Given data shows marks of 32 students in an examination. Mean of the marks will be
`{:("Class interval",0-10,10-20,20-30,30-40,40-50),("Frequency",12,6,8,4,2):}`

Options

A. 18.125
B. 21.875
C. 22.425
D. 22.545

Answer

To find the mean of the marks of 32 students based on the given class intervals and frequencies, we will follow these steps:

### Step 1: Identify Class Intervals and Frequencies
We are given the following class intervals and their corresponding frequencies:

- Class Intervals: 0-10, 10-20, 20-30, 30-40, 40-50
- Frequencies: 12, 6, 8, 4, 2

### Step 2: Calculate Midpoints of Class Intervals
The midpoint (also known as the class mark) for each class interval is calculated using the formula:
\[ \text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2} \]

Calculating midpoints:
- For 0-10: \( \frac{0 + 10}{2} = 5 \)
- For 10-20: \( \frac{10 + 20}{2} = 15 \)
- For 20-30: \( \frac{20 + 30}{2} = 25 \)
- For 30-40: \( \frac{30 + 40}{2} = 35 \)
- For 40-50: \( \frac{40 + 50}{2} = 45 \)

### Step 3: Calculate \( F \times x \) (Frequency times Midpoint)
Now we will multiply each frequency by its corresponding midpoint:
- For 0-10: \( 12 \times 5 = 60 \)
- For 10-20: \( 6 \times 15 = 90 \)
- For 20-30: \( 8 \times 25 = 200 \)
- For 30-40: \( 4 \times 35 = 140 \)
- For 40-50: \( 2 \times 45 = 90 \)

### Step 4: Sum of Frequencies and Sum of \( F \times x \)
Now, we will find the total of frequencies and the total of \( F \times x \):
- Total Frequency (\( \Sigma F \)): \( 12 + 6 + 8 + 4 + 2 = 32 \)
- Total \( F \times x \) (\( \Sigma Fx \)): \( 60 + 90 + 200 + 140 + 90 = 580 \)

### Step 5: Calculate the Mean
The mean is calculated using the formula:
\[ \text{Mean} = \frac{\Sigma Fx}{\Sigma F} \]

Substituting the values we found:
\[ \text{Mean} = \frac{580}{32} = 18.125 \]

### Final Answer
The mean of the marks of the 32 students is **18.125**.

---