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 Class Interval

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

Step 2: Calculate Mean (μ)

$$\mu =\frac{\sum (f\cdot x)}{\sum f}$$

$$\sum (f\cdot x)=(15\times 8)+(25\times 12)+(35\times 17)+(45\times 14)+(55\times 9)$$

$$=120+300+595+630+495=2140$$

$$\sum f=8+12+17+14+9=60$$

$$\mu =\frac{2140}{60}=35.67(\text{approx.})$$

Step 3: Calculate Variance (σ²)

$${\sigma }^2=\frac{\sum f(x-\mu {)}^2}{\sum f}$$

$$\sum f(x-\mu {)}^2=3417.84+1365.96+7.65+1218.70+3363.57=9373.72$$

$${\sigma }^2=\frac{9373.72}{60}=156.23$$

Step 4: Calculate Standard Deviation (σ)

$$\sigma =\sqrt{156.23}=12.50(\text{approx.})$$

Interpretation

The standard deviation of 12.50 indicates that the income values in the dataset tend to deviate from the mean income (Rs. 35.67) by approximately Rs. 12.50. A higher standard deviation suggests greater dispersion in income levels among the laborers.

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Solution 2: Finding Index Number Using Paasche’s Method

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

Paasche’s formula: $$P_{01}=\frac{\sum (p_1{q}_1)}{\sum (p_0{q}_1)}\times 100$$

$$\sum (p_0{q}_1)=90+50+36+24=200$$

$$\sum (p_1{q}_1)=150+80+60+21=311$$

Step 2: Calculate Paasche’s Index Number

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

Interpretation

The Paasche’s index number is 155.5, meaning that the price level in 2080 has increased by 55.5% compared to 2078. This indicates a significant rise in the cost of living for the given commodities.