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# Statistics lecture 12 (chapter 12)

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Index Numbers

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### Statistics lecture 12 (chapter 12)

1. 1. 1
2. 2. • On the economic and business front many concepts can be measured directly• When it is not possible, need to introduce an associated quantity to represent it• Referred to as an index number• Examples of index numbers – Production Price Index – Consumer Price Index – JSE Mining, Industrial, Gold, All Share, Bond indices – Business Confidence Index 2
3. 3. What is an index number?• A measure that summarises the change in the level of activity, price or quantity, of a single item or a basket of related items from one time period to another• Expressing the value of an item in the period for which the index is calculated as a ratio of its value in the base period• Index is a percentage value Period of reference relative to which an index is calculated 3
4. 4. What is an index number?• Index number = Value in period of interest × 100 Value in base period• When the value exceeds 100, indicates an increase in the level of activity• When the value is less than 100, indicates a decrease in the level of activity Say: Index = 108.4• Activity can indicate a change in price or quantity Say: Index = 98.6 There was a: – Price index There was a: 108.4 – 100 = 8.4% – Quantity index 100 – 98.6 = 1.4% increase decrease 4
5. 5. What is an index number?• Price indices - P – Price of the item in the period of interest – pn – Price of the item in the base period – p0• Quantity indices - Q – Quantity of the item in the period of interest – qn – Quantity of the item in the base period – q0 5
6. 6. Simple index numbers• Simple price index indicates the change in price of a single item from the base period to the period under consideration pn P 100 p0• Simple quantity index indicates the change in quantity of a single item from the base period to the period under consideration qn Q  100 q0 6
7. 7. Simple index numbers - example• The following table indicates the prices, in rand, and quantities (in 100) sold at a small supermarket for three years 2007 2008 2009 Price Quantity Price Quantity Price Quantity Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2 Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2 Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2 7
8. 8. Simple index numbers - example• Simple price index for sugar in 2008 with 2007 as base year 2007 2008 2009 Price Quantity Price Quantity Price Quantity Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2 Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2 Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2 pn 5.29 P 100  100  106.01 6.01 % increase p0 4.99 8
9. 9. Simple index numbers - example• Simple quantity index for milk in 2009 with 2007 as base year 2007 2008 2009 Price Quantity Price Quantity Price Quantity Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2 Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2 Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2 qn 59.2 Q  100  100  121.06  21.06 % increase q0 48.9 9
10. 10. Simple index numbers - example• Simple quantity index for sugar in 2009 with 2008 as base year 2007 2008 2009 Price Quantity Price Quantity Price Quantity Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2 Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2 Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2 qn 18.2 Q  100  100  97.3 2.7 % decrease q0 18.7 10
11. 11. Concept Questions• 1 – 6, p418, textbook 11
12. 12. Composite index numbers• Composite index reflect the average change in activity of a basket of items from the base period to the period under consideration – Unweighted composite indices – all items in the basket is considered to be of the same importance – Weighted composite indices – each item in the basket is weighted according to its relative importance 12
13. 13. Unweighted composite index numbers• Simple composite price index P p n 100 p 0• Simple composite quantity index Q q n 100 q 0 13
14. 14. Unweighted composite index numbers - example• Simple composite quantity index for 2009 with 2007 as base year 2007 2008 2009 Price Quantity Price Quantity Price Quantity Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2 Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2 Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2 Q q n 100  14.2  18.2  59.2 100  115.5 15.5 % inc q 0 13.1  17.3  48.9 14
15. 15. Unweighted composite index numbers - example• Simple composite price index for 2008 with 2007 as base year 2007 2008 2009 Price Quantity Price Quantity Price Quantity Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2 Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2 Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2P p n 100  15.99  5.29  8.99 100  132.4  32.4 % inc p 0 10.49  4.99  7.39 15
16. 16. Weighted composite index numbers• Weighted composite price index P  p w 100 n p w 0• Weighted composite quantity index Q  q w 100 n q w 0Where: w = weight assigned to each item in the basket 16
17. 17. Weighted composite index numbers - example• Weighted composite price index for 2008 with 2007 as base year using the profit for each item as weight 2007 2008 2009 Price Profit Price Profit Price Profit Coffee (500g) 10.49 70% 15.99 70% 17.99 70% Sugar (500g) 4.99 30% 5.29 30% 7.49 30% Milk (1 l) 7.39 20% 8.99 20% 9.39 20%P  p w 100 n p w 0 15.99(.7)  5.29(.3)  8.99(.2)  100  141.3 41.3 % inc 10.49(.7)  4.99(.3)  7.39(.2) 17
18. 18. Weighted composite index numbers- Laspeyres approach• The base period values will be assigned as weights to the items in the basket Price index:• Laspeyres price index weight is the quantity PL   pn q0 100 in the base period p q0 0• Laspeyres quantity index Quantity index: weight is the price QL  q pn 0 100 in the base period q p0 0 18
19. 19. Weighted composite index numbers- Laspeyres approach• Advantage is that indices calculated for different period using the same basket of items may be compared directly as long as the base period remains unchanged• Disadvantage is that it over estimates increases in the prices as times goes by – it is necessary to adjust the base period from time to time 19
20. 20. Weighted composite index numbers - example• Laspeyres price index for 2009 with 2007 as base year 2007 2008 2009 Price Quantity Price Quantity Price Quantity Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2 Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2 Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2PL  p qn 0 100  17.99(13.1)  7.49(17.3)  9.39(48.9) 100 p q0 0 10.49(13.1)  4.99(17.3)  7.39(48.9)  140.9  40.9 % inc 20
21. 21. Weighted composite index numbers- Paasche approach• The consumed current period values will be assigned as weights to the items in the basket• Paasche price index Price index: weight is the quantity PP  p q n n 100 in the current period p q 0 n• Paasche quantity index Quantity index: weight is the price QP  q p n n 100 in the current period q p 0 n 21
22. 22. Weighted composite index numbers- Paasche approach• Advantage is that indices calculated for different period using the same basket of items may be compared directly as long as the base period remains unchanged• Disadvantage is that it over estimates increases in the prices as times goes by – it is necessary to adjust the base period from time to time 22
23. 23. Weighted composite index numbers - example • Paasche quantity index for 2009 with 2007 as base year 2007 2008 2009 Price Quantity Price Quantity Price Quantity Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2 Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2 Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2QP  q pn n 100  14.2(17.99)  18.2(7.49)  59.2(9.39) 100 q p 0 n 13.1(17.99)  17.3(7.49)  48.9(9.39)  114.9 14.9 % inc 23
24. 24. Weighted composite index numbers- Fischer approach• Fischer price index PF  PL  PP• Fischer quantity index QF  QL  QP• May only be used if the indices for Laspeyres and Paasche have the same base period 24
25. 25. ExampleThe price of bread (rands/bread), meat (rands/kg), Cabbage (rands/cabbage) and wine(rands/bottle), as well as the quantities (in millions) consumed during 2006, 2007 & 2008are given in the following table:- Price Quantity 2006 2007 2008 2006 2007 2008 Bread 7.0 6.6 8.4 900 1000 900 Meat 44.0 46.0 59.0 600 600 700 Cabbage 7.0 7.3 9.6 5 6 5.5 Wine 30.4 30.4 32.1 90 90 100Calculate the:-1. Simple quantity index for meat in 2008 with 2006 as base year2. Simple composite price index for 2007, with 2006 as base year3. Lapeyres price index for 2008 with 2007 as base year4. Paasche price index for 2008 with 2007 as base year5. Fischer price index for 2008 with 2007 as base year6. Simple composite quantity index for 2008 with 2007 as base year7. Fischer quantity index for 2007 with 2006 as base year 25
26. 26. EXAMPLE ANSWER1. 1) Q =  q  100 = 700  100 = 116.67 n q 0 600 2) P=  p  100 = 90.3  100 = 102.15 n p 0 88.4 3) PL =  p q  100 = 46146.6  100 = 124.79 n 0 p q 0 036979.8 4) PP =  p q  100 = 52122.8  100 = 126.45 n n p q 0 41220.15 n 5) PF = PL  PP = (124.79)(126.45) = 125.62 6) Q =  q  100 = 1705.5  100 = 100.56 n q 0 1696 7) QL =  q p  100 = 36178  100 = 101.99 n 0 q p 0 35471 0 QP =  q p  100 = 36979.8  100 = 101.84 n n q p 036312.5 n 26 QF = QL  QP = (101.99)(101.84) = 101.92
27. 27. The index series• Collection of indices for the same item or basket of items constructed for a number of consecutive periods using the same base period• The base period will be the period = 100 27
28. 28. The index series - example • Construct an index series for the monthly electricity usage for a household – use June as base month Month April May June July August Useage (kw) 680 754 820 835 798 82.9 92.0 100 101.8 97.3 qnn q 100  680 100  82.9  17.1% dec 754QQ  100  100  92.0  8 % dec q00 q 820 820 28
29. 29. Important indices – Consumer price index• Composite price index of a representative basket of consumer goods and services• Serves as a measure of relative change in the prices of services and goods consumed in SA• Stats SA publish the CPI monthly• Price information in the index refers to the first 7 days of that month• Published in the second half of the next month• Info used to determine the CPI is obtained from a survey in each of 12 urban areas for each of 3 income groups and contains almost 600 items in 17 categories 29
30. 30. Important indices – Consumer price index• A weight is assigned to each item in the basket according to their relative importance p0 q0 w  p0 q0   w pn p0 CPI  w 30
31. 31. Important indices – Consumer price index• CPI is used to determine the inflation rate• Deflate other value series• Adjust prices, wages, salaries and other variables for changes in the inflation rate• It is available quickly• A disadvantage is that it is based on a household with on average 1.6 children, takes only certain good and services into account, includes indirect taxes but excluded direct taxes• Can use consecutive CPI’s as a time series to make forecasts on future values and trends 31
32. 32. EXAMPLES OF IMPORTANT INDICES• JSE all share index• JSE gold index• CPI- consumer price index – used to calculate inflation rate and cost of living• Inflation rate• PPI – Production price index• Business confidence index• New car sales index 32
33. 33. Statistics SA• Look at www.stassa.gov.za 33
34. 34. Example• Activity 1, p197 Module Manual 34
35. 35. Example• Activity 2, p199 Module Manual 35
36. 36. Classwork/Homework• Revision exercises 1,2,3 p 200 module manual 36