(a) Explain the concept of index number. Calculate the Fisher's ideal index number from the following data and verify that…

(a)

An index number is a statistical device that measures relative change in a variable or group of variables, such as prices, quantities, or production, from a base period to another period, usually expressed as a percentage with base value 100. It is dimensionless, allows comparison over time, and is used to deflate monetary values, compare cost of living, and measure productivity. Index numbers may be simple or aggregate; here an aggregate index is needed because five commodities are combined. The Fisher index is preferred because it uses both base and current quantities and satisfies the two important tests. Here base year is 2006 and current year 2007.

For the data:

• Σp₀q₀ = 400+550+420+224+180 = 1774
• Σp₁q₀ = 480+550+510+280+195 = 2015
• Σp₀q₁ = 450+572+420+232+240 = 1914
• Σp₁q₁ = 540+572+510+290+260 = 2172

Laspeyres price index L₀₁ = (2015/1774)×100 = 113.585. Paasche price index P₀₁ = (2172/1914)×100 = 113.480. Fisher’s ideal index F₀₁ = √(L₀₁P₀₁) = 100√[(2015×2172)/(1774×1914)] = 100√(364715/282953) ≈ 113.53.

Time reversal test: F₀₁F₁₀ = 100². Here F₁₀ = 100√(282953/364715) ≈ 88.08, so 113.53×88.08 ≈ 10000. Hence it satisfies the time reversal test.

Factor reversal test: Fisher quantity index Fq₀₁ = 100√[(1914/1774)(2172/2015)] = 100√(2078604/1787305) ≈ 107.84. Value index V₀₁ = (2172/1774)×100 = 122.435. In percentage form, F₀₁×Fq₀₁ = 100×V₀₁; using unrounded values, 113.532×107.843 ≈ 12243.5. Hence it satisfies the factor reversal test.

(b)

(i) Use the logistic model dP/dt = rP(1−P/K), K = 10000000 = 10000 thousand. Its solution is P(t)=K/(1+Ae^(−rt)), where A=(K−P(0))/P(0). Taking logs gives y=ln[(K−P)/P]=ln A−rt, a straight line in t. Estimate a=ln A and b=−r by least squares, minimizing squared residuals in y. This is valid because 0<P<K, so the logarithm is defined. The carrying capacity K is fixed, so only r and A are estimated.

For P in thousands:

• t=0, P=1000, y=ln(9000/1000)=2.197225
• t=2, P=1785, y=ln(8215/1785)=1.526543
• t=4, P=2575, y=ln(7425/2575)=1.059003
• t=6, P=3400, y=ln(6600/3400)=0.663294
• t=8, P=4900, y=ln(5100/4900)=0.040005
• t=10, P=6200, y=ln(3800/6200)=−0.489548

Sums: n=6, Σt=30, Σt²=220, Σy=4.996522, Σty=6.693425. The normal equations come from differentiating Σ(y−a−bt)² with respect to a and b: 6a+30b=4.996522 and 30a+220b=6.693425. Solving, b=(6×6.693425−30×4.996522)/(6×220−30²)=−109.735122/420=−0.261274. Then a=(4.996522−30b)/6=2.139124. The fitted line is y=2.139124−0.261274t. Thus r=0.261274 year⁻¹ and A=e^a=8.4920. The slope is negative, so r is positive, as required for growth toward K. Equivalently, fitted P(0)=10000/(1+8.4920)=1053.5 thousand. The fitted value at t=0 need not equal the observed value because both parameters are estimated from all six observations. At t=10, fitted y=−0.473617, close to observed −0.489548, and P≈6163 thousand, close to 6200 thousand. The estimate is sensitive to the log transformation, but it is the standard hand-calculation method for a fixed K.

(ii) For 2026, t=12. P(12)=10000/[1+8.4920e^(−0.261274×12)] thousand. Since e^(−3.135289)=0.043487, P(12)=10000/[1+8.4920×0.043487]=10000/1.369293=7303.04 thousand. Predicted population ≈ 7,303,000.

(c)

Area statistics. India's land-use statistics follow a nine-fold classification (forests, area not available for cultivation, permanent pastures, land under miscellaneous tree crops, culturable waste, current fallows, other fallows, net sown area, and so on), from which net sown area, gross cropped area and cropping intensity (gross cropped area ÷ net sown area × 100) are derived. In the temporarily settled states, the patwari or village accountant records crop-wise area plot by plot during the girdawari (complete enumeration) each season. For timeliness, the Timely Reporting Scheme (TRS) asks patwaris to report a 20% random sample of villages early, and the Improvement of Crop Statistics (ICS) scheme of the NSO's Field Operations Division checks a sub-sample of TRS villages. In Kerala, Odisha and West Bengal, which lack village-level land records, area estimates come from the EARAS sample survey. Remote sensing now complements these records: the FASAL programme of the Mahalanobis National Crop Forecast Centre gives pre-harvest forecasts, and since 2024 the Digital Crop Survey under AgriStack records geo-referenced, plot-level crop data.

Yield statistics. Average yield is estimated through crop-cutting experiments (CCEs) under the General Crop Estimation Surveys, based on the stratified multi-stage random sampling design developed by P.V. Sukhatme and P.C. Mahalanobis: a plot of fixed size is located at random in a randomly selected field, harvested and weighed, and yield is expressed in kg or quintals per hectare. CCE results also settle claims under the Pradhan Mantri Fasal Bima Yojana, and the Digital General Crop Estimation Survey now records them through a mobile app. Production is estimated as area × yield, and the Directorate of Economics and Statistics of the Ministry of Agriculture releases advance estimates of production and publishes Agricultural Statistics at a Glance. Limitations include delays, the patwaris' workload, non-sampling errors in plot marking and eye estimation, and weak coverage of horticulture and mixed cropping.

Need and importance. Reliable area and yield statistics are needed to:

• estimate foodgrain availability and plan procurement, buffer stocks and the public distribution system;
• fix minimum support prices and decide on exports, imports and stock limits;
• settle crop insurance claims (PMFBY) and assess drought or flood damage for relief;
• estimate agriculture's share of GDP and rural incomes;
• guide credit (NABARD, banks), input supply, irrigation and research priorities (ICAR);
• compare states and districts, detect productivity trends and plan crop diversification.

Because agriculture employs about 46% of India's workforce (PLFS 2023–24) and remains dependent on the monsoon, timely and accurate area-yield statistics are essential for food security and farm policy.