Updated NCERT Solutions for Class 11 Statistics Chapter 7: Index Numbers

An index number is a specialized type of average that measures the change in the level of a phenomenon (like price, quantity, or value) over a period of time, across locations, or between different categories. It simplifies complex data into a single figure for easy comparison. The base period value is typically set to 100.

Problems in the Construction of Index Numbers:

1. Purpose of the Index: The first problem is to clearly define the purpose. An index for wholesale prices will be different from one for retail prices. The choice of items, base year, and formula all depend on the purpose.
2. Selection of the Base Period: The base period should be a normal period, free from abnormalities like wars, famines, or economic depressions. It should also not be too far in the past.
3. Selection of Items: We must select a representative sample of items that are relevant to the index's purpose.
4. Selection of Price Quotations: Prices can vary from place to place. It's crucial to obtain accurate and representative price quotations from a reliable source.
5. Choice of an Average: Different averages (like arithmetic or geometric mean) can be used. The geometric mean is often considered better, but the arithmetic mean is more common.
6. Selection of Weights: All items are not equally important. Assigning appropriate weights (value or quantity) is a critical and difficult step.
7. Choice of Formula: There are several formulas (Laspeyres', Paasche's, Fisher's, etc.). The choice of formula can affect the final result.

The Consumer Price Index (CPI), also known as the Cost of Living Index, is an index number that measures the average change over time in the prices paid by a specific class of consumers for a basket of consumer goods and services. It helps in understanding the effect of price changes on the cost of living.

Construction of CPI:

1. Determine the Scope and Target Group: Decide for which group of consumers the CPI is being constructed (e.g., industrial workers, urban employees).
2. Conduct a Family Budget Survey: A survey is conducted to determine the goods, services, and quantities they consume to create a 'consumption basket' and assign weights.
3. Select a Base Year: A normal year is chosen as the base year.
4. Collect Price Data: Retail prices for the selected items are collected regularly.
5. Calculate the Index: The CPI is calculated using two main methods:
   • Aggregative Expenditure Method: Formula: CPI = (ΣP₁Q₀ / ΣP₀Q₀) × 100
   • Family Budget Method: Formula: CPI = (ΣRW / ΣW), where R = Price Relative and W = Weight.

First, we create a calculation table to find the required values: ΣP₁Q₀, ΣP₀Q₀, ΣP₁Q₁, and ΣP₀Q₁.

Now, we apply the formulas:

(i) Laspeyres' Price Index (P₀₁ (La))
Formula: P₀₁ (La) = (ΣP₁Q₀ / ΣP₀Q₀) × 100
Calculation: (2070 / 1660) × 100 = 1.2469 × 100 = 124.69

(ii) Paasche's Price Index (P₀₁ (Pa))
Formula: P₀₁ (Pa) = (ΣP₁Q₁ / ΣP₀Q₁) × 100
Calculation: (1790 / 1470) × 100 = 1.2176 × 100 = 121.76

(iii) Fisher's Price Index (P₀₁ (F))
Formula: P₀₁ (F) = √[Laspeyres' × Paasche's]
Calculation: √[124.69 × 121.76] = √[15182.25] = 123.22

Fisher's index number is called the 'Ideal Index' because it satisfies several statistical tests, making it theoretically superior.

1. Based on Geometric Mean: It is the geometric mean of Laspeyres' and Paasche's indices, which is considered the best average for index numbers.
2. Considers Both Years' Quantities: It uses both base year (Q₀) and current year (Q₁) quantities, avoiding the bias of Laspeyres' (upward bias) and Paasche's (downward bias).
3. Satisfies Time Reversal Test: This test requires that P₀₁ × P₁₀ = 1. Fisher's index satisfies this perfectly.
4. Satisfies Factor Reversal Test: This test requires that the price index multiplied by the quantity index equals the value index (P₀₁ × Q₀₁ = V₀₁). Fisher's index also satisfies this test.

Due to these properties, it provides a more accurate and unbiased measure of price changes.

A simple index number gives equal importance to all items. A weighted index number assigns different levels of importance (weights) to different items based on their relative significance (e.g., quantity consumed). Weighted indices are more realistic.

Laspeyres' Price Index P₀₁(La) = (ΣP₁Q₀ / ΣP₀Q₀) × 100
P₀₁(La) = (400 / 320) × 100
P₀₁(La) = 1.25 × 100 = 125.

The Aggregative Expenditure Method for CPI is the same as Laspeyres' Method.
Formula: CPI = (ΣP₁Q₀ / ΣP₀Q₀) × 100

Now, calculate the CPI:
CPI = (2150 / 1780) × 100
CPI = 1.2078 × 100 = 120.78

Conclusion: The Consumer Price Index is 120.78, indicating that the cost of living has increased by approximately 20.78%.

1. Laspeyres' Price Index:
   P₀₁(La) = (ΣP₁Q₀ / ΣP₀Q₀) × 100 = (10000 / 8000) × 100 = 125
2. Paasche's Price Index:
   P₀₁(Pa) = (ΣP₁Q₁ / ΣP₀Q₁) × 100 = (10375 / 8300) × 100 ≈ 125
3. Comparison and Reason:
   In this case, both indices are 125, showing the same price change. This is a coincidence. Generally, Laspeyres' has an upward bias as it ignores consumer shifts away from pricier goods, while Paasche's has a downward bias.