The frequency distribution of the marks obtained by 100 students in a test carrying 50 marks is given below. Then the mean is
`{:("Marks ",0-9,10-19,20-29,30-39,40-49),("No. of students "," "8," "15," "20," "45," "12):}`

Options

A. `28 . 3`
B. 28
C. `27. 3`
D. `26 . 4`

Answer

To find the mean of the frequency distribution of marks obtained by 100 students, we will follow these steps:

### Step 1: Create a Frequency Table
We will create a table with the midpoints (x_i), frequencies (f_i), and the product of midpoints and frequencies (x_i * f_i).

| Marks | No. of Students (f_i) | Midpoint (x_i) | f_i * x_i |
|------------|-----------------------|----------------|-----------|
| 0 - 9 | 8 | 4.5 | 36 |
| 10 - 19 | 15 | 14.5 | 217.5 |
| 20 - 29 | 20 | 24.5 | 490 |
| 30 - 39 | 45 | 34.5 | 1552.5 |
| 40 - 49 | 12 | 44.5 | 534 |
| **Total** | **100** | | **2830** |

### Step 2: Calculate the Midpoints (x_i)
To find the midpoints for each class interval, we use the formula:
\[ x_i = \frac{\text{Lower limit} + \text{Upper limit}}{2} \]

- For 0 - 9: \( x_1 = \frac{0 + 9}{2} = 4.5 \)
- For 10 - 19: \( x_2 = \frac{10 + 19}{2} = 14.5 \)
- For 20 - 29: \( x_3 = \frac{20 + 29}{2} = 24.5 \)
- For 30 - 39: \( x_4 = \frac{30 + 39}{2} = 34.5 \)
- For 40 - 49: \( x_5 = \frac{40 + 49}{2} = 44.5 \)

### Step 3: Calculate the Product of Frequencies and Midpoints (f_i * x_i)
Now we multiply the frequency (f_i) by the corresponding midpoint (x_i):

- For 0 - 9: \( 8 \times 4.5 = 36 \)
- For 10 - 19: \( 15 \times 14.5 = 217.5 \)
- For 20 - 29: \( 20 \times 24.5 = 490 \)
- For 30 - 39: \( 45 \times 34.5 = 1552.5 \)
- For 40 - 49: \( 12 \times 44.5 = 534 \)

### Step 4: Calculate the Summation of f_i and f_i * x_i
Next, we sum up the frequencies and the products:

- \( \sum f_i = 8 + 15 + 20 + 45 + 12 = 100 \)
- \( \sum (f_i \cdot x_i) = 36 + 217.5 + 490 + 1552.5 + 534 = 2830 \)

### Step 5: Calculate the Mean
Finally, we use the formula for the mean:
\[ \text{Mean} = \frac{\sum (f_i \cdot x_i)}{\sum f_i} \]

Substituting the values we calculated:
\[ \text{Mean} = \frac{2830}{100} = 28.3 \]

### Conclusion
The mean of the frequency distribution is **28.3**.

---