line strictly corresponds to a straight line.

Each of the three methods discussed has its own advantages, but in most cases the analytical alignment method is preferable. However, its application is associated with large computational work: solving a system of equations; checking the validity of the selected function (form of communication); calculating the levels of the aligned series; plotting. To successfully complete such work, it is advisable to use a computer and appropriate programs.

The growth rate is one of the dynamic, that is, changing indicators of the 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 absolute growth is quite simple: to do this, calculate the difference between the latest economic indicators and the 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:

• For the reporting period, revenue is 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 of the last year and the 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.