Find out the mean and mode from the following table: Marks 0 20 20 40 40 60 60 80 80 100 : : : : : : No. of students 6 10 16 8 3 OR Compute Laspeyre's and Paasche's price index number from the…

Solution 1: Mean and Mode

Assumption: The data is continuous. We convert the class intervals to true class intervals by adjusting the boundaries (e.g., 0-20 becomes -1 to 21, 20-40 becomes 19 to 41, etc.) to ensure continuity.

Step 1: Construct the Frequency Table

We need the mid-values ($x$) and the cumulative frequencies ($cf$) to find the mean and mode.

Step 2: Calculate the Mean

The formula for the mean of grouped data is: $$\bar{x}=\frac{\sum fx}{\sum f}$$

Substituting the values from the table: $$\bar{x}=\frac{1990}{43}$$ $$\bar{x}\approx 46.28$$

The Mean is 46.28.

Step 3: Calculate the Mode

The modal class is the class with the highest frequency. Here, the highest frequency is 16, which corresponds to the class 39 - 61.

Formula for Mode: $$Mode=L+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h$$

Where:

• $L$ = Lower limit of the modal class = 39
• $f_1$ = Frequency of the modal class = 16
• $f_0$ = Frequency of the preceding class = 10
• $f_2$ = Frequency of the succeeding class = 8
• $h$ = Class width = $61-39=22$

Substituting the values: $$Mode=39+\frac{16-10}{2(16)-10-8}\times 22$$ $$Mode=39+\frac{6}{32-18}\times 22$$ $$Mode=39+\frac{6}{14}\times 22$$ $$Mode=39+0.4286\times 22$$ $$Mode=39+9.43$$ $$Mode=48.43$$

The Mode is 48.43.

---

Solution 2: Laspeyre's and Paasche's Price Index

Given Data:

• Base Year ($0$): 2020
• Current Year ($1$): 2024
• $p_0$ = Price in Base Year
• $q_0$ = Quantity in Base Year
• $p_1$ = Price in Current Year
• $q_1$ = Quantity in Current Year

Step 1: Calculate Aggregates

We need to calculate $\sum p_0{q}_0$, $\sum p_1{q}_0$, $\sum p_0{q}_1$, and $\sum p_1{q}_1$.

Step 2: Calculate Laspeyre's Price Index Number ($P_{01}^L$)

Laspeyre's index uses base year quantities ($q_0$) as weights.

$$P_{01}^L=\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100$$

$$P_{01}^L=\frac{1475}{1100}\times 100$$ $$P_{01}^L=1.3409\times 100$$ $$P_{01}^L=134.09$$

Laspeyre's Price Index Number is 134.09.

Step 3: Calculate Paasche's Price Index Number ($P_{01}^P$)

Paasche's index uses current year quantities ($q_1$) as weights.

$$P_{01}^P=\frac{\sum p_1{q}_1}{\sum p_0{q}_1}\times 100$$

$$P_{01}^P=\frac{2150}{1600}\times 100$$ $$P_{01}^P=1.34375\times 100$$ $$P_{01}^P=134.38$$

Paasche's Price Index Number is 134.38.