Ratios

Lecture, the 22d of February

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Statistical Indicators

Statistical indicator is a numeric characteristic of social and economic processes.

All indicators can be classified as individual and summary

Individual & Summary

Individual indicators characterize only one unit of population

Summary indicators characterize one class of population or the whole population

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Individual & Summary

1 2

Summary Individual

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Individual

Individual absolute value characterizes one unit of the population investigated. It reflects the size of quantitative traits in individual units of the studied population.

Individual absolute values are obtained in the process of statistical observation and characterize the individual units of a population (a man's height, weight, volume of production, etc.)

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Summary The summary, or total absolute value characterizes the

group of units together, or population as a whole. It expresses the size, amount of quantitative traits in the whole studied population.

Summary indicator gives us the characteristic size of the phenomenon analyzed on a given set of objects or any part of the set. The total values are obtained by direct counting of units of observation or as a result of summation of the values of quantitative traits, which have a unit (for example, the population of the country, a separate branch of production).

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Individual & Summary

When we study wages the individual absolute indicator is a specific amount of wage for each worker, and the summary absolute value is the wages fund for the entire company, for some classes of workers or the payroll of a structural unit (for example, the first shop)

Summary indicators can also be classified as volumetric and calculated

Volumetric & Calculated

Volumetric indicators are received by adding values of population separate units

Calculated indicators can be received by making different calculations

All indicators can also be classified as absolute, ratio and average

Absolute, Ratio & Average

Absolute indicators are measured in natural units (ton, kg, meter), cost units (USD, ruble, euro), and labour units (man-hour, man-day)

Absolute values are the basis for the calculation of various statistical ratios

Average indicators will be described in the next lecture

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Example 1 302. What is it?

Possible answers:a) The date – the 2nd of March;

b) personal number of Dalaloyan Anait;

c) a digital; г) a number;d) population of Surgut on the

1st of January, 2010; e) something else – point out

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1.Absolute indicators

In statistics, bare numbers can not exist without a specific reference to the unit of measurement, time and place

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Ratios

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Ratio

Any relative value is the result of comparison of two variables 

Ratio, or relative indicator RI represents the result of dividing one absolute indicator A by another B and expresses a ratio between two quantitative indicators

LOGORatio

RI=A/B

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Ratios

The ratio of quantities A and B can be expressed as:

the ratio of A to B

as B is to A

A:B. The quantities A and B are sometimes called

terms with A being the antecedent and B being the consequent

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Ratios

The proportion expressing the equality of the ratios A:B and C:D is written A:B=C:D or A:B::C:D.

Again, A, B, C, D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means. The equality of three or more proportions is called a continued proportion

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Ratios

A ratio that has integers for both quantities and that cannot be reduced any further (using integers) is said to be in simplest form or lowest terms

Sometimes it is useful to write a ratio in the form   1:n or n:1    to enable comparisons of different ratios.

For example, the ratio 4:5 can be written as   1:1.25 (dividing both sides by 4)

Alternatively, 4 : 5 can be written as  

0.8: 1   (dividing both sides by 5)

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Ratios

Where the context makes the meaning clear, a ratio in this form is sometimes written without the 1 and the colon, though, mathematically, this makes it a factor or multiplier

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Ratios

Ratios express numeric relation specific to particular social phenomena or processes. 

The indicator A is called compared value.The indicator B, that is compared with indicator A, is called the base or a base of comparison.

 When both indicators namely A and B have the same unit of measure, the result is expressed in coefficient (e.g. 0.3), percentage (30%) or per mil (from Latin pro mille) (300 ‰) 

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Ratios

In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient

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Quotient

A quotient is the result of a division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the divident, and 3 the divisor. The quotient can also be expressed as the number of times the divisor divides into the dividend.

A quotient can also mean just the integer part of the result of dividing two integers. For example, the quotient of 13 ÷ 5 would be 2 while the remainder would be 3

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Properties of RatioRatio shows how many times the compared value

A is more or less than the base B, or what proportion of A is in relation to B. In some cases, the relative value indicates how many units of A corresponds per unit of B.

Another important property - the ratio abstracts from absolute values and allows to compare indicators, the absolute amounts of which are not directly comparable

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Expression forms of ratios

LOGOExpression forms of ratios

A comparison of the absolute values with the same name gives us unnamed ratios. They can be expressed in the form of shares, times, percentages, per mils, etc.

A comparison of values with different names gives us named ratios. Their name is formed as combination of the names of A and B.The choice of form depends on the nature of analytical problems: just to express the ratio most clearly

LOGOSimple Division

RI=A/BThis case of Ratio shows: 1 - how many times the compared value A is more or

less than the base B2 - what proportion of A is in relation to B3 - in some cases, the Ratio indicates how many units

of A corresponds per unit of B

LOGOPercentage

RI=A*100/BTo switch from % to coefficient, RI should be

divided by 100To obtain % from coefficients, we’ll multiply

RI by 100

LOGOPer mil

RI=A*1000/BPer mil: Latin pro mille, i.e. per one

thousand. This form is used in demographic statistics

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Examples of RatioThe quantities being compared in a ratio might

be physical quantities such as speed, or may simply refer to amounts of particular objects. A common example of the latter case is the weight ratio of water to cement used in concrete, which is commonly stated as 1:4. This means that the weight of cement used is four times the weight of water used. It does not say anything about the total amounts of cement and water used, nor the amount of concrete being made

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Display parameters

The ratio of width to height of typical computer displays

Older televisions have a 4:3 ratio which means that the height is 3/4 of the width. Widescreen TVs have a 16:9 ratio which means that the width is nearly double the height

LOGOPer mil

RI=A*1000/BTo pass from per mil to coefficients, RI

should be divided by 1000To obtain per mil from coefficients, multiply

RI by 1000To go from per mil to per cent, RI divide by

10To move from per cent to per mil,

multiply RI by 10

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There are seven kinds of ratios

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Kinds of Ratios

Kinds of RI

Plan RatioPR

Intensity Ratio IR

Ratio of Plan FulfillmentPF

Dynamics Ratio DR

Coordination Ratio CR

Structure Ratio SR

Ratio of Comparison RCom

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Ratio of Plan target RP

Plan Ratio PR is a ratio between the planned value of future period and real achieved level of basic period (the previous or past or base value):

where - plan indicator;

- real level of basic period

0

100%,plxPR

x

0

100%.plxPR

x

0x

0x

plx

plx

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PRPlan Ratio is the ratio between the value of indicator set at the

planned period and the value of indicator achieved by the planned period or by the period taken as the basis of comparison

PR is expressed in coefficients or percentages after additional multiplication by 100%

In case of coefficients PR shows by how many times the plan is larger or smaller than achieved values by the planned period

In case of percentage PR shows by how many percent the planned value is larger or smaller than the actual value in previous or past value

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PF

Ratios of plan fulfillment PF characterize the extent of accomplishment of plan target

PF is the ratio between the current or reporting value and the planned value :

where - achieved indicator ;

- planned indicator.

It shows how the plan has been fulfilled

1 100%,pl

xPF

x

1x plx

1x

plx

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DR Dynamics Ratio or Time Ratio DR – represents

the ratio of values of the same indicator during different periods of time). It is a ratio between the current or reporting value x1 and the past or base meaning x0 and expressed in percentages

where x1 – real, achieved indicator ; x0 – basic indicator.

There are two kinds of DR – chain and basic

1

0

100%,x

DRx

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Basic DR

Basic ratio of dynamics – ratio between the value of indicator of current period and the value considered as the basis of comparison

where x1 – current level; x0 – basic level

1

0

100%,Bs

xDR

x

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Chain DR

Chain ratio of dynamics – ratio between the current value and the past period value. Shows the change of the indicator from one period to another or from one moment of time to another.

where xi – current level; xi-1 – previous adjacent level

1

100%,iCh

i

xDR

x

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Chain method

While using chain calculation method we should compare each consequent level with the previous adjacent.

Time series analysis indicates its levels by letter Y instead of X

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Basic vs Chain

Year У DRBs DRCh

2006 У0 1 -

2007 У1 У1/ У0 У1/ У0

2008 У2 У2/ У0 У2/ У1

2009 У3 У3/ У0 У3/ У2

2010 У4 У4/ У0 У4/ У3

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Basic vs Chain - 1There is a connection between chain and basic

DRs1.Multiplying each chain DR we’ll get basic DR

of the last period:

0

4

3

4

2

3

1

2

0

1

y

y

y

y

y

y

y

y

y

y

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Basic vs Chain - 2 2.Dividing the following basic DR by the

previous DR we’ll get the chain DR of the following period:

3

4

0

3

0

4

y

y

y

y

y

y

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Basic vs Chain - 3

3. Dividing the following basic DR by the chain DR of the same period we will get the previous basic DR:

4 4 3

0 3 0

y y y

y y y

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Example 3

The sale of cotton fabric by a section of department store in January totaled 3,956,000 rubles, in February – 4,200,000 rubles, in March – 4,700,000 rubles

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Example 1Rates of growth

Basic DRs (basis – level of sales in January)

DRF/J = 4200 * 100%

=106,3%

3950

DRMJ = 4700 * 100%

=118,9%

3950

Chain RDs

DRF/J = 4200 * 100%

=106,3%

3950

DRM/F = 4700 * 100%

=111,9%

4200

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Correlation of 3 ratios

DR = PR * PF

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Example 2In the third quarter the turnover was 150 million rubles.

Plan for the fourth quarter was 180 million rubles. Real turnover in the fourth quarter was 202,5 million rubles. Calculate DR, PR, PF and show their interconnection

y0=150;

y1pl=180;

y1=202,5

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Example 2

1801,2;

150202,5

1,125;180

202,51,35

150

PR

PF

DR

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Example 2

Interconnection of DR, PR & PF:

DR= PR* PF

1,35 =1,2 х 1,125

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Example 3The increase of the output of a branch during

2010 was planned to be 7.5%. Real increase during 2010 was 109,5%. Determine the ratio of plan fulfillment of the output.

PF = 109,5 * 100% = 102%

107,5

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Ratio of comparisonRCom

LOGORCom

Ratio of comparison is the ratio of similar indicators related to different objects. RCom is a ratio between two identical characteristics describing

different populations:

1

1

parameter of population ARCom

parameter of population B

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Example 4

The water reserves in Lake Baikal - 23000 cu. km, and in Lake Ladoga 911cu. km

1

2300025,25

911RCom times

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Example 4

Another way is to calculate a share in per cent, it gives us an idea of the

next ratio

2

9110.0396

23000RCom

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Structure Ratio SR

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SR

Structure Ratio is a ratio of parts and the whole characterizing the structure of the population, i.e. a share of each part in the population. SR is expressed in unit shares or in per cent:

The sum of SRs calculated for all parts of a population is equal to 1 or 100% depending on the unit of measure

100%.part of total

SRtotal

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Example 5

The total number of Russian population at the beginning of 2009 was  equal to 141.9 million, 103.7 million were urban residents, 38.2 million - rural.Calculating SR, we can determine the structure of the population by place of residence:

103.7*100% 73%

141.938.2

*100% 27%141.9

urban

rural

SR

SR

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Intensity Ratio IR

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IR

Intensity Ratio IR shows how much a process under analysis is spread (birth rate, death rate, GDP per capita). IR characterizes the distribution of the process in a certain environment (density, intensity of a certain event)

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IR

Intensity Ratio IR is always a ratio of absolute values with different units of measure. For instance density of population in persons per one km² we receive dividing number of population in thousands by square in thousand km²:

population density = the total number of people / area of land (measured in km² or sq miles)

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IR

IR is the ratio of different indicators relating to the same object

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Example 6

Number of retailers in the region at the end of the year was 6324. The population of the region on the same date amounted to 234.2 thousand. 

IR = 6324 * 10 000 / 234 200 = 27.003                Unit of measure – number of retailers per 10 thousand people living in the region

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IR

IR characterizes the distribution of the process in a certain environment

For example, production per capita is calculated as the ratio of annual production by the average annual population, the fertility rate is obtained by dividing the number of births during a year by the average number of women in the fertility age (15-49 years)

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Coordination Ratio CR

LOGOCoordination Ratio CR

CR is a ratio between two parts of one population

_1_ _ _1.

_1_ _ _ 2

parameter of partCR

parameter of part

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CR

Coordination Ratio is the ratio of parts of the whole between each other

RC = Part of the whole/ Another part of the same population

CR is put into times or unit shares. Multiplication by 10 and 100 is allowed if logic requires that – there cannot be a ratio between people: 1 to 1.5, there can be 10 to 15

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RCRC is used for additional characteristic of structure (e.g. number of women for per 1000 men and vice versa)

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Example 7

At the beginning of the year the number of employees with higher education working in “Trade house” association was equal to 53, while the number of employees with specialized secondary education was 106

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Example 7

Принимаем за базу сравнения численность специалистов с высшим образованием: We take the number of employees with higher education as the base of comparison

RC = 106 / 53 = 2.0 : 1.0,

i.e. for every two employees with specialized secondary education there is one with higher education

LOGO Ratio of level of economic

development LED

LOGOLED ratio

Ratio of level of economic development characterizes the size of different types of production per capita

It characterizes the size of output per capita. We put “capita” in the denominator – average population size

.volume of production of a certain commodity

LED ratioaverage annual population

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LED ratio

LED ratio is a case of intensity ratio

Your Task 3

• 1.Send a request for a Ratio Puzzle on email asidenko@fa.ru or oknedis@bk.ru

• 2.Solve the puzzle: explain all calculations below the table

3.The number of points is equal to the number of cells filled (number of steps described below the table)

Task 3

• 4. Bonuses:- If you think the puzzle has no solution you

add a desirable number to any empty cell and add 10 points for each wrong cell

- You may invent a new ratio puzzle and get a prize of minimum 50 points

- 5.Calculate the total desired number of your points gained

- 6.Send solved puzzle to my email

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