STATISTICS TICS. Index calculation (1)

Transcription

1 STATSTCS TCS ndex calculation ()

2 Questions to be answered during index calculation How has the outut value, sales revenue and sales turnover changed? How has the volume of roduction and sales changed? How has the rice of roducts and the rice level changed?

3 Basic concets, uue indices ndex calculation ndex calculation exresses joint average change of uantities / comarable ratios that cannot be aggregated directly. Purose: Analysis of the roduction, service, trade, consumtion and use, as socio-economic categories by a joint examination of oulations reresenting different uality. Dynamic ratios: analysis of the rice and uantity of only - roduct; ndex calculation: analysis of the simultaneous imact of rice and uantity change of several roducts; Change can be exressed, as relative change ( index), absolute change (K difference).

4 Recently, enterrises roduce and sell not only one tye of goods aggregated indicators for examing of the temoral evolution of their activities, comaring their data with those of other businesses and areas. The simlest solution for a joint investigation of the volume and rice changes of roducts is comaring of the roduction values. The uts are the same the roduction values can be aggregated, can be comarable in the form of both uotient and difference.

5 The comarison can be made: n sace: comare data of 2 countries, or 2 comaes, n time: comare data of 2 years. Comarison in values is called aggregation, while the cumulative value data are called aggregates. Notations: i volume (uantity) i rice v i value v i i i v value index K v value difference rice index K value differences arising from differences in rices volume index K value differences arising from differences in uantities i custom indices k secific derogations

6 ndex calculation ndex numbers serve to comare uantity ( ), rice () and the value of data (v) over time, or in sace for an aggregate oulation being coherent, but disarate (different uts). The value of a commodity (grou) is determined by the uantity and ut rice v A Non-summarizable roducts (having different measures) can be analyzed on their value amount. The total value of a roduct grou of being coherent but consisiting of disarate roducts (i.e., heterogenous roduct grou) is called aggregate value (A): ndices: uotients of two aggregates that are different either in time or sace, concerng the same range of roducts.

7 ndividual value index and individual value difference t shows the rate of change in the value of individual roducts (i v ), or deviation (k v ) that is, how the roduction value for a given roduct (traffic) changed, from the base eriod to the current eriod. i v k v v v v v i i Changes in the value of the roducts examined are due to the change in rices and/or the change in volumes.

8 ndividual rice index and individual rice difference t shows rice changes of an individual roduct. i k

9 f we resent the imact of both rices and volumes in the form of value, the following relationshis can be written: The imact of the current eriod rice and the base eriod rice to the changes in the value: the imact of the current eriod volume, and the base eriod volume to the changes in the value : k i ndividual volume index and individual volume difference

10 ndividual indices (indices, i.e. ratios for a given tye of roduct ndividual rice index: where: : ut rice of the current eriod : ut rice of the base eriod ndividual volume index: where: : ut volume of the current eriod : ut volume of the base eriod ndividual value index: where: v : value of the roduct in the current eriod v : value of the roduct in the base eriod

11 Aggregate value data Four kinds of aggregates are used in index calculation real aggregate fictive aggregate fictive aggregate real aggregate Aggregation: value summation of a heterogeneous grou of roducts. The current aggregate value (aggregate) is called nominal value data.

12 ndices for various roducts - heterogeneous commodity grous. Aggregate forms of combined indices a Value index: Price index (volumes, i.e. data are constant) Volume index (rices, i.e. data are constant)

13 Value index and value difference A t termelés, shows thea simultaneous forgalom, a fogyasztás and average értékének change együttes, in the value átlagos of változását roduction, mutatja, sales and vagyis consumtion, két olyan namely aggregátum it is a uotient hányadosa, of two melyek a mennyiségi aggregates és that az differ áradatokban from eachis other eltérnek in both egymástól. volume and rice data. v K v Azaz That is, az the értékindex value index megmutatja, shows that hogy in the a mennyiség case of simultaneous és az ár együttes changes változása of volume esetén, and rice, hányszorosára how many times változott the value az érték has changed, az összes taking terméket into figyelembe account of all véve. the Az roducts. érték különbség At the same(érték time, eltérés) the value edig difference azt mutatja, shows hogy that how mennyivel much the változott value has az changed. érték.

14 Task: Sales of a market seller Products December 28 December 29 Sales volume Ut rice Sales volume Ut rice A-roduct (db) B-roduct (kg) C-roduct (kg) How has the turnover changed? Value index and value difference v ,362 36,2% K v Ft

15 Price index and rice difference When examing the effect of rice changes, the volume is assumed to be constant. Different statisticians use different weighting, so we can calculate in the following way. Current year weighting: Paashe rice index: P Base year weighting: Laseyres rice index: L The geometric mean of two rice indices: Fisher s rice index: F

16 Price differences: K K Price index: how many times the value changes exclusively on the effect of rice change; Price difference: how much the value changes; The sum of multilications in the formulae and are fictitious aggregates.

17 Task (continuation of the revious task):, P + +, L + +

18 6,8%,68,6479,5887 F 95F K 8525Ft K

19 Volume index and volume difference n this case, the rice is considered to be constant, so it is also ossible two weightings. Current year weighting: Paashe volume index: P Base year weighting: Laseyres volume index: L The geometric mean of two volume indices: Fisher s volume index: F

20 Volume differences: K K Volume index: how many times the value changes exclusively on the effect of volume change; Volume difference: how much the value changes.

21 Task (continuation of the revious task): Volume index and volume difference: P,826 L ,8568

22 F,826,8568,843 84,3% K Ft K Ft Textual analysis of the indicators that have been calculated (): Analysis of the turnover of a market seller by comaring data of December 28 and December 29. Prices of all three roducts increased, ut rice of C-roduct showed the greatest increase by 86.67%, while ut rice of B-roduct and A-roduct increased the smallest degree and by 25%, resectively. The ut rice of B-roduct increased by HUF Sales volume of A-roduct was only 4.5% higher in December 29, however the B-roducts have been sold more than 2%. Sales volume of C-roduct decresead by 28,57%, comared to 28.

23 Textual analysis of the indicators that have been calculated (2): An examination of market sales of a merchant by comaring his data for December 28 and December 29. The growth from revenues from each roduct is the highest for B- roduct, namely 47%. Revenue from sales of A- and C-roducts was by 3.2%, and 32.33% higher in 29 than in 28, resectively. Due to the changes in ut rices and uantities sold, turnover of the merchant in 29 was by 36.2% higher than in 28, which corresonds to HUF 6 5. The average rice change was 6.8%, while sales volumes decreased by an average of 5.78%.

24 Given the rice index and the volume index exresses the change of only one factor, while their multilication rovides the value index, because this index shows the joint change of rice and volume effect. Associations between indices v F F v K K K K K + +

25 Calculating averages with indices ndices can be calculated not only in aggregate form, but also in average of the individual indices. n this case, any of the aggregates or their uantitative ratios reresent as weight in the index. The way of the weighting deends on whether the given aggregate occurs in the numerator or denominator of the index.. f the value of the amount is available only for the base eriod, or the current eriod, then the indices can be calculated in average form.

26 Average forms of the value index f only the base year rice data and volume data are known, as well as the uue value index of the individual roduct grous, then the aggregate value index can be calculated as a weighted arithmetic average. The weight in this case, is the aggregate of the base eriod. v i vi

27 f only the current eriod rice data and volume data are known, as well as the uue value index of the individual roduct grous, then the aggregate value index can be calculated in a harmoc mean form. The weight in this case, is the aggregate of the current eriod. v Σ i vi

28 Average forms of the rice index Similarly to the value index, two kinds of calculations can be erformed in this case, as well: n the form of arithmetic average: i i i i n the form of harmoc mean: Σ i i Σ i i

29 Average forms of the volume index Volume index can also be calculated by both kinds of averaging: n the form of arithmetic average : i i i i n the form of harmoc mean: Σ i i Σ i i

30 Examle: Data on the commercial activity of a comany : Product grou Turnover in 27 (thousand HUF) Sales volume Turnover in 28, as ercentage of that in 27 A B C D Task: Calculate the value-, rice- and volume indices for the four roduct categories! Revenues in 27 are the base-year data, namely are known to every roduct category. i is the sold volume change, while i v is the change in turnover.

31 , % v i i vi i % The individual rice indices can be calculated as follows: i A.45/.5.26 i B.25/..4 i C.4/.25.2 i D.2/.98.2

32 i i.36.36% ,2732,36,2732,78,537,879 We rove that: Then Fisher s rice index and volume index can be calculated easily: F F

33 Use of aggregate indices in case of areal comarison Aggregate tye indices can also be used for i areal comarison. However, only in the case, if: we comare data for identical date or identical eriod. the base of the comarison deends on the study, or it is decided by the analyser. when drafting the numerical results, the exressions increase or decrease can not be used. nstead, the larger, smaller, higher, different words are used.

34 The secific case of regional comarison is comarison and analysis of data of different currencies of two countries. Value index has no meang, since currencies are included in the numerator and the denominator. For rice index and volume index only the Fisher's formulae are interreted, since large differences can occur both in rices and the volumes. The meang of rice index and its form of exression changes. Price levels are comared here, so it cannot be described in ercentage, but it aears as the ratio of the analyzed currencies. The currencies are used by their international signs, e.g. Hungarian Forint is marked by HUF. The volume index kees its original meang, as the consumtion uotient of the study roducts of the eole in the two countries.

35 Task: Food consumtion of two countries is characterized by the following data: Product D country G country Per caita consumtion Sale rice (in delta currency) Per caita consumtion Sale rice (in gamma currency). roduct roduct Comare ) er caita consumtion of the two countries and 2) urchasing ower of the currencies of the two countries. Benchmark is D country. ) 2)

36 Looking at the range of the roducts, er caita consumtion in G country was 7.4% smaller than in D country.

37 Purchasing ower of the delta currency of D country euals to times the urchasing ower of the gamma currency of G country.

38 ndex series t is often necessary to comare data of not two, but several eriods (time). Then, the indices should be calculated for each eriod (time). Values obtained in this way form a index series. Deending on the comarisons, we can seak of base index series and chain index series, similarly to the dynamic ratios. According to the weighting, the rice index series and the volume index series can be fixed-weighted index series variable weighted index series.

39 Base value index series: Chain value index series: 2i 2i ;... ; ;,i n,i n 2i 2i ;... ; ;

40 Base rice index series with fixed-weighting According to Paasche (current year weighting): ; ; 2i ;... According to Laseyres (base year weighting): ; ; 2i ;...

41 Base rice index series with variable weighting ; ; 2i 2i 2i ;...

42 Chain rice index series with fixed-weighting According to Paasche (current year weighting): ; ; 2i ;... n,i According to Laseyres (base year weighting): ; ; 2i ;... n,i

43 Chain rice index series with variable weighting ; ; 2i 2i 2i ;... n,i

44 Base volume index series with fixed-weighting According to Paasche (current year weighting): ; ; 2i ;... According to Laseyres (base year weighting): ; ; 2i ;...

45 Base volume index series with variable weighting ; ; 2i 2i 2i ;...

46 Chain volume index series with fixed-weighting According to Paasche (current year weighting): ; ; 2i ;... n,i According to Laseyres (base year weighting): ; ; 2i ;... n,i

47 Chain volume index series with variable weighting ; ; 2i 2i 2i ;... n,i

48 Task: Sales data of a haberdashery store Product Sales volume Ut rice Sales volume Ut rice Sewing thread (kg) Ktting yarn (iece) Zier (iece) Task: Analyse the changes of the store revenue, and determine how much the revenue changed due to the changes in rice and in volume sold!

49 Task: Data of three sections of a deartment store Section Percentage of Prices Turnover revenues in 29 in 29, as ercentage of 28 Slowork Piece goods Shoes Total - - Task: Calculate the ercentage change in sales of the store! Calculate the rice index and the volume index!

50 Task: Some data on accommodation Tye Commercial accommodation Non-commercial accommodation Győr-Moson-Soron county No. of ghts Price (iece) (HUF/ght) No. of ghts (iece) Veszrém county Price (HUF/ght) Task: Comare two counties using aggregate tye index calculation!

51 Task: Turnover data of a shoe store Product Breakdown of turnover (%) Price index (%) women s shoes men s shoes children s shoes The turnover increased by HUF 2 million, i.e. by 2% from 28 to 29. Analyse revenue changes and its factors using index calculation!

52 Always look on the bright side of things! We fished for today, goodbye!

53 ямарваа нэг зүйлийн гэгээлэг талыг нь үргэлж олж харцгаая өнөөдөртөө ингээд дуусгацгаая, баяртай 让我们总是从光明的一面来看待事 物吧! 今天的课程到此结束, 谢谢! دHQR BOPH داDM إKE اGHJE اUVWء! اLWSVXW VTذا اSTوم وداLM!