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Welcome. To. A Session. On. Index Numbers. Continued………. Your Text Book. BUSINESS. STATISTICS. By. S.P. GUPTA & M.P.GUPTA. (Fourteenth Enlarged edition). Chapter 9. Continued………. What is an Index Numbers ?.

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  1. Welcome To A Session On Index Numbers Continued……….

  2. Your Text Book BUSINESS STATISTICS By S.P. GUPTA & M.P.GUPTA (Fourteenth Enlarged edition) Chapter 9 Continued……….

  3. What is an Index Numbers ? It is a number that expresses the relative change in price, quantity, or value compared to a base period. According to Wheldom, “ An index number is a device which shows by its variation the changes in a magnitude which is not capable of accurate measurement in itself or of direct valuation in practice’ According to Karmel, “An index number represents the general level of magnitude of the changes between two or more situations of a number of variables taken as a whole”. Continued……….

  4. What is an Index Numbers ? An index number may be described as a specialized average designed to measure the change in the level of a phenomenon with respect to time, geographic location or other characteristics, such as income etc. “An index number is a device which shows by its variation the changes in a magnitude which is not capable of accurate measurement in itself or of direct valuation in practice” __ Wheldom: Business Statistics. “ An index number represents the general level of magnitude of the changes between two or more situations of a number of variables taken as a whole” __Karmel Index numbers are quantitative measures of the general level of growth of prices, production, inventory and other quantities of economic interest. ____Ronold.

  5. Example The price of a fall weekend package (including lodging and all means) at the Tyron Mountain Lodge in Western North Carolina in 2000 was $450. The price rose to $795 in 2004. What is the price index for 2004 using 2000 as the base period and 100 as the base value? Here, P1=$795 , Po=$450 Price – index for 2004 = 176.7

  6. Example This means that the price of the fall weekend package increased 76.7 per cent from 2000 to 2004.

  7. According to Ronold, “Index numbers are quantitative measures of the general level of growth of prices, production, inventory and other quantities of economic interest.” Example: According to the Bureau of Labour Statistics (in a country), in January 1987 the average of hourly earnings of production workers was $ 8.90. In January 2003 it was $ 15.07. What is the index of hourly earnings of production workers for January 2003 based on January 1987? In this case, Continued…..

  8. Thus, the hourly earnings in January 2003 compared to January 1987 were 169.3 per cent. This means there was a 69.3 per cent increase in hourly earnings during the period (found by subtracting 100 from 169.3 ie. 169.3–100 =69.3)

  9. Example 2 According to a research organization in U.S.A, the mean daily cost for a private hospital room in the United States in $ 643.80. The mean price for a private hospital room in Las Vegas in $ 713.17. What is the index for Las Vegas based on the national average? This indicates that the average private hospital room in Las Vegas costs 10.8 per cent more than the national average.(110.8–100 =10.8). Continued…..

  10. Example 3 The population of the Canadian province of British Columbia in 2002 was 4,141,272 and for Ontario it was 12,068,301. What is the population index of British Columbia compared to Ontario? This indicates that the population of British Columbia is 34.3 per cent (about one – third) of the population of Ontario, or the population of British Columbia is 65.7 percent less than the population of Ontarieo (100 – 34.3 =65.7)

  11. What is simple index ? If the index number is used to measure the relative change in just one variable, such as hourly wages in manufacturing, we refer to this as a simple index. How is simple price index calculated? To calculate the simple price index P using 100 as the base value for any given period, the following formula is used:

  12. Exercise The price of a fall weekend package (including lodging and all means) at the Tryon Mountain Loge in Western North Carolina in 2000 was $ 450. The price rose to $ 795 in 2004. What is the price index for 2004 using 2000 as the base period and 100 as the base value? the price of the all weekend package increased 76.7(=176.7-100 )per cent from 2000 to 2004

  13. How are index numbers classified? • Index numbers may be classified in terms of what they measure. In economics and business, the classifications are : • Price • Quantity • Value and • Special purpose Continued…….

  14. What are the methods of constructing index numbers? • A large number of formulae have been derised for constructing index numbers. They can be grouped under two heads: • Unweighted indices • Weighted indices • In the unweighted indices , weights are not expressly assigned whereas in the weighted indices, weights are assigned to the various items. Each of these items may further be divided under heads: • Simple Aggregative ; and • Simple Average of Price Relatives.

  15. What are the methods of constructing index numbers? The following chart illustrates the variance methods Index Numbers Unweighted Weighted Simple Average of Price Relatives Weighted Aggregative Simple Aggregative Weighted Average of Price Relatives

  16. Simple Aggregative Method In this method, the total of current year prices for the various commodities in question is divided by the total of base year prices and the quotient is multipled by 100. Symbolically, Where ∑p1=Total of current year prices for various commodities, and ∑po=Total of base year prices for various commodities.

  17. Simple Average of Price Relatives Method

  18. Example A From the following data, construct an index number for 2005 taking 2004 as base:

  19. Solution Simple Aggregative Method

  20. This mans that as compared to 2004, in 2005 there is a net increase in price of commodities included in the index to the extent of 2.51%.

  21. Example: Other measures of central tendency are not in common use for averaging common relatives. Example: • From the data given in example A, compute price index by simple average of price relatives methods on : • Arithmetic mean and • Geometric mean

  22. Solution: Price index based on Simple Average of price relatives.

  23. Price Index based on Geometric Mean of Price Relatives The arithmetic mean is often used because it is easier to compute and much better known.

  24. The following are the prices of six commodities for 2004 and 2005. Compare a price index by • Simple aggregative method and • Average of price relative method by using both arithmetic mean as well as geometric mean.

  25. Solution:

  26. a) Price index by Simple aggregative method b) Simple average of price relative method( using Arithmetic Mean) Continued…….

  27. Simple Average by price relative method (using Geometric Mean)

  28. There are various methods of assigning weights and consequently a large number of formulae for constructing index numbers have been devised of them, some important formulae are: Lespeyres method (L) Paasche method (P) Dorbish and Bowley’s method

  29. Fishers’ Ideal method Marsahll – Edgeworth method Kelly’s method

  30. Problem: • Compute the index numbers of prices from the following data by applying • Laspeyres’, • Paasche’s • Fisher’s • Bowley’ methods.

  31. Solution Calculation of various indices

  32. Pasche’s Method Fisher’s Ideal Method

  33. Bowley’s method Continued….

  34. The following are the group index numbers and the group weights of an average working class family’s budget. Construct the cost of living index number by assuming the weight. Index number Weight Group 152 110 130 100 90 48 6 8 12 15 Food Fuel and lighting Clothing House rent Miscellaneous

  35. Solution:

  36. Formulae for Quantity Index Numbers 1. 2. Fisher’s Ideal Formula 3.

  37. What are the tests to be applied for choosing an appropriate index? • The following tests are used to choose an appropriate index: • Time Reversal Test • Factor Reversal Test • Circular Test

  38. What is Time Reversal Test? Time reversal test is a test used to determine whether a given method works both ways in time, forward or backward. According to Prof. Irving Fisher – “The test is that the formula for calculating the index number should be such that it will give the same ratio between one point of comparison and the other, no matter which of the two is taken as base.”

  39. In other words, when the data for any two years are treated by the same method, but with the bases reversed, the two index numbers secured should be reciprocal of each other so that their product is unity. P10= Index for time “1” on time “0” as base and P10= Index for time “0” on time “1” as base. there is said to be a time bias in the method

  40. What are the methods which satisfy the time reversal test? • There are five methods which satisfy the time reversal test. These are the following: • The Fisher’s Ideal formula • Simple geometric mean of price relatives • Aggregate with fixed weights • The weighted geometric mean of price relatives with fixed weights. • Marshall – Edgeworth method

  41. What is Factor Reversal test? It holds that the product of price index and the quantity. In the words of Fisher, should be equal to the corresponding value index. In the words of Fisher, “Just as each formula should permit the interchange of the two items without giving inconsistent results, so it ought to permit interchanging the prices and quantities without giving inconsistent results i.e. the two results multiplied together should give the true value ratio.”

  42. If p, and p0 represent prices, and q1, and q0 the quantities in the current year and the base year, respectively, then If the product is not equal to the value ratio, there is an error in one or both of the index numbers (according to this test).

  43. Prove that the factor reversal test is satisfied by the Fisher’s Ideal Index. Proof:

  44. Why is Fisher’s Ideal index called “ideal”? The fisher’s Ideal index is given by the formula • The above formula is called ideal because of the following reasons: • It is based on the geometric mean which is theoretically considered to be the best average for constructing index number • It takes into account both current year as well as base year prices and quantities. Continued……..

  45. Why is Fisher’s Ideal index called “ideal”? The fisher’s Ideal index is given by the formula • The above formula is called ideal because of the following reasons: • It is based on the geometric mean which is theoretically considered to be the best average for constructing index number • It takes into account both current year as well as base year prices and quantities. Continued……..

  46. Why is Fisher’s Ideal index called “ideal”? • It is free from bias. The two formulae (Laspeyres’ and Paasche’s) that embody the opposing types and weight biases are, in the ideal formula, crossed geometrically, i.e. by an averaging process that of itself has no bias. The result is the complete cancellation of biases of the kinds revealed by time reversal and factor reversal tests. Continued……..

  47. Using the following data, show that Fisher’s Index formula for Satisfies the factor Reversal test and Time Reversal Factor test. Continued…….

  48. Solution:

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