The growth rate has decreased. The average growth rate is calculated using the formula

(Tr) is an indicator of the intensity of changes in the level of a series, which is expressed as a percentage, and the growth coefficient (Kr) is expressed in shares. Kr is defined as the ratio of the subsequent level to the previous one or to the indicator taken as the basis of comparison. It determines how many times the level has increased compared to the base level, and in the case of a decrease, what part of the base level is being compared.

We calculate the growth rate, multiply by 100 and get the growth rate

Can be calculated using the formulas:

Also, the growth rate can be determined as follows:

The growth rate is always positive. There is a certain relationship between the chain and base growth rates: the product of the chain growth coefficients is equal to the base growth rate for the entire period, and the quotient of dividing the subsequent base growth rate by the previous one is equal to the chain growth rate.

Absolute increase

Absolute increase characterizes the increase (decrease) in the level of a series over a certain period of time. It is determined by the formula:

where уi is the level of the period being compared;

Уi-1 - Level of the previous period;

Y0 is the level of the base period.

Chain and basic absolute increases are related each other in this way: the sum of successive chain absolute increases is equal to the base, i.e., the total increase for the entire period of time:

Absolute increase can be a positive or negative sign. It shows how much the level of the current period is higher (lower) than the base one, and thus measures the absolute rate of growth or decline in the level.

(Tpr) shows the relative magnitude of the increase and shows by what percentage the compared level is greater or less than the level taken as the basis of comparison. It can be either positive or negative or equal to zero, it is expressed in percentages and shares (growth rates); calculated as a ratio absolute growth to the absolute level taken as the base:

The growth rate can be obtained from the growth rate:

The growth rate can be obtained as follows:

Absolute value of 1% increase

The absolute value of 1% growth (A%) is the ratio of absolute growth to the growth rate, expressed as a percentage and shows the significance of each percentage of growth over the same period of time:

Absolute value of one percent increase equal to one hundredth of the previous or base level. It shows what absolute value is hidden behind relative indicator- one percent increase.

Examples of calculations of dynamics indicators

Before studying the theory on the topic of dynamics indicators, you can look at examples of problems on finding: growth rate, growth rate, absolute growth, average dynamics

About dynamics indicators

When studying the dynamics of social phenomena, the difficulty arises in describing the intensity of change and calculating the average indicators of dynamics that are asked to students.

Analysis of the intensity of change over time occurs using indicators that are obtained by comparing levels. These indicators include: growth rate, absolute growth, absolute value of one percent growth. To generalize the dynamics of the phenomena under study, the following are determined: the average levels of the series and the average indicators of changes in the levels of the series. Dynamics analysis indicators can be determined using constant and variable comparison bases. Here it is customary to call the comparable level the reporting level, and the level from which the comparison is made is the base level.

For calculation dynamics indicators on a permanent basis, you need to compare each level of the series with the same basic level. As a base level, only the initial level in the dynamics series or the level from which it begins is used new stage development of the phenomenon. The indicators that are calculated in this case are called basic. To calculate dynamics analysis indicators on a variable basis, you need to compare each subsequent level of the series with the previous one. The calculated dynamics analysis indicators will be called chain indicators.

Growth rate - relative rate of change in the level of a time series per unit of time.

Growth rate is the ratio of one level of a time series to another, taken as a basis for comparison; expressed as a percentage or growth rates.

Absolute increase - the difference between two levels of a time series, one of which (the one under study) is considered as current, the other (with which it is compared) as the base one. If each current level (yt or y(t)) is compared with the immediately preceding one (yt-1) or y(t-1)), then chain absolute increases are obtained. If the level yt is compared with the initial level of the series (y0) or another level taken as the basis of comparison (yt), then basic absolute increases are obtained. Increases are expressed either in absolute values ​​or as a percentage, in units.

1. Rate of increase

TP growth rate is defined as the ratio of absolute growth this level to the previous or basic one.

Rate of increase - the ratio of the growth of the indicator under study to the corresponding level of the time series taken as the basis of comparison.

1. Averages

Absolute value of one percent increase in Ai serves as an indirect measure of the baseline level. It represents one hundredth of the base level, but at the same time it also represents the ratio of absolute growth to the corresponding growth rate.

To characterize the dynamics of the phenomenon being studied over a long period, a group of average dynamics indicators is calculated. We can distinguish two categories of indicators in this group: a) average levels of the series; b) average indicators of changes in the levels of the series.

Average series levels are calculated depending on the type of time series.

For an interval series of dynamics of absolute indicators, the average level of the series is calculated using the simple arithmetic average formula.

Average level of moment series with unequal intervals is calculated using the weighted arithmetic mean formula, where the duration of time intervals between time points of changes in the levels of the dynamic series is taken as weights.

Average absolute increase (average growth rate) is defined as the arithmetic average of the growth rate indicators for individual periods of time.

Average growth rate calculated using the geometric mean formula from the growth coefficients for individual periods.

Average growth rate expressed as a percentage:

Average growth rate , for the calculation of which the average growth rate is initially determined, which is then reduced by 100%. It can also be determined by decreasing the average growth rate by one.

Section 7 Indexes in Statistics

7.1. The concept of statistical indices and their role in economics

1. Individual indices

Statistical science has in its arsenal a method that allows one to compare the indicators of a phenomenon in time and space and compare actual data with any standard, which can be a plan, forecast or some kind of standard. This is an index method that operates with relative indicators, called indices in statistics.

In the practice of statistics, indices, along with average values, are the most common statistical indicators. With their help, the development of the national economy as a whole and its individual sectors is characterized, the role of individual factors in the formation of the most important economic indicators is studied, indices are also used in international comparisons of economic indicators, determining the standard of living, monitoring business activity in the economy, etc.

Index (Latin index) is a relative value showing how many times the level of the phenomenon being studied under given conditions differs from the level of the same phenomenon under other conditions. Differences in conditions can manifest themselves in time (dynamic indices), in space (territorial indices) and in the choice of any conditional level as a basis for comparison.

According to the coverage of the elements of the population (its objects, units and their characteristics), indices are distinguished individual e (elementary) and summary (complex), which, in turn, are divided into general and group.

In statistics, an index is understood as a relative indicator that expresses the ratio of the magnitudes of a phenomenon in time, space, or a comparison of actual data with any standard.

The following tasks are solved using indexes:

measuring the dynamics of a socio-economic phenomenon over two periods of time or more;

measuring the dynamics of the average economic indicator;

measuring the ratio of indicators in different regions;

determining the degree of influence of changes in the values ​​of some indicators on the dynamics of others.

In international practice, indices are usually denoted by the symbols i and I (the initial letter of the Latin word index). The letter “i” denotes individual (private) indices, the letter “I” denotes general indices.

In addition, certain symbols are used to denote index structure indicators:

q - quantity (volume) of any product in physical terms;

p - unit price of goods;

z is the cost per unit of production;

t is the time required to produce a unit of product;

w - production output in value terms per worker or per unit of time;

v - production output in physical terms per worker or per unit of time;

T - total time expenditure (tq) or number of workers;

pq - cost of production or turnover;

zq - production costs.

The sign at the bottom right of the symbol means the period: 0 - base; 1 - reporting.

All indices can be classified according to the following criteria:

extent of coverage of the phenomenon;

comparison base;

type of scales (co-meter);

form of construction;

object of study

composition of the phenomenon;

calculation period.

According to the degree of coverage of the phenomenon, indices are individual And summary (are common).

Individual indices serve to characterize changes in individual elements of a complex phenomenon. For example, a change in the volume of production of certain types of products (TVs, electricity, etc.), as well as in the prices of shares of an enterprise.

Summary (complex) indexes serve to measure a complex phenomenon, the component parts of which are directly incommensurable. For example, changes in the physical volume of products, including goods of different names, the price index of shares of regional enterprises, etc.

According to the comparison base, indices are dynamic And territorial.

Dynamic indexes serve to characterize changes in a phenomenon over time. For example, the product price index in 1996 compared to the previous year. When calculating dynamic indices, the value of the indicator is compared in reporting period with the value of the same indicator for the previous period, which is called the base period. Dynamic indexes can be basic or chain.

Territorial indices serve for interregional comparisons. They are usually used in international statistics.

According to the type of scales, indices come with permanent And variable scales.

According to the form of construction they distinguish aggregate And average indices . The aggregate form is the most common. Average indices are derived from aggregate indices.

Depending on the nature of the object of study, indices can be labor productivity, cost, physical volume of production, etc.

According to the composition of the phenomenon, indices are permanent (fixed) composition and variable composition.

According to the calculation period, indices are annual, quarterly, monthly, weekly.

Depending on the economic purpose, individual indices are: physical volume of production, cost, prices, labor intensity, etc.

individual index of physical volume of products shows how many times the output of any one product increased (decreased) in the reporting period compared to the base period or what percentage is the increase (decrease) in the output of the product; if you subtract 100% from the index value expressed as a percentage, the resulting value will show how much production output has increased (decreased);

individual price index characterizes the change in the price of one specific product in the current period compared to the base;

an individual unit cost index shows the change in the cost of one specific type of product in the current period compared to the base period;

labor productivity can be measured by the quantity of products produced per unit of time (v), or the cost of working time to produce a unit of output (t); therefore, it is possible to construct an index of the quantity of products produced per unit of time;

labor productivity index based on labor costs;

the individual product cost (turnover) index reflects how many times the cost of a product has changed in the current period compared to the base period or what percentage is the increase (decrease) in the cost of the product.

When analyzing the development of phenomena, there is often a need to give a generalized description of the intensity of development on a long period. What are average dynamics used for:

1. Average absolute increase is found by the formula:

Where n- number of periods (levels), including the base one.

2. Average growth rate is calculated using the formula for the geometric mean simple of chain growth coefficients:

, .

When it is necessary to calculate average growth rates for periods of different lengths (unequally spaced levels), then a geometric mean weighted by the duration of the periods is used. The weighted geometric mean formula will look like:

where t is the time interval during which this growth rate is maintained.

3. Average growth rate cannot be determined directly from successive growth rates or average absolute growth rates. To calculate it, you must first find the average growth rate and then reduce it by 100%:

Example 7.1. There is data on increases in sales volumes by month (as a percentage of previous month): January – +4.5, February – +5.2, March – +2.4, April – -2.1.

Determine the growth and gain rates for 4 months and the monthly averages.

Solution: we have data on chain growth rates. Let's convert them into chain growth rates using the formula: T r = T r + 100%.

We get the following values: 104.5; 105.2; 102.4; 97.9

For calculations only growth factors are used: 1.045; 1.052; 1.024; 0.979.

The product of chain growth coefficients gives the base growth rate.

K = 1.045 1.052 1.024 0.979 = 1.1021

Growth rate for 4 months T r= 1.1021·100= 110.21%

Growth rate for 4 months T pr= 110,21 – 100 = +10,21%

The average growth rate is found using the simple geometric mean formula:

Average growth rate for 4 months = 1.0246·100= 102.46%

Average growth rate for 4 months = 102.46 – 100 = +2.46%

4. Average level of interval series is found by the simple arithmetic average formula if the intervals are equal, or by the weighted arithmetic average if the intervals are not equal:

, .

where t is the duration of the time interval.

5. Average level of the moment series of dynamics It is impossible to calculate this way, since individual levels contain elements of repeated counting.

a) Average torque level equidistant row dynamics is found using the average chronological formula:

.

Where at 1 And y n- level values ​​at the beginning and end of the period (quarter, year).

b) Average level of moment series of dynamics with unequally spaced levels determined by the chronological weighted average formula:

Where t- duration of the period between adjacent levels.

Example 7.2. The following data are available on production volumes for the first quarter (thousand units) - January - 67, February - 35, March - 59. Determine the average monthly production volume for the 1st quarter.

Solution: according to the conditions of the problem, we have an interval series of dynamics with equal periods. The average monthly production volume is found using the simple arithmetic average formula:

thousand pieces

Example 7.3. The following data are available on production volumes for the first half of the year (thousand tons) - average monthly volume for the 1st quarter - 42, April - 35, May - 59, June - 61. Determine the average monthly production volume for the half year.

Solution: according to the conditions of the problem, we have an interval series of dynamics with unequal periods. The average monthly production volume is found using the weighted arithmetic average formula:

Example 7.4. The following data is available on the balance of goods in the warehouse, million rubles: 1.01 – 17; on 1.02 – 35; on 1.03 – 59; at 1.04 – 61.

Determine the average monthly balance of raw materials and supplies in the enterprise's warehouse for the first quarter.

Solution: According to the conditions of the problem, we have a moment series of dynamics with equally spaced levels, therefore the average level of the series will be calculated using the average chronological formula:

million rubles

Example 7.5. The following data is available on the balance of goods in the warehouse, million rubles: 1.01.11 – 17; at 1.05 – 35; on 1.08 – 59; on 1.10 – 61, on 1.01.12 – 22.

Determine the average monthly balance of raw materials and supplies in the enterprise's warehouse for the year.

Solution: According to the conditions of the problem, we have a moment series of dynamics with unequally spaced levels, therefore the average level of the series will be calculated using the chronological weighted average formula.

The most important indicator of production efficiency in analyzing the financial situation of a company is the growth rate indicator. Let's talk about the features of its calculation.

How to Calculate Growth Rate: Formula

This term shows the change in the value of any economic or statistical indicator in the current period to its initial value (which is the base value) over a certain time period. It is measured in percentages or coefficients.

For example, when comparing the volume of goods output at the end of the year (let’s say, in the value of 100,000 rubles) to the volume indicator at the beginning of the year (70,000 rubles), the growth rate is found by the ratio of the final value to the initial value: 100,000 / 70,000 = 1.428. The growth index in the example was 1.429. This means that at the end of the year the output volume was 142.9%.

TR = P t / P b x 100%,

where P k and P b are indicators of the values ​​of the current and base periods.

The growth rate shows the intensity of changes in any process in relation to its initial (base) value. The result of the calculations is one of three options:

TP is more than 100%, therefore, the final value has increased compared to the initial value, i.e. there is an increase in the indicator;

TP = 100%, i.e. there were no changes either up or down - the indicator remained at the same level;

TR is less than 100%, which means that the analyzed indicator decreased by the beginning of the period.

This growth rate is called the base rate, since the basis for comparison across periods remains unchanged - the indicator at the beginning of the period. If the comparative base changes, and the growth rate is calculated by the ratio of the current value to the previous one (and not the base one), then this indicator will be a chain one.

How to calculate chain growth rates

Let's look at an example of calculating the base and chain growth rates:

Chain growth rates characterize the saturation of changes in levels from quarter to quarter, while basic ones reflect it as a whole for the entire time interval (the indicator for the 1st quarter is the comparison base).

Comparing the indicators in the above example, it can be noted that a number of values ​​calculated at the beginning of the period have a smaller amplitude of fluctuations than chain indicators, the calculations of which are tied not to the beginning of the year, but to each previous quarter.

How to calculate growth rate

In addition to calculating growth rates, it is customary to calculate growth rates. These values ​​are also basic and chain. The base increase is determined as the ratio of the difference between the indicators of the current and base periods to the value of the base period according to the formula:

∆ TR = (P current – ​​P base) / P base x 100%

Chain growth is calculated as the difference between the current and previous indicators divided by the growth rate previous period:

∆ TR = (P flow – P pr.p) / P pr.p x 100%.

More in a simple way calculation is the formula: ∆ TR = TR – 100%, where the calculated growth rate indicators are reduced by 100%, i.e. the original value. The growth rate indicator, in contrast to the growth rate values, can have a negative value, since the growth rate (or decrease) shows the dynamics of changes in the indicator, and the growth rate indicates what nature they are.

Continuing the example, let’s calculate the volume increases in the periods under consideration:

Analyzing the calculation results, an economist can conclude:

The increase in volumes was observed in the 2nd and 4th quarters, and in the 2nd it was the smallest (3.3%). In the 3rd quarter, output volume decreased by 6.7% compared to the beginning of the year;

The chain growth rates revealed deeper fluctuations: the volumes of the 3rd quarter decreased in relation to the indicators of the 2nd by 9.7%. But the output of goods in the 4th quarter increased by almost a third compared to the results of the 3rd quarter. Such significant changes in production volumes may indicate seasonality of manufactured products, interruptions in the supply of necessary raw materials, or other reasons that the analyst is investigating.

How to calculate average growth rate

The average growth rate is a general characteristic of the level of change. The calculation of average growth and increment rates is also divided into basic and chain. To determine the average growth rate, the calculated indicators for periods are added and divided by the number of periods. The average growth rates are found in the same way. Let's return to the previous example, calculating the average values ​​of the basic growth and increment rates, as well as similar chain indicators.

The obtained figures indicate that, on average, since the beginning of the year, output volumes have increased by 5.5%, and on a quarterly basis the growth was 7.4%.

Growth and growth rates are calculated very often, and not only in statistics, but also in economics, manufacturing and even sociology and law, and every student is faced with the task of understanding what these indicators are, how they are calculated and how they differ. Often students begin to get confused about them, let's try to prevent this.

Growth rate– relative economic indicator, showing the percentage growth of one indicator over the same indicator of the previous period.

For example, using growth rate you can calculate how much your wage this year was a percentage compared to last year.

If the result is more than 100, there is an increase; if less than 100, there is a decrease.

Example No. 1.1 The average wage of a worker in 2016 was 33,000 rubles, and in 2015 it was 31,500 rubles; let’s calculate the growth rate based on the given conditions. You can find out more about how to calculate the growth rate.

Growth rate = 33000/31500 * 100 - 100 = 104.76-100 = 4.76%. Thus, the average salary increased by 4.76% (+4.76%).

Example No. 2.2

Growth rate = 139000/142000 *100 -100 = 97.89-100 = -2.11%. The value turned out to be negative, which means the rate of decline in profit was 2.11%, or, more simply put, the profit of the reporting year decreased by 2.11% compared to the profit of 2015.

How else can you calculate the growth rate?

If in the task you calculated the absolute deviation, then you can use this value and divide it by the value of the base year, consider example No. 1.1

Absolute deviation = 33,000 – 31,500 = 1,500 rubles.

Growth rate = 1500 / 31500 * 100% = 4.76%. We see that changing the calculation method left the result unchanged, so choose the method that you like best.

Let's return to the topic of the article and summarize what the difference is between the growth rate and the growth rate. The difference between the indicators is as follows:

1. Calculation method.
2. The growth rate shows what percentage one indicator is relative to another, and the growth rate tells how much it has grown.
3. The growth rate is calculated based on the growth rate, but not vice versa.
4. The growth rate cannot be negative, and the growth rate can be both positive and negative.

If, after reading the material, it is not clear to you how to calculate the indicator or you still have questions on the topic, do not hesitate to ask them in the comments.