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Statistics — syllabus topic (viii) of 10 · 🎯 PYQs from this topic: 2024 Q78

Part of the full combined notes. · 1 PYQ callout inside.

🔷 TOPIC (viii) — THEORY OF INDEX NUMBERS

Syllabus topic (viii) of 10 · runs until Topic (ix)

Covers: Characteristics · problems in construction · types · simple (unweighted) index · quantity index · value index · weighted aggregative — Laspeyres, Paasche, ⭐ Fisher's Ideal · weighted average of price relatives · CPI · WPI · CPI vs WPI · ⭐ Tests of Adequacy (Unit, Time Reversal, Factor Reversal, Circular) 🎯 Asked in the papers: 2024 · Q78 — Fisher Ideal Value Index — the hardest question on the 2024 paper

Statistical measure that shows changes in variables with respect to time, geography or other characteristics.

First time calculated by Italian statistician "Giovanni Rinaldo carli" $\downarrow$ Calculated ratio of prices for (grain, wine and oil). 1500 and 1750
Also known as "Economic Barometer"

1. Introduction

1.1 Characteristics

(1) Expressed in percentage form. (2) Relative or comparative measurement of a group of commodities. (3) Represent the specialised averages. (4) e.g; consumer Price Index, cost of living index, Industrial production Index.

1.2 Problems in Construction

Purpose of Index number should be pre-defined, every index number has its specific use.
Selection of the base year:
- Base year should not be too near or too far.
- Should be calamity free.

Selection of commodities.
Choosing the source of data.
Choice of average.
Choice of calculation method.

1.3 Limitations

Index numbers give approximate values.
Based on samples only.
Quality of the products is not taken in consideration.
For every purpose, different index numbers are to be constructed.

1.4 Types of Index Numbers

Price Index numbers
Quantity Index Numbers.
Value Index numbers

Price Index measures change in price b/w Base year and current year. $\hookrightarrow$ Whole sale Price Index Number $\hookrightarrow$ Retail Price Index Number
Quantity Index numbers show average changes in quantities, produced, consumed or sold. e.g Imports, Exports, production in Industries etc.
Value Index Number represent the product of commodity and the quantity.

2. Methods of Constructing Index Numbers

graph TD
    A[Index Numbers] --> B["==Unweighted / Simple Index Numbers=="]
    A --> C["==Weighted Index Numbers=="]
    
    B --> D["==Simple Aggregative Method=="]
    B --> E["==Simple Average of Price Relatives=="]
    
    C --> F["==Weighted Aggregative Method=="]
    C --> G["==Weighted Average of Price Relatives=="]

3. Simple (Unweighted) Index Numbers

3.1 Simple Price Index

Each item has got the same weight therefore no individual weights are assigned. (i) Simple Aggregative method (ii) Simple Average of Price Relatives.

(i) Simple Aggregative Method. Procedure (i) Add all the current year prices of various commodities. (ii) Add all the base year prices of various commodities.

Formula Price of "1" on "0" $\rightarrow$ $P_{01} = \frac{\sum P_1}{\sum P_0} \times 100$

$P_{01} \rightarrow$ Index number of current year
$\sum P_1 \rightarrow$ total (sum) of the current year prices.
$\sum P_0 \rightarrow$ total (sum) of the Base year prices.

Commodity	Base year Price (Rs.)	Current year Price (Rs.)
A	10	20
B	20	30
C	25	35
D	30	40
E	35	45

Sol $P_{01} = \frac{\sum P_1}{\sum P_0} \times 100 = \frac{20+30+35+40+45}{10+20+25+30+35} \times 100$ $P_{01} = \frac{170}{120} \times 100 = \frac{1700}{12} = \text{==141.66==}$ Price index of current year is 141.66 (or) Price of current year has increased by 41.66%.

3.2 Quantity Index

Formula $q_{01} = \frac{\sum q_1}{\sum q_0} \times 100$

$q_{01} \rightarrow$ quantity index no. of current year.
$\sum q_1 \rightarrow$ sum total of current year quantities.
$\sum q_0 \rightarrow$ sum total of Base year quantities.

Q. Calculate the quantity index number for 2019 when the base year is 2011.

Commodities	Quantity (2019) (tons) $q_1$	Quantity (2011) (tons) $q_0$
A	10	5
B	20	10
C	30	15
D	20	10
E	15	5
	$\sum q_1 = 95$	$\sum q_0 = 45$

Sol $q_{01} = \frac{\sum q_1}{\sum q_0} \times 100 = \frac{95}{45} \times 100 = \frac{9500}{45}$ $= 211.11$ $\therefore$ Quantity index for current year (2019) = 211.11 (or) Quantity of current year has increased by 111.11.

(ii) Simple Average of Price Relatives Method

Procedure (i) calculate relative price of current year (current year price by Base year price) $\left( \frac{P_1}{P_0} \times 100 \right)$ (ii) obtain the sum of Relative prices / Price Relative $\sum \frac{P_1}{P_0} \times 100$ (iii) Divide the sum of Relative prices by total number of commodities. $P_{01} = \frac{\sum \left( \frac{P_1}{P_0} \times 100 \right)}{N}$

Q. Using Price Relative Method, for the year 2020 find out index values from the given data.

Sol

Name	2010 Price (Rs.) $P_0$	2020 Price (Rs.) $P_1$	Price Relative = $\frac{P_1}{P_0} \times 100$
A	10	20	$\frac{20}{10} \times 100 =$ 200
B	15	25	$\frac{25}{15} \times 100 =$ 166.67
C	20	30	$\frac{30}{20} \times 100 =$ 150
D	25	35	$\frac{35}{25} \times 100 =$ 140
E	20	30	$\frac{30}{20} \times 100 =$ 150

N = 5 | $\sum \left( \frac{P_1}{P_0} \times 100 \right) =$ 806.67

$P_{01} = \frac{\sum \left( \frac{P_1}{P_0} \times 100 \right)}{N} = \frac{806.67}{5} = \text{==161.33==}$

The Price index for the year 2020 is 161.33 or there is increase in the price of 2020 by 61.33% compared to 2010.

H/W calculate the Quantity Index Number for the Previous question Assuming prices in (Rs.) as quantity in quintals.

Formula $q_{01} = \frac{\sum \left( \frac{q_1}{q_0} \times 100 \right)}{N}$

$P_{01} = \frac{\sum \left( \frac{P_1}{P_0} \times 100 \right)}{N} = \frac{806.67}{5} = \text{==161.33==}$

The Price index for the year 2020 is 161.33 or there is increase in the price of 2020 by 61.33% compared to 2010.

H/W calculate the Quantity Index Number for the Previous question Assuming prices in (Rs.) as quantity in quintals.

Formula $q_{01} = \frac{\sum \left( \frac{q_1}{q_0} \times 100 \right)}{N}$

Ans Same as above question.

3.3 Value Index Number

$V_{01} = \frac{\sum P_1 q_1}{\sum P_0 q_0} \times 100$

$\Rightarrow V_{01} = \frac{39}{36} \times 100 = 108.33$ (or) Prices increase by 8.33% from base year.

Commodity	Base year		Current year		$P_1 Q_1$	$P_0 Q_0$
Commodity	Quantity $q_0$	Price $P_0$	Quantity $q_1$	Price $P_1$	$P_1 Q_1$	$P_0 Q_0$
A	2	3	2	4	8	6
B	3	1	3	2	6	3
C	4	2	1	3	3	8
D	5	3	2	3	6	15
E	2	2	4	4	16	4
					$\sum P_1 q_1 = 39$	$\sum P_0 q_0 = 36$

4. Weighted Index Numbers

Appropriate weights are assigned to various commodities to show their relative importance.

4.1 Weighted Aggregative Method

There are lot of methods to calculate weighted index numbers viz; Laspeyre's, Paasche's, Fisher's, Kelley's, Bowley's, Dorbish and few more methods.

Laspeyres' Method

It is used to compare the expenditure of basket of commodities of Base year and current year.

Price Index $P_{01} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$ * Quantities of base year are taken as weights.

Quantity Index $q_{01} = \frac{\sum q_1 p_0}{\sum q_0 p_0} \times 100$ * Prices of Base year as weights.

Paasche's Method

Used to know the cost of basket of commodities in current year when the same basket cost ₹100 in Base year.

Price Index $P_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$

Quantity Index $q_{01} = \frac{\sum q_1 p_1}{\sum q_0 p_1} \times 100$

Quantities of current year and Prices of current year are used as weights.

Fisher's Ideal Method

🎯 THIS CAME IN THE EXAM — 2024 · Q78 — THE HARDEST QUESTION ON THE PAPER

"Commodity Y: price index = 10 for 1993 with base 1990; quantity index = 0.5 for 1990 with base 1993. Find the Fisher Ideal VALUE index for 1993 with base 1990." a) 5 · b) 10.5 · c) 20 ✅ · d) 15 ⭐ Step 1 — spot the reversal. The quantity index is given the wrong way round (1990 on base 1993). Flip it with the Time Reversal Test, $q_{01} \times q_{10} = 1$ : $q_{01} = \frac{1}{0.5} = 2$ ⭐ Step 2 — apply the Factor Reversal Test, Value Index = Price Index × Quantity Index: $V_{01} = 10 \times 2 = \mathbf{20}$ ⚠️ This is why the two tests below matter. The question is unsolvable unless you know that Fisher satisfies BOTH — the whole item is really a test of the Time Reversal and Factor Reversal rules, not of arithmetic.

Also called as Ideal method.

Price Index $P_{01} = \sqrt{L \times P}$ $P_{01} = \sqrt{ \frac{\sum P_1 q_0}{\sum P_0 q_0} \times \frac{\sum P_1 q_1}{\sum P_0 q_1} } \times 100$

It is the geometric mean of Laspeyre's and Paasche's method.
Both Base and current year Quantities are used as weights.

**Quantity Index ** $q_{01} = \sqrt{L \times P}$ $q_{01} = \sqrt{ \frac{\sum q_1 p_0}{\sum q_0 p_0} \times \frac{\sum q_1 p_1}{\sum q_0 p_1} } \times 100$

Both Base year and current year Prices are used as weights.

	Laspeyre's (Base)	Paasche's (Current)	Fisher's method
Price	$P_{01} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$	$P_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$	$P_{01} = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}} \times 100$
Quantity	$q_{01} = \frac{\sum q_1 p_0}{\sum q_0 p_0} \times 100$	$q_{01} = \frac{\sum q_1 p_1}{\sum q_0 p_1} \times 100$	$q_{01} = \sqrt{\frac{\sum q_1 p_0}{\sum q_0 p_0} \times \frac{\sum q_1 p_1}{\sum q_0 p_1}} \times 100$

Q. Calculate Price and quantity Index numbers using Paasche's, Lapeyere's and Fisher's method.

Commodities	$p_0$	$q_0$	$p_0 q_0$	$p_1$	$q_1$	$p_1 q_1$	$p_1 q_0$	$p_0 q_1$
A	2	3	6	6	2	12	18	4
B	3	4	12	3	4	12	12	12
C	4	2	8	2	3	6	4	12
D	5	3	15	4	7	28	12	35
			$\sum = 41$			$\sum = 58$	$\sum = 46$	$\sum = 63$

Worked example — Laspeyres

Price Index $\rightarrow$ $P_{01} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$ $= \frac{46}{41} \times 100$ $= 112.19$ | $12.19\% \uparrow$

Quantity Index $\rightarrow$ $q_{01} = \frac{\sum q_1 p_0}{\sum q_0 p_0} \times 100$ $= \frac{63}{41} \times 100$ $= 153.65$ | $53.65\% \uparrow$

Worked example — Paasche

**Price Index ** $P_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$ $= \frac{58}{63} \times 100$ $= 92.06$ / $7.94\% \downarrow$ Fallen

Quantity Index $q_{01} = \frac{\sum q_1 p_1}{\sum q_0 p_1} \times 100$ (Correction: formula written in notes uses $p_1$ for weights) $= \frac{\sum q_1 p_1}{\sum q_0 p_1} \times 100$ $= \frac{58}{46} \times 100$ $= 126.08$ / $26.08\% \uparrow$

Worked example — Fisher

Price Index $P_{01} = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}} \times 100$ $= \sqrt{\frac{46}{41} \times \frac{58}{63}} \times 100$ $= \sqrt{\frac{2668}{2583}} \times 100$ $= \sqrt{1.032} \times 100$ $= 1.015 \times 100$ $= 101.5$ or $1.5\% (\uparrow)$

Quantity Index $q_{01} = \sqrt{\frac{\sum q_1 p_0}{\sum q_0 p_0} \times \frac{\sum q_1 p_1}{\sum q_0 p_1}} \times 100$ $= \sqrt{\frac{63}{41} \times \frac{58}{46}} \times 100$ $= \sqrt{\frac{3654}{1886}} \times 100$ $= \sqrt{1.937} \times 100$ $= 1.391 \times 100$ $= 139.1$ or $39.1\% (\uparrow)$

4.2 Weighted Average of Price Relatives

Procedure (i) Calculate Price Relatives of current year $R = \frac{p_1}{p_0} \times 100$

(ii) calculate the value weights $V = (p_0 q_0)$

Price Index $P_{01} = \frac{\sum RV}{\sum V}$

Q. calculate the weighted Average Price Relatives Index for given data.

Commodities	Price 2011 $p_0$	Quantity $q_0$	Price 2018 $p_1$	Price Relative $(R) = \frac{p_1}{p_0} \times 100$	$V = (p_0 q_0)$	$(RV)$
A	8	20	10	$10/8 \times 100 = 125$	160	20,000
B	6	10	8	$8/6 \times 100 = 133.33$	60	7999.8
C	4	8	6	$6/4 \times 100 = 150$	32	4800
D	2	6	4	$4/2 \times 100 = 200$	12	2400
					$\sum V = 264$	$\sum RV = 35199.8$

$P_{01} = \frac{\sum RV}{\sum V} = \frac{35,199.8}{264} = \text{==133.33==}$

(or) $\frac{133.33 - 100}{100} \times 100 = 33.33\%$ Increase in prices of 2018 on 2011.

5. Consumer Price Index (CPI)

Also known as:

Real Price Index Number.
Cost of Living Index number.
Price of Living Index Number.
Retail Price Index Number.

CPI is used to measure the price of basket of goods and services of a particular class or region at particular point of time in comparison to Base year.

5.1 Applications of CPI

(1) To know increase or decrease in cost of living. (2) Used by government to make salary and (D.A). (3) Used to find purchasing power of money and real income/wages.

$\text{Purchasing power of money} = \frac{1}{\text{Cost of living Index}}$
$\text{Real wages} = \frac{\text{Money wages}}{\text{Cost of Living Index}} \times 100$

(4) Used by government in framing price policy and income policy.

5.2 Types of CPI

(CPI - IW) - Industrial workers (1982 B.Y) $\rightarrow$ 2001 $\rightarrow$ 2016 [Ministry of Labour, Labour Bureau (Shimla)]
(CPI - AL) - Agricultural Labours (1986-87 B.Y)
(CPI - RL) - Rural Labours (1986-87 B.Y)
(CPI - UNME) - Urban Non-Manual employees (1984-85 B.Y) $\rightarrow$ (CSO) $\downarrow$ Now (NSO) [MOSPI - Ministry of Statistics & Program Implementation]

2011 - CPI (R), CPI (U), CPI (Combined)

🔴 OUTDATED — THE CPI BASE YEAR CHANGED IN 2026 (verified online, 28 Aug 2026)

The CPI (Rural / Urban / Combined) series above ran on the 2012 base year. That is no longer current.

⭐ The new CPI series has base year 2024, first released on 12 February 2026

Built on the Household Consumption Expenditure Survey (HCES) data
Published by MoSPI / NSO
⭐ Remember the pair for 2026: CPI base = 2024 · WPI base = 2022-23

(The sub-index base years above — CPI-IW 2016, CPI-AL/RL 1986-87 — are separate series and remain as stated.)

5.3 Problems in Construction of CPI

Difference in price of goods and services.
Difference in the standard of living.
Choice of Base year.
Difference in proportion of expenditure.

5.4 Methods of Construction of CPI

Aggregate Expenditure Method
Family Budget Method

**(1) Aggregate Expenditure method / Weighted Aggregate Method ** Quantities of base year are taken as weights $CPI = \frac{\text{Expenditure in current year}}{\text{Expenditure in Base year}} \times 100$ $CPI = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$ ** Laspeyere's method

**(2) Family Budget Method / Weighted Average of Price Relatives ** Expenditure in the base period are taken as weights. $P = \frac{p_1}{p_0} \times 100$ Weights " $W$ " = $p_0 q_0$ $CPI = \frac{\sum WP}{\sum W}$

Q. Find the cost of living index in the given Data.

Commodities	$p_0$	$q_0$	$p_1$	$p_1 q_0$	$p_0 q_0$
A	5	4	10	40	20
B	3	2	6	12	6
C	2	3	4	12	6
D	1	4	2	8	4
				$\sum p_1 q_0 = 72$	$\sum p_0 q_0 = 36$

Sol Cost of Living = $\frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100 = \frac{72}{36} \times 100 = \text{==200==}$

** Based on Lapeyre's Price Index.

Q. Data about the middle class family is as follows.. Expense on food 30%, Rent 15%, clothing 20%, Fuel 10%, others 25% on base year. Price (₹) 2001: 100, 20, 70, 20, 40 Price (₹) 2011: 90, 20, 140, 15, 60 Find Cost of living.

Sol

Commodities	Expenses (%) (W)	$P_0$	$P_1$	$P = \frac{P_1}{P_0} \times 100$	WP
Food	30	100	90	$90/100 \times 100 = 90$	2700
Rent	15	20	20	$= 100$	1500
Clothing	20	70	140	$= 200$	4000
Fuel	10	20	15	$= 75$	750
Others	25	40	60	$= 150$	3750
					$\sum WP = 12700$

$\text{Cost of Living} = \frac{12700}{100} = \text{==127==}$

6. Wholesale Price Index (WPI)

It measures the general changes in the whole sale price of goods in the country.

Based on the commodities produced and distributed (first stage of transaction).
Rise or fall in prices at wholesale level spills over to the retail level after lag.
Published by Economic Advisor, Ministry of Commerce and Industry.
First time published 10th January 1942 (1939 B.Y).
7th revision is with Base year (2011-2012) $\downarrow$ chaired by Dr. Sumitra Chaudhari. $\hookrightarrow$ Education, Health, etc not included. * taxes are also not included.
WPI Food Index (CSO) separately presented.
697 commodities included in WPI.

🔴 OUTDATED — THE WPI BASE YEAR CHANGED IN 2026 (verified online, 28 Aug 2026)

The 2011-12 base year and the 697 commodities above are the OLD series. They were superseded before your exam.

	Old series	⭐ NEW series
Base year	2011-12	⭐ 2022-23
Effective from	—	⭐ June 2026 (with the May 2026 indices)
Number of items	697	⭐ 957
Weights basis	—	GVO-based; renewable energy now included

⭐ A new PRODUCER PRICE INDEX (PPI) was introduced alongside it, intended to eventually replace the WPI. Released by DPIIT, Ministry of Commerce & Industry (unchanged).

⚠️ The three-group weights were also rebased, so the 22.60 / 13.20 / 64.20 split below is superseded. Reported figures for the new series put Manufactured ≈ 65%, Primary Articles ≈ 20%, Fuel & Power ≈ 5% — but confirm the exact percentages against the DPIIT release before memorising them. The base-year change is the part that gets asked.

6.1 WPI Basket

graph TD
    A[WPI Basket] --> B["Primary Articles<br>(22.60%)<br>117 items<br>Rice, wheat, fruits etc."]
    A --> C["Fuel & Power Articles<br>(13.20%)<br>16 items<br>Power, coal petroleum etc."]
    A --> D["Manufactured Articles<br>(64.20%)<br>564 items<br>oil (edible), sugar, chemicals etc."]

6.2 Uses of WPI

(1) Estimation of Inflation: $\frac{X_n - X_{n-1}}{X_{n-1}} \times 100$ $X_n = \text{WPI for } n^{\text{th}} \text{ week.}$ $X_{n-1} = \text{WPI for } (n-1)^{\text{th}} \text{ week.}$

$\text{Yearly inflation rate} = \left( \frac{\text{Current year}_{WPI}}{\text{Previous year}_{WPI}} \times 100 \right) - 100$

(2) Estimation of Monetary value and Real value. (3) Used for estimating GDP by CSO. (4) Used by Business contractors (Demand & supply). (5) By Global investors for investment decisions.

6.3 Method of Calculation

Stage 1: Elementary price Indices using Jevon's index (Geometric mean for price). Stage 2: Elementary aggregated using Laspeyre's index formula.

6.4 WPI vs CPI

WPI	CPI
1. Released by Office of Economic Advisor (Ministry of Commerce & Industry)	1. NSO (Ministry of statistics and Program Implementation)
2. Measures Goods only	2. Both Goods & services.
3. Items — 697	3. Items — 448 (Rural Basket) 460 (Urban Basket)
4. Base year — 2011-2012	4. Base year: 2012.
5. 3 categories Manufactured Products (64.20%) Fuel & power (13.20%) Primary Articles (22.60%)	5. Many categories Food & Beverages (45.86) Housing (10.07) Fuel & light (6.84) Clothing & Footwear (6.53) Pan, tobacco, intoxicants (2.38) Miscellaneous (28.32)

7. Tests of Adequacy

(1) Unit Test (2) Time Reversal Test (3) Factor Reversal Test (4) Circular Test (extention of TRT)

(1) Unit tests Index number formulae should be independent of the units in which prices or quantities are used. * Satisfied by all index methods except simple (unweighted) aggregative method.

(2) Time Reversal Test Interchanging of time subscripts of price/quantity gives the reciprocal of the original formula. $\rightarrow P_{01} \times P_{10} = 1$ (Base year '1', Base year '0') $\rightarrow q_{01} \times q_{10} = 1$ * Not satisfied by Laspeyre & Pasche.

(3) Factor Reversal Test If $p$ and $q$ factors in price/quantity index formula are interchanged so that a quantity/price index formula is obtained, the product of two indices should give true value ratio. $P_{01} \times q_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_0} = V_{10}$ * Satisfied by Fisher Index only.

(4) Circular Test (extension of the Time Reversal Test) If the index for year 2018 is based on 2017 and Another index for 2017 based on 2016 then index for 2018 with base year 2016 should be directly obtained.

$P_{01} \times P_{12} \times P_{20} = 1$ * Satisfied by:
simple geometric mean of price relatives
Kelly's fixed Base method (simple)

⬛ END OF TOPIC (viii) — Topic (ix) begins below.