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 situations that differ in time, geographical location or other characteristics — and then extends the same measurement idea to data recorded over time as a time series.

Index numbers. An index number is a pure, unit-free number; the index for a time period $n$ is written $I_n$, and a list of such indexes across periods forms an index series. Constructing an index involves four choices: selection of the data, the base period, the weights, and the variables to be tracked.

Methods of construction. The unit covers nine ways to build an index (price forms shown; a quantity index replaces $p$ by $Q$):

• Relative index number: $P = \dfrac{p_1}{p_0} \times 100$
• Simple (unweighted) aggregative: $I_n = \dfrac{\sum p_n}{\sum p_0} \times 100$
• Simple average of relatives: $I_n = \dfrac{1}{N}\sum\left(\dfrac{p_n}{p_0}\times100\right)$
• Weighted aggregative: $I_n = \dfrac{\sum p_n Q}{\sum p_0 Q} \times 100$
• Laspeyres' method (base-period quantities $q_0$ as weights): $I_n^{L} = \dfrac{\sum p_n q_0}{\sum p_0 q_0} \times 100$
• Paasche's method (current-period quantities $Q_n$ as weights): $I_n^{Pa} = \dfrac{\sum p_n Q_n}{\sum p_0 Q_n} \times 100$
• Fisher's ideal method (geometric mean of Laspeyres and Paasche): $I_n^{F} = \sqrt{\dfrac{\sum p_n q_0}{\sum p_0 q_0} \times \dfrac{\sum p_n Q_n}{\sum p_0 Q_n}} \times 100$
• Marshall–Edgeworth's method: $I_n^{ME} = \dfrac{\sum (q_0 + q_n)\,p_n}{\sum (q_0 + q_n)\,p_0} \times 100$
• Weighted average of relatives (value weights $p_0 Q_0$): $I_n = \dfrac{\sum (p_0 Q_0)\, R}{\sum p_0 Q_0}$, where $R = \dfrac{p_n}{p_0}\times100$

Types of index numbers: value index, quantity index and price index (the Consumer Price Index, CPI, being the familiar price index; the Index of Industrial Production, IIP, a quantity index).

Tests of adequacy. Four consistency tests exist — the unit test, the time-reversal test, the factor-reversal test and the circular test. The time-reversal test requires $P_{01} \times P_{10} = 1$, where $P_{01}$ is the index for the current year "1" on base year "0" and $P_{10}$ is the reverse comparison. Laspeyres' and Paasche's methods do not satisfy this test; Fisher's ideal index does.

Time series. A time series is numerical data for a variable recorded sequentially at successive, distinct time intervals for a specified period; its purpose is to reveal how the variable moves — often a growth pattern — over time. Related data structures are cross-sectional data (one or more variables collected at the same point in time) and pooled data (a combination of time-series and cross-sectional data). A series in which only one variable varies over time is univariate; one collecting several variables over time is multivariate.

Components of a time series. A time series is made up of four components:

• Secular trend — the smooth, regular, long-term variation observed over a long period.
• Seasonal — regular periodic variability captured within one-year periods.
• Cyclical — an oscillatory movement whose period of oscillation is more than a year (one complete swing being a cycle; real GDP is a classic example). …