Indexes in statistics Indexes are indicators of size comparison of any socio-economic process. Index number measures how much a variable changes over time or in space
All indexes can be classified as: n single and composite n time and territorial n aggregate and average n with chain base and fixed base n with chain weight and fixed weight n with variable structure and constant structure
Indexes in statistics Index number can be calculated as the ratio of the current value to the base value. The resulting number is usually multiplied by 100 to express the index as a percentage
Indexes in statistics n To calculate an index number it is necessary to compare two identical periods of time, for example month with month, year with year
The index method is applied to solve the following problems: n study the change of the event in time n carry out spatial comparisons n measure plan fulfillment n indicate the degree of structural changes influence
Indexes are subdivided into individual (single) and composite (summary)
Individual (single) index is applied to find out the change degree of a separate element of a complex socio-economic process
Individual (single) index n It turns out to be a result of a comparison of a simple commodity process. Single index is the dynamic ratio DR, plan ratio PR, plan fulfillment ratio PF or ratio of comparison RCom
Notations P – price of a good, q – quantity of sold (produced) goods; pq – value (monetary worth) of sold (produced) goods, turnover; z - unit cost (себестоимость); zq – total cost, t - labor intensiveness (трудоемкость) of producing a good; w – labor productivity; T=tq - labor inputs, or number of employed persons; h = zq/pq – cost per one money unit of good production…
Notations f – individual wage or salary; F –wages & salaries fund (compensation of employees); S – cultivated area; y - crop yield; Sy - total harvest, Q – total amount of production or total sales, in natural units or in cost units
Individual (single) index n There are many single indexes, first of all we will consider single price index, single quantity index, single value index, single unit cost index, and single labor intensiveness index
Single price index n Single price index is expressed by the ratio of price per one unit for one period to the price per one unit of the same commodity for another period. After additional calculations it shows by how many percent the price has been reduced or increased n
Individual price index where P 1 – the price of a commodity for the accounting period; P 0 – the price for the basic period
Single quantity index (Individual index of physical volume of goods turnover) where q 1, q 0 – quantities of sold (produced) goods in current and base periods, respectively
The Single quantity index (Individual index of physical volume of goods turnover) n shows after additional calculations by how many percent the quantity volume has been increased or reduced n Multiplying the single price index and the single quantity index gives us the single value index:
Individual index of goods turnover (the single value index) where P 1 q 1, P 0 q 0 – values for current and base periods of time, respectively
. The single value index (individual index of goods turnover) after additional calculations shows by how many percent the turnover (of total monetary worth) for one commodity has been increased or reduced
Price per unit, rub. Quantity Price per unit, rub. 1,58 1,25 1,00 1,27 1,25 1,5 1,25 1,0 0, АBCАBC 2nd period1st periodGoods Quantity Find the change of prices, sales volumes and turnover for each commodity and for all commodities
Single price index for commodity A: Price of commodity A has increased by 25%
Single quantity index for commodity A: Sales volume of commodity A has increased by 26.7%
Single value index for commodity A: Total monetary worth (turnover) of commodity A has increased by 58.3%
We can say that the turnover of commodity A has increased by 58.3% because the sales volume has increased too by 26.7%, and the price per unit has increased by 25%
Thus, the change of turnover consists of the price change and quantity change: Turnover, pq 58.3% 26.7% Volume, q Price, p 25%
Single cost indexes The next indexing system characterizes the change of total cost, unit cost and quantity volume, it includes single unit cost index, single quantity index and single total cost index
Single unit cost index shows by how many percent the cost per one unit has changed: where z 1, z 0 – unit cost in current and base periods, respectively. Single quantity index has been considered above
Single total cost index is calculated by the following equation: where z 1 q 1, z 0 q 0 – - total costs (expenses) for current and base periods, respectively
Single labor intensiveness index (индекс трудоемкости) and single quantity index are joined by the single labor inputs index
Single labor intensiveness (трудоемкости) index shows by how many percent the labor intensiveness increased or reduced
This index is an exception to the rule because for constructing it, the labor intensiveness for base period has to be divided by the labor intensiveness for the reporting period:
Single labor inputs index could be calculated dividing the single quantity index by the single labor intensiveness index:
The above individual, single indexes are used for analysis of one commodity process. However, to analyze commodity set the composite (общие) indexes are required. Composite index combines the individual indicators such as price, quantity and value. Composite index can be expressed by aggregate or average formulas
General (composite) index is the relative indicator characterising change of the complex phenomenon, consisting of the elements which can not be added directly
The simplest form of a composite index is an aggregate index. An aggregate index is calculated by adding the elements in the composite for the given time period and then dividing this result by the sum of the same elements during the base period
Aggregate value index can be calculated using the following equation:, where - total turnover or total monetary worth of all commodities for base and current periods of time, respectively
The aggregate quantity index and the aggregate price index are named weighted aggregate indexes,
The aggregate quantity index is calculated by adding the volume of all elements in comparable prices for the given time period and then dividing this result by the sum of the same elements during the base period:, where - comparable price (prices for the same period, usually base period)
The General index of physical volume of goods turnover The index shows how the total receipts change as the result of change in quantity of goods sold
weighting n The weighting allows us to improve the accuracy of the general quantity estimate based on our data set. For constructing the aggregate quantity index, the influence of price should be excluded by means of price fixing during the base period
The idea of constructing the general price index The general price index shows how the prices on all considered commodity groups vary on average. It is impossible to add directly prices of various goods, it is necessary to choose a certain indicator to make economic sense in summation action. We use goods turnover or a profit in the role of such an indicator
The goods turnover size is influenced by two factors: n price level; n quantity of goods sold. As we are interested to measure only the change of prices, so influence of the second factor should be eliminated. For this purpose the quantity of goods sold is fixed at constant level:
Two variants are possible: 1.The quantity of the goods sold is fixed at level of the accounting period: where - Paashe index
2. The quantity of the goods sold is fixed at level of the base period: where - Laspeyres price index
Aggregate price index in Russian statistics is calculated using the Paashe formula: where - quantity volume of each element for the current period and weighting factor
n For constructing the aggregate price index the influence of volume should be excluded by quantity fixation during the current or reporting period. The numerator of the aggregate quantity index and the denominator of the aggregate price index are the same. It expresses the possible total n monetary worth without price changing
To get a uniform result Fisher's index is used which is calculated as simple geometric mean from indexes of Paashe and Laspeyres:
Quant -ity Price per unit, rub. Quan- tity Price per unit, rub Total АBCАBC 2nd period1st period Goods
The aggregate value index: The total monetary worth (total turnover) has increased by 45.6%. Why? The prices and sales volume have changed
The aggregate quantity index: Thus, the total monetary worth (total turnover) has increased by 45.6% while the sales volume has increased by 27.8%
The aggregate price index: Thus, prices on three commodities have increased by 13.9%
Weighted aggregate quantity index and weighted price index turn into aggregate value index: Thus, the total monetary worth (total turnover) has increased by 45.6% while prices increased by 13,9%, and the sales volume has increased by 27.8%
To construct the aggregate indexes it is recommended to apply the following rule: n Constructing the index quantitative indicator, the weights of base period should be used. The example of this index is aggregate quantity index. Constructing the index of qualitative indicator, the weights of reporting period should be used. The examples of this index are aggregate price index, aggregate unit cost index, and aggregate labor intensiveness index
The factorial analysis n For the analysis of influence of separate factors on a goods turnover we take a difference between numerator and a denominator of a corresponding general index
1. Absolute change of goods turnover (numerator minus denominator from the value index of goods turnover):
Factorial analysis n The goods turnover change occurs under the influence of two factors: changes of quantity of the goods sold and price change for a commodity unit. The sum of increases under the influence of these factors should be equal to the total increase in value of goods turnover
Factorial analysis n For reception of comparable results it is recommended to observe such sequence of inclusion of factors in the analysis: in the beginning there are quantitative factors (in our case q), then qualitative (P).
2. Absolute change in goods turnover at the unfluence of quantity change of the goods sold (numerator minus denominator of the general physical volume of goods turnover index according to Laspeyres):
3. Absolute change of goods turnover at the influence of prices change (numerator minus denominator of price index according to Paashe): Here two cases are possible: n economy or n buyers over-expenditure at the expense of the price change
In this case the general index of physical volume of goods turnover is calculated as the average arithmetic value from individual indexes of physical volume of goods turnover where as scales act values of goods turnover of the basic period:
Total ,936 0,918 1,013 -6,4 -8,2 +1, АБВАБВ Change of the physical volume of realization, % Realization during the current period, rub. Goods
Physical volume of realization of given goods on average decreases by 3,6%
Indexes of average levels (indexes of variable structure, constant structure and structural shifts)
Sale of goods by several firms is considered. Each firm has certain volume of sale and the price. It is required to analyse, how the average price of the goods changes
The average price index (the index of variable structure)
From the formula of an index of variable structure changes it is visible that the average price changes as a result of action of two factors: n change of the prices in different firms; n change of relative density of firms in total amount of realisation of the goods. Hence, the index of variable structure can be spread out on two subindexes, each of them characterises actions of one of these factors
1. A subindex - an index of constant structure. It shows, how the average price changes as a result of price changes in different firms:
where n d i – a share of commodities sold by firm i in the total quantity of sales. We used this share in the modified formula of weighted arithmetic mean:
2. A subindex - an index of structural shifts. It shows, how the average price changes as a result of relative density change of firms in total sale of the goods (as a result of structural shifts):
During construction of territorial indexes there are questions on base of comparison and object at which level it is necessary to fix index weight. These questions can be answered, proceeding from research specific targets. For example, it is necessary to compare price levels of two regions (regions A and B).
As scales we take quantity of the goods sold in region A.
As scales we take quantity of the goods sold in region B.
The given indexes are NOT interconnected among themselves:
For reception of uniform result the total sales volume of two regions act as scales.
It is possible the construction of a price index on the basis of method called indirect standardization.
If indexes are calculated for the value which is more than two numbers of the periods of time, so they can be received in the basic and chain way. We will consider construction of basic and chain indexes on an example of physical volume of goods turnover index.
The interrelation noted above is unconditional only for individual indexes. For the general indexes this interrelation will be fair only when the general indexes will be calculated with so-called constant scales.
Let there is data about realisation of the several goods for four periods of time.
System of basic indexes First period is considered as a basic.