Applied Mathematics · Class 12 Commerce
Ch 8Index Numbers and Time-based Data — Class 12 Applied Mathematics, concept-first.
This concept map shows how the chapter fits together. It has two halves. Constructing an index number can be done by simple (unweighted) methods — the relative, aggregative and simple-average-of-relatives methods — or by weighted methods, chiefly the weighted aggregative family (Laspeyres', Paasche's, Marshall-Edgewort…
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Price Level Comparison
Imagine you have ₹1000. In India, that might buy you a nice dinner for two. Now take that same ₹1000 to Japan — it might get you a bowl of ramen. Take it to Switzerland, and you might not even afford a coffee.
Most relevant Q&A
- Suppose we want to compare price level of crude oil for the years 2005 through 2020. | Period | Price (thousand ₹ / gallon) | | --- | --- |…Free
- A departmental store paid annually for newspaper and television advertisements in 1990 and 2000 as shown below: | Expenditure | 1990 | 2000…Preview
- A manufacturer purchases four distinct raw materials, that differ in unit price as given below: | Commodity | Unit Price (₹) Year 2000 | Uni…Preview
- Construct the index number by the Simple Average of relatives Method for the following data: | Commodity | Unit Price (₹) Year 2000 | Unit P…Preview
- With reference to example above, what will happen to the index number for year 2000 if the commodities are used in different weights (quanti…Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Concept Map
This concept map shows how the chapter fits together. It has two halves. Constructing an index number can be done by simple (unweighted) methods — the relative, aggregative and simple-average-of-relat…
Introduction
When a teacher says your test scores improved by 25%, or the news reports that the cost of everyday goods rose 30% over a decade, both are talking about the same underlying idea — an index number.
Index Number
An index number is a single figure that measures how a group of related variables — such as the prices of a basket of goods — has changed between two different situations.
Use of Index Numbers
Index numbers matter because they turn scattered data into decisions. They measure changes in the standard of living and in price fluctuations, giving a clear read on whether people are, on average, p…
Construction of Index Number
Building a reliable index number is not just a computation — several judgement calls have to be made carefully first.
Relative Index Number
A relative index number (or price relative) measures the price change of a single commodity between the base period and the current period, expressed as a percentage:
+−Worked Examplesi2 questions
- Example 1Suppose we want to compare price level of crude oil for the years 2005 through 2020. | Period | Price (thousand ₹ / gallon) | | --- | --- |…Free
- Example 2A departmental store paid annually for newspaper and television advertisements in 1990 and 2000 as shown below: | Expenditure | 1990 | 2000…Preview
Simple (Unweighted) Aggregative Method
Once you need to track a group of items together — say a full basket like housing, food, medical care and transport — rather than one commodity at a time, the simple aggregative method builds a single…
Simple Average of Relatives Method
The simple aggregative method has a drawback: a commodity priced in large numbers (a car, say) dominates the aggregate over one priced in small numbers (salt), even though neither is necessarily more…
Weighted Aggregative Method
The simple methods above treat every commodity as equally significant, which rarely reflects reality — a household spends far more on staple food than on stationery, for example.
Laspeyres' Index
Laspeyres' Index, named after the German economist Etienne Laspeyres, is a weighted aggregative price index that fixes the weights at base-period quantities — it asks how much the original "basket" of…
Paasche Index
Where Laspeyres' Index freezes the weights at the base year, the Paasche Index — proposed by German statistician Hermann Paasche in 1874 — instead uses current-period quantities as weights, revising t…
Fisher's Ideal Method
Fisher's Ideal Index is built to combine the strengths of the two methods above rather than choose between them: it is defined as the geometric mean of the Laspeyres and Paasche indices.
Marshall-Edgeworth's Method
The Marshall-Edgeworth Index, named after statisticians Alfred Marshall and Francis Edgeworth, takes a more direct route to the same goal as Fisher's method — balancing base-period and current-period…
Weighted Average of Relatives
The weighted average of relatives method combines two ideas seen earlier in this chapter — price relatives and weighting — into a single index.
Types of Index Numbers
Beyond the price indexes built so far, index numbers are also classified by what they measure. A Value Index compares the average value of a variable in one period against its average value in the bas…
Limitations of an Index Number
For all their usefulness, index numbers are not perfect measures, and their limitations should be kept in mind when interpreting them.
Index Series
When index numbers are computed for several different periods, all measured against the same base period, the resulting list of values is called an index series.
Test of Adequacy of Index Numbers
With several different methods available for constructing the same index, a natural question follows: how do we know a particular method is actually reliable? This is what the tests of adequacy answer…
Unit Test
The unit test checks whether an index-number formula's result depends on the physical units used to record prices or quantities — for example, wheat priced per kilogram and milk priced per litre.
Time-reversal Test
The time-reversal test checks whether an index-number method gives internally consistent results no matter which period is chosen as the base.
Time Series
A time series is a sequence of numerical data points for a given variable, recorded at successive points or periods of time and arranged in chronological order.
Time Series Analysis
A time series can involve just one variable or several. A univariate time series tracks a single variable over time — a temperature sensor logging one reading every second is a simple example.
Trend Analysis by Fitting Linear Trend Line
Of the four components that make up a time series, the secular trend is usually of the greatest practical interest, since it captures the long-term direction the series is actually heading in — the si…
Trend Analysis by Moving Average Method
The moving average method draws a smooth curve through a time series by systematically averaging out the noise around the underlying trend.
+−Examples 6.9.2(i)i2 questions
- Example 9Calculate the 3-year moving averages for the loans (In lakh ₹) issued by co-operative banks for farmers in different states of India based o…Free
- Example 10Compute the trends by the method of moving averages, assuming that 4-year cycle is present in the following series. | Year | 1980 | 1981 | 1…Preview
Computation of Straight-line Trend by Using Method of Least Squares
The method of least squares fits the mathematically best-possible straight line to a time series — "best" in the specific sense that it minimises the total squared distance between the actual values a…
+−Examples 6.9.2(ii)i2 questions
- Example 11Given below are the consumer price index numbers (CPI) of the industrial workers. | Year | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 |…Free
- Example 12Based on the data available for the sales of an item in a district, by the method of least squares (i) tabulate the trend values (ii) find t…Preview
Unit Summary
This CBSE Class 12 Applied Mathematics unit builds the idea of an index number — a single figure measuring how a group of related variables (prices, quantities or values) changes between two situation…
Check Your Understanding — Reflective Questions
+−Reflective Questions15 questions
- Q1Judge the correctness or otherwise of the following statements: i) An index number is a pure number ii) Index numbers are independent of cho…Free
- Q2A price index which is based on the prices of the items in the composite, weighted by their relative index is called: i) price relatives ii)…Free
- Q3A weighted aggregate price index in which the weight for each variable is considered its current-period quantity is: i) Aggregative index ii…Free
- Q4An index constructed to measure changes in quantities over a period of time is: i) Quantity index ii) Time series index iii) Quality index i…Preview
- Q5For calculating the weighted index number, which of the following uses quantities consumed in the base period as weights: i) Fisher's method…Preview
- Q6What is the index number of the base period? i) 200 ii) 300 iii) 10 iv) 100Preview
- Q7Index number is a special type of : i) Average ii) Dispersion iii) Correlation iv) None of the abovePreview
- Q8Index number is always expressed in i) Percentage ii) Ratio iii) Proportion iv) None of the abovePreview
- Q9Which index number is called as ideal index number i) Laspeyres ii) Paasches iii) Fisher iv) None of the abovePreview
- Q10In Laspeyres price index number weight is considered as i) Quantity in base year ii) Quantity during current year iii) Prices in base year i…Preview
- Q11In Paasche's price index number weight is considered as i) Quantity in base year ii) Quantity in current year iii) Prices in base year iv) P…Preview
- Q12Fishers price index number is the i) A.M. of Laspeyres and Paasche's ii) G.M. of Laspeyres and Paasche's iii) Difference between Laspeyres a…Preview
- Q13When the prices of rice are to be compared, we compute: i) Volume index ii) Value index iii) Price index iv) Aggregative indexPreview
- Q14Purchasing power of money can be accessed through: i) Simple index ii) Fisher's index iii) Consumer price index iv) Volume indexPreview
- Q15Cost of living at two different cities can be compared with the help of: i) Value index ii) Consumer price index iii) Volume index iv) Un-we…Preview
Practice Exercise
+−Practice Exercise13 questions
- Q1Calculate index numbers from the following data by simple aggregate method taking prices of 1995 as base period. | Commodity | Year | A | B…Free
- Q2Construct price index number from the following data using i) Laspeyre's Method and ii) Paasche's method iii) Fisher's Ideal method | Commod…Free
- Q3Taking 1995 as base year calculate relative index number for the years 1997-2005 | Year | 1995 | 1997 | 1999 | 2001 | 2003 | 2005 | | --- |…Free
- Q4Compute the weighted aggregative index number for the following data: | Variable | Price Current year | Price Base year | Weights | | --- |…Preview
- Q5Calculate price index number for 2004 taking 1994 as the base year from the following data by simple aggregative method: | Item | Rice | Whe…Preview
- Q6Based on the data on the expenses of middle-class families in a certain city, calculate the cost-of-living index during the year 2003 as com…Preview
- Q7From the data given below, obtain the index of retail sales in India for years 1982, 1983, 1984 with the year 1981 as base period. | Year |…Preview
- Q8Calculate the price index number for the following data using weighted aggregative method: | Commodity | Unit | Weight | Price Base year | P…Preview
- Q9Based on the given data, check whether i) Paasche's formula and, ii) Fisher's formula will satisfy the time reversal test: | Commodity | Bas…Preview
- Q10The annual rainfall (in mm) was recorded for Cherrapunji, Meghalaya: | Year | Rainfall (in cm) | | --- | --- | | 2001 | 1.2 | | 2002 | 1.9 |…Preview
- Q11Compute the seasonal indices by 4-year moving averages from the given data of production of paper (in thousand tons) | Year | 1980 | 1981 |…Preview
- Q12Given below is the data of workers welfare expenses (in lakh ₹) in steel industries during 2001 - 2005. Use method of least squares to: i) t…Preview
- Q13Fit a straight-line tend by method of least squares for the following data and also find the trend value for year 1998: | Year | 1992 | 1993…Preview