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Unformatted text preview: MN1025 – Business Statistics 5 Lecture 2—Friday 18/1/2008 INDEX NUMBERS INTRODUCTION TO PROBABILITY Reference: Lind et al. , Chapters 5,18. 2.1 Index numbers Index numbers turn various measurements into a standard scale and help us to see the pattern of change. Typically they try to produce a single fig- ure which shows the combined change due to several components. For example, the Retail Price Index for each month is a single figure putting together hun- dreds of component prices, some of which may rise and some may fall. Share indexes give an overall fig- ure representing the performance of many different companies. There are also specialized indexes (e.g. universities’ costs index). All these indexes are usu- ally weighted averages of sets of costs, prices, wages, etc. Normally they start at 100 (= base) and move with time. 2.2 Example: Indexing a single quantity Prices in dollars of a bushel of wheat over a ten year period are: year 1 2 3 4 5 6 price 1.33 1.34 1.76 3.95 4.09 3.56 year 7 8 9 10 price 2.73 2.33 2.97 3.78 Write these prices with year 1 as base (= 100): year 1 2 3 4 5 6 price 100 101 132 297 308 268 year 7 8 9 10 price 205 175 233 284 To obtain the index numbers from the original prices, multiply each price by the same factor, which in this example is 100 / 1 . 33. For instance, 1 . 33 → 100, so 4 . 09 → 100 1 . 33 × 4 . 09 = 308 1 . 33 → 100, so 3 . 56 → 100 1 . 33 × 3 . 56 = 268 (rounded, in this example, to the nearest integer). 2.3 Example: Re-basing and joining indexes In Example 2.2, 1.33 was set to 100; in fact any data or index figure can be set to 100, or indeed to any value, with consequent changes to the other figures. Suppose that another index for wheat prices had year 10 as its base for the dollar prices, so we had, say: year 10 11 12 13 index 100 120 141 169 This can be run on from the previous index values. First we re-base this index: 100 → 284, so 120 → 284 100 × 120 = 341, 141 → 284 100 × 141 = 400, etc. year 10 11 12 13 index 284 341 400 480 Now, writing these as a single sequence, we can join the indexes to run from Year 1 to Year 13: year 1 2 3 4 5 6 7 index 100 101 132 297 308 268 205 year 8 9 10 11 12 13 index 175 233 284 341 400 480 This gives a single index running from Year 1 to Year 13. 2.4 Different price changes in one index Putting different price changes together in one index: Suppose we have several items and ask how over- all prices have changed (on average). In index- ing price changes, the problem is that along with price changes there are changes in the correspond- ing quantities, usually in the opposite direction. So if we just measure total expenditure the price rises are distorted. To focus on the changes in prices we need to compare like quantities with like quantities. But should we use the present quantities or the original quantities? We see how this can be done in different ways with the Laspeyres and Paasche indexes....
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