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Index No in Statistics

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### I ndex no_stats

1. 1. INDEX NUMBERS PRESENTED BY: Mansi Doomra (130550034) Mehak Puri (130550035) Nidhi Aggarwal (130550039) Nikhil Khera (130550040) Shashank Kapoor (130550052) Vineet Kumar (130550062)
2. 2.  An index number measures the relative change in price, quantity, value, or some other items of interest from one time period to another. A simple index number measures the relative change in one or more than one variable.
3. 3.  Index Numbers are a specialized type of averages. - M. Blair  Index Numbers are devices for measuring differences in the magnitude of a group of related values. - Croxten and Cowden  An index number is a statistical measure designed to show changes in a variable or group of related variables with respect to time, geographical location or other characteristics. -Spigel
4. 4. HELPFUL IN PREDICTIONS •Index numbers give the knowledge as to what changes have occurred in the past. HELPFUL IN COMPARISONS •By Index numbers relative changes occurring in the variables are determined, which simplifies the comparison of Data. USEFUL IN BUSINESS •Index numbers measures the changes taking place in the Business World and are useful in making a comparative study of the changes.
5. 5. Choice of the base period. Choice of an average. Purpose of index numbers. Selection of commodities. Data collection.
6. 6. Simple Average of Price Relatives Numbers Index Un-weighted Simple Aggregative Weighted Weighted Aggregative Weighted Average of Price Relatives
7. 7. In this method, sum of current year’s prices is divided by sum of base year’s prices and the quotient is multiplied by 100. Its formula is: P01 p1 p0 100 Where, P01= Index number of the current year. p1 = Total of the current year’s price of all commodities. p0 = Total of the base year’s price of all commodities.
8. 8. PRICE (Rs) 2008 COMMODITIES UNITS PRICE (Rs) 2007 Sugar Quintal 2200 3200 Milk Quintal 18 20 68 71 Oil Litre Wheat Quintal 900 1000 Clothing Meter 50 60
9. 9. PRICE (Rs) 2008 COMMODITIES UNITS PRICE (Rs) 2007 Sugar Quintal 2200 3200 Milk Quintal 18 20 Oil Litre 68 71 Wheat Quintal 900 1000 Clothing Meter 50 60 p0 3236 p1 4351 Index Number for 2008: P01 p1 p0 100 4351 100 34.45 3236 It means the prize in 2008 were 34.45% higher than the previous year.
10. 10. In it, initially the price relatives of all the commodities are found out. To calculate price relatives, price of current year (p1) is divided by price of base year (p0) and then, the quotient is multiplied with 100. 1. When ARITHMETIC MEAN is used: P01 2. p1 100 p0 N Where N is Numbers Of items. When GEOMETRIC MEAN is used: P01 Anti log p1 log 100 p0 N
11. 11. 3. When MEDIAN is used: P01 N 1 Size of 2
12. 12. From the data given below construct the index number for the year 2008 taking 2007 as base year by using arithmetic mean. Commodities Price (2007) Price (2008) P 6 10 Q 2 2 R 4 6 S 10 12 T 8 12
13. 13. Index number using arithmetic mean Price (2007) P 6 10 166.7 Q 12 2 16.67 R 4 6 150.0 S 10 12 120.0 T 8 12 150.0 p0 Price (2008) Price Relative Commodities p1 p1 p0 100 p1 100 = 603.37 p0 P01 p1 100 p0 N 603.37 120.63 5
14. 14. When index numbers is constructed taking into consideration the importance of different commodities, then they are called weighted index numbers. There are two methods of contructing weighted index numbers. 1. Weighted Aggregative Index Numbers. 2. Weighted Average of Price Relative Methods.
15. 15. In it, commodities are assigned weights on the basis of the quantities purchased. Different statisticians have used different methods of assigning weights, which are as follows:       Laspeyre’s method. Paasche’s method. Fisher’s ideal method. Dorbish and Bowley method. Marshall-Edgeworth’s method. Kelly’s method.
16. 16. This method was devised by Laspeyres in 1871. In this method the weights are determined by quantities in the base. p01 p1q0 p0 q0 100 Paasche’s Method: This method was devised by a German statistician Paasche in 1874. The weights of current year are used as base year in constructing the Paasche’s Index number. p01 p1q1 p0 q1 100
17. 17. This method is a combination of Laspeyre’s and Paasche’s methods. If we find out the arithmetic average of Laspeyre’s and Paasche’s index we get the index suggested by Dorbish & Bowley. p01 p1q0 p0 q0 p1q1 p0 q1 2 100 Fisher’s Ideal Method: Fisher’s ideal index number is the geometric mean of the Laspeyre’s and Paasche’s index numbers. P01 p1q0 p1q1 p0 q0 p0 q1 100
18. 18. In this index the numerator consists of an aggregate of the current years price multiplied by the weights of both the base year as well as the current year. p01 p1q0 p1q1 p0 q0 p0 q1 100 Kelly’s Method: Kelly thinks that a ratio of aggregates with selected weights (not necessarily of base year or current year) gives the base index number. p1q p01 100 p0 q Where q refers to the quantities of the year which is selected as the base. It may be any year, either base year or current year.
19. 19. Given below are the price quantity data,with price quoted in Rs. per kg and production in qtls. Find: (1) Laspeyre’s Index (2) Paasche’s Index (3)Fisher Ideal Index. 2002 2007 ITEMS PRICE PRODUCTION PRICE PRODUCTION BEEF 15 500 20 600 MUTTON 18 590 23 640 CHICKEN 22 450 24 500
20. 20. ITEMS PRICE PRODUCT ION PRICE PRODUC TION p1q0 p0 q0 7500 p1q1 p0 q1 p0 q0 p1 q1 BEEF 15 500 20 600 10000 MUTTON 18 590 23 640 13570 10620 14720 11520 CHICKEN 22 450 24 500 10800 TOTAL 9900 12000 9000 12000 11000 34370 28020 38720 31520
21. 21. 1.Laspeyre’s index: p01 p1q0 p0 q0 100 34370 100 122.66 28020 100 38720 100 122.84 31520 2. Paasche’s Index: p01 p1q1 p0 q1 3. Fisher Ideal Index: P01 p1q0 p1q1 p0 q0 p0 q1 100 34370 28020 38720 31520 100 122 .69
22. 22. In weighted Average of Price relative, the price relatives for the current year are calculated on the basis of the base year price. These price relatives are multiplied by the respective weight of items. These products are added up and divided by the sum of weights. Weighted arithmetic mean of price relative is given by: P01 Where: P PV V P 1 100 P0 P=Price relative V=Value weights = p 0 q0
23. 23. Quantity index numbers are designed to measure the change in physical quantity of goods over a given period. These index numbers represents increase or decrease in physical quantities of goods produce or sold. The method of construction of quantity index is same as that of price index. (1) Simple quantity index numbers (a) Simple Aggregative Method: Q01 q1 q0 100
24. 24. (b) Simple Average of Relative Method: (i) Using A.M. Q01 q1 100 q0 N (ii) Using G.M. log Q01 Anti log q1 100 q0 N
25. 25. (2) Weighted quantity index numbers I. Weighted Aggregate Method (a) Laspeyre’s quantity index no.: Q01 q1 p0 q0 p0 100 (b) Paasche’s quantity index no.: Q01 q1 p1 q0 p1 100
26. 26. (c) Fisher’s quantity index numbers: Q01 q1 p0 q1 p1 q0 p0 q0 p1 100 (d) Bowley’s quantity index: Q01 q1 p0 q0 p0 q1 p1 q0 p1 2 100 (e) Marshall’s quantity index: Q01 q1 p0 q1 p1 q0 p0 q0 p1 100
27. 27. II. Weighted Average of Relative Method: QW Q01 Where, Q W q1 100 q0 and W q0 p0
28. 28. Value is the product of price and quantity. A simple ratio is equal to the value of the current year divided by the value of base year. If the ratio is multiplied by 100 we get the value index number. V p1q1 p0 q0 100
29. 29. Various formulae can be used for the construction of index numbers but it is necessary to select an appropriate/suitable formula out of them. Prof. Fisher has given the following tests to select an appropriate formula:   TIME REVERSAL TEST (TRT) FACTOR REVERSAL TEST (FRT)
30. 30. According to this test, if considering any year as a base year, some other year’s price index is computed and for another price index, time subscripts are reversed, then the both price indicies must be reciprocal to each other. TRT is satisfied when: P01 1 or P01 P 10 P 10 1 Where, P01 is price index for the year 1 with 0 as base and P10 is the price index for the year 0 with 1 as base.
31. 31. Time reversal test permits interchange of price and quantities without giving inconsistent results, i.e. the two results multiplied together should give the true value ratio: FRT is satisfied when: Price Index x Quantity Index = Value Index OR P01 Q01 p1q1 p0 q0