Index Numbers

Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Based on NCERT Economics Class XI (Statistics for Economics), Chapter 7.

Have prices really doubled?

Your grandparents will tell you that a cup of tea cost a rupee when they were young, a train ticket a few rupees and a month of rent a few hundred. Today each has a different price. Tea has multiplied several times over, rent perhaps many more, a mobile call has become cheaper. So when a newspaper says "prices rose by 6 per cent this year", what does it mean? Thousands of goods each changed by a different amount.

An index number squeezes all those changes into one figure. It is a statistical device for measuring the average change in a group of related variables between two situations. This chapter shows how that one number is built, why the way you build it matters, and which Indian indices CUET expects you to know.

Reading an index: base year equals 100

An index is expressed as a percentage of a base period, whose value is fixed at 100. Any other period is read against it.

An index of 150 means a 50 per cent rise, not a 150 per cent rise. When you compare two non-base years, use the percentage change formula, not the difference of the index values.

$$\text{percentage change} = \dfrac{I_1 - I_0}{I_0} \times 100$$

If the index goes from 120 to 150, it has risen by 30 index points, but the percentage rise is $(150 - 120)/120 \times 100 = 25$ per cent. The same slip works the other way. An index that goes 100, 120 and then 96 has fallen 20 per cent from the peak (a drop of 24 on 120), yet it stands 4 per cent below the base.

Indices come in two main families. A price index measures changes in prices, and is the more widely used. A quantity index measures changes in physical volume, such as the output of factories or farms. Price indices answer "what happened to the cost of buying?" and quantity indices answer "what happened to production?"

Two ways to build an index

The first is the aggregative method: add up prices (or the cost of a basket) in each period and compare the totals. The second is the method of averaging relatives: first turn each item's price into a ratio to its base price, then average the ratios. Each can be unweighted (every item counts equally) or weighted (important items count more).

Simple aggregative index: where it goes wrong

The simplest idea is to add up the prices in the current period, add up the prices in the base period, and divide.

$$P_{01} = \dfrac{\sum P_1}{\sum P_0} \times 100$$

It has two weaknesses. It ignores how much of each good people buy, and it depends on the units in which each price is quoted.

Worked example: units change the answer. Rice costs ₹40 per kg in the base year and ₹50 now. Wheat costs ₹30 per kg and ₹36. Cooking oil costs ₹120 per litre and ₹132. Compute the simple aggregative index. Then quote wheat per quintal (100 kg) instead.

The prices did not change, only the way wheat was measured, yet the index moved from 114.7 to 119.7. The quintal price swamps the others in the sum.

Common wrong step. Trusting the first answer because "the arithmetic was right". The arithmetic is right, but the method is flawed. Averaging the price relatives ($125$, $120$ and $110$ for rice, wheat and oil) gives $(125 + 120 + 110)/3 = 118.3$ under either unit, because a ratio of two prices has no unit.

Weighted aggregative index: Laspeyres and Paasche

To fix the problem of importance, fix a basket of goods with their quantities. Then ask how its cost has changed. The only decision is which quantities to use, and that gives the two classic formulas.

Memory hook: the Laspeyres index looks long back (base-year quantities). Paasche looks at the present.

Worked example: Laspeyres and Paasche. Prices (₹) and quantities of four goods are given.

Both exceed 100, so prices rose on average. Laspeyres says 32.4 per cent and Paasche says 26.0 per cent.

Common wrong step. Mixing the quantities, for example dividing $\sum P_1 q_1$ by $\sum P_0 q_0$. That compares two different baskets, so the change in the index is no longer purely a change in price. The whole point of a weighted index is to hold the basket fixed.

Beyond the textbook. For the exam: Laspeyres uses base quantities and Paasche uses current quantities. In reality: when a good becomes costlier, buyers shift away from it. In the table, vegetables doubled in price and purchases fell from 15 kg to 8 kg. Laspeyres keeps pricing the old basket and so tends to overstate the rise in the cost of living, while Paasche prices the new basket, already tilted toward cheaper goods, and tends to understate it. Why it matters: here Laspeyres (132.4) is above Paasche (126.0), and that is the usual pattern.

Pause and predict: if consumers shift away from the goods whose prices have risen most, which index will come out higher, Laspeyres or Paasche?

Check your answer: Laspeyres. It still weights the goods with their old, larger quantities, so the expensive goods keep their big weight.

Method of averaging price relatives

A price relative is the current price as a percentage of the base price, $\dfrac{P_1}{P_0} \times 100$. Because it is a ratio, it has no unit, which fixes the problem in the earlier example.

$$\text{Simple: } P_{01} = \dfrac{1}{n}\sum \dfrac{P_1}{P_0} \times 100, \qquad \text{Weighted: } P_{01} = \dfrac{\sum W R}{\sum W}, \quad R = \dfrac{P_1}{P_0} \times 100$$

The weights $W$ are normally the shares of expenditure on each item. The simple average treats a rice price rise and a salt price rise as equally important. The weighted one does not.

Worked example: relatives with weights. Use the four goods above. Weights are base-year expenditures $W = P_0 q_0$.

The simple average of the relatives is $(125 + 120 + 110 + 200)/4 = 138.75$, which is inflated by the 100 per cent jump in vegetables. The weighted index is $298{,}000 / 2250 = 132.4$.

This equals the Laspeyres index from the previous example. That is no accident: when the weights are base-year expenditures $P_0 q_0$, the weighted relatives formula collapses to $\dfrac{\sum P_1 q_0}{\sum P_0 q_0}$ because $P_0 q_0 \times \dfrac{P_1}{P_0} = P_1 q_0$.

Common wrong step. Reporting the simple average (138.75) when the question gives expenditure weights. Vegetables are a small part of the budget (₹450 out of ₹2250), so they should not pull the index so hard.

The Consumer Price Index (CPI)

The Consumer Price Index, also called the cost-of-living index, measures the average change in retail prices of a fixed basket bought by a group of consumers. The question it answers is a personal one: how much more must I spend today to buy what I bought in the base year? It is built with weighted relatives, using household expenditure shares as weights.

The Reserve Bank of India uses the All-India Combined CPI as its main measure of consumer price change. The Consumer Food Price Index (CFPI) is the CPI food and beverages group excluding alcoholic beverages and prepared meals, snacks and sweets. The weights of the major groups in the 2012 based CPI are:

Food and beverages carry the largest weight, nearly half the basket, which is why food price swings move the CPI so strongly.

Worked example: a CPI basket. A simplified household basket has five groups, with weights and price relatives as shown. Find the CPI.

Common wrong step. Dividing $\sum WR$ by the number of groups (5) instead of by $\sum W$. That gives 2,615, an index that makes no sense. The simple average of the relatives, $(150 + 120 + 110 + 130 + 115)/5 = 125$, is also wrong because it gives rent the same importance as food, which has twice the weight.

WPI, IIP and SENSEX

The CPI is about what consumers pay. The other indices answer other questions.

Within IIP, the eight core industries (coal, crude oil, natural gas, refinery products, fertilisers, steel, cement and electricity) carry a combined weight of 40.27 per cent. The IIP can also be grouped by use. Its weights are primary goods 34.1, capital goods 8.2, intermediate goods 17.2, infrastructure and construction goods 12.3, consumer durables 12.8 and consumer non-durables 15.3.

Worked example: IIP as a weighted mean. In a month the output index relative to the base (the quantity relative) is 105 for mining, 110 for manufacturing and 108 for electricity. Use the sector weights above to find the IIP.

The IIP is $10{,}912 / 100 = 109.1$. Industrial output is 9.1 per cent above the base.

Common wrong step. Taking the simple average of the three relatives, $(105 + 110 + 108)/3 = 107.7$. It ignores that manufacturing is more than three quarters of the index, so its 110 should dominate.

Beyond the textbook. For the exam: the bases above are the ones to remember. In reality: base years are revised from time to time as spending patterns change. For example, the CPI for Industrial Workers is now published with 2016 as the base year rather than 2001. Why it matters: an index with an old base prices a basket that people no longer buy.

Beyond the textbook. For the exam: the RBI's main measure of consumer price change is the All-India Combined CPI. In reality: since 2016 India has followed flexible inflation targeting. The framework set a target of 4 per cent CPI inflation with a tolerance band of 2 percentage points on either side. Why it matters: it is CPI, the figure that reflects what households pay, that is used as the policy anchor.

What index numbers are used for

Two formulas translate a price index into what money can actually buy. Here CPI is the index with the base year equal to 100.

$$\text{Real wage} = \dfrac{\text{Money wage}}{\text{CPI}} \times 100, \qquad \text{Purchasing power of one rupee} = \dfrac{100}{\text{CPI}}\ \text{rupees}$$

Worked example: a raise that is not a raise. A worker earns ₹40,000 a month when the CPI is 200. A year later she earns ₹45,000 and the CPI is 250. Compare her real wages and find the inflation rate.

Her money wage rose 12.5 per cent, but prices rose 25 per cent, so her real wage fell by 10 per cent.

Common wrong step. Putting the CPI in the numerator (money wage times CPI). The real wage is the money wage deflated by prices, so the CPI is in the denominator. Also, calling the 12.5 per cent a gain without checking the CPI.

Pause and predict: a worker's money wage rises by 10 per cent in a year in which the CPI also rises by 10 per cent. What happens to the real wage?

Check your answer: it stays the same. Both the wage and the price level rise by the same factor, so the ratio is unchanged, and the worker can buy exactly the same basket.

Issues in constructing an index

Exam traps

Formula sheet

60-second revision

Where this chapter connects

• Previous: Correlation moved from one variable to two. Index numbers go back to summarising one group of variables over time.
• Averages at work: Measures of Central Tendency defines the weighted mean, which is exactly how the CPI and the IIP are built.
• Presentation: Presentation of Data shows how an index series is drawn as an arithmetic line graph, and Use of Statistical Tools puts the whole statistics unit to work.