Find the Standard Deviation from the given data and interpret the result. [4+1] Income 10 20 20 30 30 40 40 50 50 60 : : : : : : Labour 8 12 17 14 9 OR Find the index number by using Paasche's…

Solution 1: Finding Standard Deviation

Step 1: Find the Midpoint (x) of Each Income Class

The given data is grouped, so we calculate the midpoint of each income range.

Step 2: Calculate $f\times x$ and $f\times x^2$

We compute the necessary values for standard deviation.

Step 3: Calculate Mean ($\bar{x}$)

$$\bar{x}=\frac{\sum fx}{\sum f}=\frac{2140}{60}=35.67$$

Step 4: Calculate Variance (${\sigma }^2$)

$${\sigma }^2=\frac{\sum f{x}^2}{N}-{\left(\bar{x}\right)}^2$$ $${\sigma }^2=\frac{85600}{60}-(35.67{)}^2=1426.67-1272.23=154.44$$

Step 5: Calculate Standard Deviation ($\sigma$)

$$\sigma =\sqrt{154.44}=12.43$$

Interpretation

The standard deviation of Rs. 12.43 indicates that the income values of laborers deviate from the mean income (Rs. 35.67) by approximately Rs. 12.43 on average. A higher standard deviation suggests greater variability in income distribution among laborers.

---

Solution 2: Finding Index Number Using Paasche’s Method

Step 1: Calculate Base Year (2078) and Current Year (2080) Expenditures

Paasche’s index uses current year quantities as weights.

Step 2: Compute Paasche’s Index Formula

$$P_{01}=\frac{\sum (P_1\times Q_1)}{\sum (P_0\times Q_1)}\times 100$$

$$P_{01}=\frac{311}{200}\times 100=155.5$$

Interpretation

The Paasche’s index of 155.5 indicates that the price level in 2080 has increased by 55.5% compared to 2078, considering the current year’s consumption pattern. This suggests inflationary pressure on consumers.