Calculation of growth and growth rates. The average growth rate is calculated by the formula

If you have ever dealt with time series analyzes, 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 its calculation. 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 growth for you and tell you how to find the growth rate.

Growth Rate vs Growth Rate: What's the Difference?

The growth rate is an indicator that is needed in order to determine how much one value of the series occupies in another. As the latter, as a rule, the previous value is used, or the base value, 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 that is 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 report period to the value of the base or previous time period.

Unlike the growth rate, the growth rate allows you to calculate how much the value that we are studying has changed. When calculating, the obtained positive value may indicate the presence of a growth rate, at the same time, a negative value indicates that there is a rate of decrease in the value relative to the previous or base period.

How is the growth rate calculated? For this calculation, you must first find the ratio of the indicator to the previous one, and then subtract one from the result and multiply the resulting amount by one hundred. Multiplying a number by 100 will give you 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, while the actual value of the indicator that we analyze is not known to us. Is it possible to calculate the growth rate in this case? It is possible, but the standard formula will no longer help us in this, it is necessary 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 the absolute increase can be both positive and negative. Having learned this information, you can determine whether the selected indicator is increasing or decreasing for a certain period.

How to calculate the growth rate

Since the growth rate is a relative value, it is calculated in shares or percentages, and acts as a growth rate. 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 final value by one hundred to get the figure as a percentage.

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 (for example, 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 the first year. Then we calculate the growth rate for the entire period by dividing this difference by the indicator of the first year.

Calculation of the growth rate on the calculator

Of course, the growth rate formula is not at all complicated, but even with such calculations, difficulties can sometimes arise. During the latest technologies, of course, you can find ways that will make life easier for us and help with calculations even of such complexity. Now on the Internet you can find special calculators designed to calculate the 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 growth, it is enough to enter the available data into the appropriate fields of the calculator and it will make all the calculations.

After we have dotted the i’s and found out what formulas can be used to find out the rate of growth and growth, 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 the value of the absolute increase in profits in an enterprise gradually increases, but at the same time, development slows down. This suggests that any signs of dynamics need a comprehensive analysis.

It would seem, how can the growth and growth rates differ, because these are the same-root words, which, most likely, denote the same phenomenon? But, no matter how it might seem at first glance, these are two economic indicators, which, although interconnected, still have a different purpose and method of determination. To understand what their distinctive features are, it is necessary to get acquainted with their economic essence.

Definition

Growth rate is designed to show how many percent one indicator is from another, that is, it can be used to compare the indicator under study with the base or previous value. If the obtained value is less than 100%, then there is a decrease rate of the studied indicator in relation to the base or previous one.

Rate of increase shows by what percentage one or another indicator has increased or decreased compared to the base or previous value. If the result obtained has a negative value, then there is not a growth rate, but a rate of decrease in the analyzed indicator compared to the base or previous value.

Comparison

The most important difference is in their calculation method, since they use different formulas. So, in order to calculate the growth rate, it is necessary to find the ratio of the studied value to the previous or base value, and then multiply it by 100%, since this indicator is measured as a percentage. And then the conclusion will be as follows: indicator A compared to indicator B was X%.

To calculate the growth rate, you need to use the same formula, just subtract 100% from it. In addition, the formula will look simpler if you subtract 100% from the growth rate. In this case, you can find out exactly how many percent the studied indicator has changed. The conclusion according to this formula will sound as follows: indicator A is more than indicator B by X%.

Findings site

The growth rate shows how many percent one indicator is from another, and the growth rate shows how many percent one indicator differs from another.
The growth rate can be used to calculate the growth rate, but vice versa it cannot.
If not the growth rate is observed, but its opposite, then the value of the result will be less than 100%; if there is not a growth rate, but a decline rate, then the value of the effective indicator will be negative.

Task

The following data is available:

	Availability of flower beds in the city of Arkhangelsk

Determine by basic and chain methods:

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

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

Computing

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 every 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 1990 is taken as a comparison base, 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.

Rate of growth (decrease) (T p)

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 by 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 (%)

The rate of growth (decrease) in the "basic" way.

If 1990 is taken as a comparison base, then in relation to it, the growth rate (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 (%)

Rate of increase (decrease) (T pr)

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 by 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(%)

The rate of growth (decrease) in the "basic" way.

If 1990 is taken as a comparison base, then in relation to it, the growth rate (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 p)

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 will be:

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 growth (decrease), growth rate (decrease), growth rate (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

Availability of flower beds in the city of Arkhangelsk, units	Absolute increase (decrease) Availability of flower beds in the city of Arkhangelsk, units		Growth rate (decrease) Availability of flower beds in the city of Arkhangelsk, %		Growth rate (decrease) in the presence of flower beds in the city of Arkhangelsk,
		Basic method		Basic method		Basic method

findings

In 1990, the presence of flower beds in the city of Arkhangelsk amounted to 16226.

In 1991, the presence of flower beds in the city of Arkhangelsk amounted to 17159 units. The absolute increase in the availability of flower beds in the city of Arkhangelsk compared to 1990 amounted to 933 units. The growth rate of the availability of flower beds in the city of Arkhangelsk in 1991 compared to 1990 was 105.7 percent. The growth rate of the availability 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 15833 units. The absolute decrease in the availability of flower beds in the city of Arkhangelsk in 1992 compared to 1991 amounted to 1326 units. The absolute decrease in the availability of flower beds in the city of Arkhangelsk in 1992 compared to 1990 amounted to 393 units. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1992 compared to 1991 was 92.3 percent. The rate of decrease in the availability 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 availability of flower beds in the city of Arkhangelsk in 1993 compared to 1992 amounted to 4378 units. The absolute decrease in the availability of flower beds in the city of Arkhangelsk in 1993 compared to 1990 amounted to 4771 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 decrease in the availability 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 amounted to 1213 units. The absolute increase in the presence of flower beds in the city of Arkhangelsk in 1994 compared to 1990 amounted to 3558 units. The growth rate of the availability of flower beds in the city of Arkhangelsk in 1994 compared to 1993 was 110.6 percent. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1994 compared to 1990 was 78.0 percent. The growth rate of the availability 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 availability of flower beds in the city of Arkhangelsk in 1995 compared to 1994 amounted to 458 units. The absolute decrease in the presence of flower beds in the city of Arkhangelsk in 1995 compared to 1990 amounted to 3100 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 decrease 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 availability 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 14553 units. The absolute increase in the availability of flower beds in the city of Arkhangelsk in 1996 compared to 1995 amounted to 1427 units. The absolute decrease in the availability of flower beds in the city of Arkhangelsk in 1996 compared to 1990 amounted to 1673 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 decrease in the availability 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 availability of flower beds in the city of Arkhangelsk in 1997 compared to 1996 amounted to 433 units. The absolute decrease in the availability of flower beds in the city of Arkhangelsk in 1997 compared to 1990 amounted to 2106 units. The rate of decrease in the availability of flower beds in the city of Arkhangelsk in 1997 compared to 1996 was 97.0 percent. The rate of decrease in the availability 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 availability of flower beds in the city of Arkhangelsk in 1998 compared to 1997 amounted to 1543 units. The absolute decrease in the availability of flower beds in the city of Arkhangelsk in 1998 compared to 1990 amounted to 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 decrease in the availability 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 17290 units. The absolute increase in the availability of flower beds in the city of Arkhangelsk in 1999 compared to 1998 amounted to 1627 units. The absolute increase in the availability of flower beds in the city of Arkhangelsk in 1999 compared to 1990 amounted to 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 availability 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 18115 units. The absolute increase in the availability of flower beds in the city of Arkhangelsk in 2000 compared to 1999 amounted to 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 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 19220 units. The absolute increase in the availability of flower beds in the city of Arkhangelsk in 2001 compared to 2000 amounted to 1105 units. The absolute increase in the availability of flower beds in the city of Arkhangelsk in 2001 compared to 1990 amounted to 2994 units. The growth rate of the availability 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 flower beds in the city of Arkhangelsk in 2001 compared to 1990 was 18.5 percent.

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Series of dynamics- these are series of statistical indicators characterizing the development of natural and social phenomena in time. Statistical collections published by the State Statistics Committee of Russia contain a large number of time series in tabular form. Series of dynamics allow revealing patterns of development of the studied phenomena.

Time 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 time series can be expressed in absolute values (production in tons or rubles), relative values (share of the urban population in%) and average values (average wages of industry workers by years, etc.). A row of dynamics contains two columns or two rows.

The correct construction of time series involves the fulfillment of a number of requirements:

all indicators of a series of dynamics must be scientifically substantiated, reliable;
indicators of a series of dynamics should be comparable in time, i.e. must be calculated for the same time periods or on the same dates;
indicators of a number of dynamics should be comparable across the territory;
indicators of a series of dynamics should be comparable in content, i.e. calculated according to a single methodology, in the same way;
indicators of a series of dynamics should be comparable across the range of farms considered. All indicators of a series of dynamics should be given in the same units of measurement.

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

The initial series of dynamics can be converted into a series of average values and a series of relative values (chain and base). Such time series are called derived time series.

The method of calculating the average level in the series of dynamics is different, due to the type of series of dynamics. Using examples, consider the types of time series and formulas for calculating the average level.

Interval time series

The levels of the interval series characterize the result of the process under study over a period of time: production or sales of products (for a year, quarter, month, and other periods), the number of people hired, the number of births, etc. The levels of the interval series can be summarized. At the same time, we get the same indicator for longer time intervals.

The average level in the interval series of dynamics() is calculated by a simple formula:

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

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

	Sugar sold, thousand tons

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

Moment series dynamics

The levels of moment series of dynamics characterize the state of the phenomenon under study at certain points in time. Each subsequent level includes all or part of the previous indicator. Thus, 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 will get a repeated account of those workers who worked throughout the month. The amount received has no economic content, it is a calculated indicator.

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

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

Consider the methodology for such a calculation according to 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 - enterprises:

The calculation is made according to the chronological average formula. The average payroll number of employees of the enterprise for the 1st quarter was 155 people. In the denominator - 3 months in a quarter, and in the numerator (465) - this is an estimated number, it 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 duration of time (t- days, months) is taken as the average weight. Let's do 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 and dismissal of workers. This information can be presented in the following form:

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 an economic content. In the above example, the numerator (6665 person-days) is the employees of the enterprise for October. The denominator (31 days) is the calendar number of days in a month.

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

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

A series of averages

First, we convert the above moment series of dynamics with equal time intervals into a series of average values. To do this, we calculate the average payroll 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 put it in tabular form.

Average level in derived series average values is calculated by the formula:

Note that the average payroll number of employees of the enterprise for the 1st quarter, calculated by the chronological average formula on the database on the 1st day of each month and by the arithmetic average - according to the data of the derived series - are equal to each other, i.e. 155 people. Comparison of calculations makes it possible to understand why in the chronological average 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 averages derived from moment or interval time series should not be confused with time series in which the levels are expressed as an average. For example, the average wheat yield by year, the average wage, etc.

Series of relative values

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

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

Time series analysis

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

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

Analysis of the dynamics of production of product "A" by the enterprise for 1994-1998.

Produced, thousand tons	Absolute gains,		Growth factors		pace growth, %		Growth rate, %		The value of 1% increase, thousand tons
		basic		basic		basic		basic
	3	4	5	6	7	8	9	10	11

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

With a decrease in the absolute values of the series, there will be a "decrease", "decrease", respectively.

The indicators of absolute growth indicate that, for example, in 1998 the production of product "A" increased by 4,000 tons compared to 1997, and by 34,000 tons compared to 1994; for other years, see table. 11.5 gr. 3 and 4.

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

Rates of growth show how many percent the next level of the series is in comparison with the previous one (column 7 - chain growth rates) or in comparison with the initial level (column 8 - basic growth rates). The calculation formulas can be written as follows:

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

Growth rates show how many percent 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 \u003d T p - 100% or T pr \u003d absolute increase / level of the previous period * 100%

So, for example, in 1996, compared to 1995, the product "A" was produced more by 3.8% (103.8% - 100%) or (8:210) x 100%, 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 (growth rate with a minus sign).

Absolute value of 1% increase(column 11) shows how many units must be produced in a given period in order for the level of the previous period to increase 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.

There are two ways to determine the magnitude of the absolute value of 1% growth:

the level of the previous period divided by 100;
chain absolute increments divided by the corresponding chain growth rates.

Absolute value of 1% increase =

In dynamics, especially over a long period, it is important to jointly analyze growth rates with the content of each percentage increase or decrease.

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

Along with the considered analytical indicators calculated for each year in comparison with the previous or initial level, when analyzing the time 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 series of dynamics we are considering, the average level of the series is calculated by a simple formula:

The average annual output of the product for 1994-1998. amounted to 218.4 thousand tons.

The average annual absolute increase is also calculated by the formula of the simple arithmetic mean:

Annual absolute increments varied over the years from 4 to 12 thousand tons (see gr. 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 the average growth rate require more detailed consideration. Let's consider them on the example of the annual indicators of the series level given in the table.

Average annual growth rate and average annual growth rate

First of all, we note that the growth rates given 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 rate (decrease) according to the geometric mean formulas can be calculated in two ways:

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

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

2) based on annual growth rates according to the formula

m is the number of coefficients.

The results of the calculation by 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 root expression is the coefficient of growth of the indicator for the entire period of time (see Table 11.5, column 6, for the line for 1998).

The average annual growth rate is

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

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

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

Seasonal Analysis

The study of seasonal fluctuations is carried out in order to identify regularly repeating differences in the level of time series depending on the time of year. So, for example, the sale of sugar to the population in the summer period increases significantly due to the canning of fruits and berries. The need for labor force in agricultural production is different depending on the season. The task of statistics is to measure seasonal differences in the level of indicators, and in order for the identified seasonal differences to be regular (and not random), it is necessary to build an analysis based on data for several years, at least not less than three years. In table. 11.6 the initial data and a technique of the analysis of seasonal fluctuations by a method of simple average arithmetic are resulted.

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

Seasonality index(Table 11.5, group 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 \u003d 3280 tons) or by dividing by 12 the sum of the average monthly, i.e. total for gr. 6 (2022 + 2157 + 2464 etc. + 2870): 12.

Table 11.6 Seasonal fluctuations in fuel consumption in agricultural enterprises of the region for 3 years

	Fuel consumption, tons	Amount for 3 years, t (2+3+4)	Average monthly for 3 years, t	seasonality index,
		Amount for 3 years, t (2+3+4)	Average monthly for 3 years, t	seasonality index,









September

Rice. 11.1. Seasonal fluctuations in fuel consumption in agricultural enterprises for 3 years.

For clarity, on the basis of seasonality indices, a seasonal wave graph is constructed (Fig. 11.1). Months are placed along the abscissa, and seasonality indices in percent are placed along the ordinate (Table 11.6, column 7). The overall average monthly for all years is at the level of 100%, 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 y-axis.

The points are connected to each other by a smooth broken line.

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

for each year we calculate the average monthly value;
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%;
for the entire period, we calculate the average seasonality indices according to the formula of the arithmetic mean simple of the monthly seasonality indices calculated for each year. So, for example, we get the average seasonality index for January if we add the January values of the 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 trend of growth (decrease) in the series of dynamics and measure seasonal fluctuations more accurately.

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

Bringing time series to the same base

In economic practice, it often becomes necessary to compare several series of dynamics with each other (for example, indicators of the dynamics of electricity production, grain production, car sales, etc.). To do this, you need to convert the absolute indicators of the compared time series into derivative series of relative base values, taking the indicators of any one year as a unit or as 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 base of comparison, but in economic research, the base of comparison must be 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 level of 1990 as a base for comparison.

Time series alignment methods

To study the patterns (trends) in the development of the phenomenon under study, data are needed for a long period of time. 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, dry years, etc.). Almost all series of dynamics of economic indicators on the chart have the form 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 the actual data of a series of dynamics and from the schedule. But statistics should not only determine the general trend in the development of the phenomenon (growth or decline), but also give quantitative (numerical) characteristics of development.

Trends in the development of phenomena are studied by the methods of leveling the series of dynamics:

Interval coarsening method
moving average method

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

The method of enlargement of time intervals (column 3).

Considering that the series of dynamics is small, the intervals are taken for three years and the averages are calculated for each interval. The average annual volume of grain production for three-year periods is calculated by the formula of the simple arithmetic mean and is related 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). For 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 results of the calculation 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 columns 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 above example, five-bar (for five-year periods) moving averages are calculated and referred to the middle year in the corresponding five-year period. So, for the first five years (1981-1985), using the simple arithmetic mean formula, the average annual volume of grain production was calculated and recorded in Table. 11.7 vs. 1983 (73.8+ 98.0+ 104.3+ 85.1+ 98.6): 5= 92.0 Mt; 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 \u003d 493.5: 5 \u003d 98.7 million tons

For subsequent five-year periods, the calculation is made in a similar way by deleting 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 blank.

How long should the time periods be? Three, five, ten years? The question is decided by the researcher. In principle, the longer the period, the more smoothing occurs. But we must take into account the length of the series of dynamics; do not forget that the moving average method leaves the 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.

Table 11.7 Adjustment of data on grain production in Russia for 1981-1992

Produced, million tons	Average for 3 years, million tons	Rolling amount for 5 years, million tons	Estimated indicators
		Rolling amount for 5 years, million tons

Analytical alignment method

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

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

The required quantities for solving the system of equations are calculated and given in the table (see columns 6 - 8), we substitute them into the equation:

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

Substitute the value of the parameters and get the equation of the straight line:

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

Rice. 11.2. Grain production in Russia for 1981-1982.

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

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

For clarity, the indicators of the levels of the actual series of dynamics and the aligned series will be plotted on a graph (Fig. 11.2). The actual data is represented by a broken line in black, indicating rises and falls in grain production. The remaining lines on the chart show that the use of the moving average method (a line with truncated ends) allows you to significantly align the levels of the dynamic series and, accordingly, make the broken curved line on the chart smoother and smoother. However, aligned lines still remain curved lines. Constructed on the basis of the theoretical values of the series obtained by mathematical formulas, the line strictly corresponds to a straight line.

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

Analytical indicators of changes in the levels of the series

Name of indicator	Basic
Absolute growth
Growth rate, %		;
Growth rate, %
Absolute value 1st % 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 growth() - this is the difference between the next level of the series and the previous (or base). If the difference between the next and the previous is chain absolute growth:

if between the next and the base, then basic:

Substituting the values of cement output 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 growth calculated in two ways:

1) as the simple arithmetic mean of annual chain increments:

Substituting into the formula (1.10.3) the values from column 2 (Table 1.10.4) into the numerator and n\u003d 11 (number of compared years or number of periods) into the denominator, we get:

2) as the ratio of basic 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 in percent, as in our case:

Substituting in the formula (1.10.5) the corresponding data of column 1 of Table. 1.10.4, we obtain the values of the chain growth rate, see column 4 of Table. 1.10.4.

Base growth rate is the ratio of each subsequent level to one level, taken as the base of comparison:

(1.10.6)

Substituting in formula (1.10.6) the same data 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 basic growth rates. Knowing the basic rates, you can calculate the chain by dividing each subsequent basic rate by the previous one.

Average growth rate is calculated by the formula of the geometric mean of the chain growth coefficients:

(1.10.7)

To do this, the indicators of column 4, expressed as a percentage, will be converted into coefficients, substituting into formula (1.10.7), we get:

Average growth rate can be calculated 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, which characterizes the relative rate of change in the level of the series per unit time. The growth rate shows by what share (or percentage) the level of a given period or point in time is more (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 one, 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 basic one - between the basic 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 similarly to the growth rate by the formula:

Thus, the production of cement for the studied years increased by an average of 0.75% per year.

In statistical practice, instead of calculating and analyzing growth and growth rates, 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 of 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).

Middle level a number of dynamics () is calculated from the chronological average. Average chronological called the average, calculated from values that change over time. Such averages summarize chronological variation. The chronological average reflects the totality of the conditions under which the phenomenon under study developed in a given period of time.

Methods for calculating the average level of the interval and moment series of dynamics are different. For interval equally spaced series, the average level is found by the simple arithmetic mean formula and for unequally spaced series by the weighted arithmetic mean:

(1.10.11)

where - the level of a series of dynamics;

n - number of levels;

Thus, Table 1.10.4 shows an interval series of dynamics with equally spaced levels. Based on 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 counting.

The average level of the moment equidistant series of dynamics is found by the formula of the average chronological:

(1.10.12)

The average level of moment series of dynamics with unequally spaced levels is determined by the formula of the average chronological weighted:

where , - levels of a series of dynamics;

The duration of the time interval between levels.

Time series alignment methods

An important task of statistics in the analysis of time series is to determine the main development trend inherent in a particular time series. For example, behind fluctuations in the yield of any agricultural crop in certain years, the trend of growth (decrease) in yield may not be directly visible, and therefore must be identified by statistical methods.

Methods for analyzing the main trend in the series of dynamics are divided into two main groups:

1) smoothing or mechanical alignment of individual members of the time series using the actual values of neighboring levels;

2) alignment using a curve drawn between specific levels in such a way that it reflects the trend inherent in the series and at the same time frees it from minor fluctuations.

Consider the methods of each group.

Interval coarsening method. If we consider the levels of economic indicators for short periods of time, then due to the influence of various factors acting in different directions, in the series of dynamics there is a decrease and increase in these levels. This makes it difficult to see the main trend in the development of the phenomenon under study. In this case, for a visual representation of the trend, the method of enlargement of intervals is used, which is based on the enlargement of the time periods to which the levels of the series belong. For example, the daily output series is replaced by the monthly output series, and so on.

Simple moving average method. Smoothing a series of dynamics using a moving average consists in calculating the average level from a certain number of the first levels in 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 one next. 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 than the actual one subject to fluctuations due to random reasons, and more clearly expresses the main trend in the growth of yields over the period under study, associated with the action of long-term existing causes and development conditions.

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 shortcoming is eliminated by using the method of analytical alignment for the analysis of the main trend.

Analytical alignment involves the representation of the levels of a given series of dynamics as a function of time - y=f(t).

Various functions are used to display the main trend in the development of phenomena over time: degree polynomials, exponents, logistic curves, and other types. The polynomials have the following form:

polynomial of the first degree:

polynomial of the second degree:

third degree polynomial:

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