The Table given below shows the number of students having obtained different marks. `{:("Marks","Number of Students"),(9-11,6),(11-13,5),(13-15,2),(15-17,2),(17-19,5):}`
What is the mean marks per student ?

Options

A. 13.5
B. 12.25
C. 15.5
D. 14.25

Answer

To find the mean marks per student from the given table, we will follow these steps:

### Step 1: Identify the mid-values for each class interval.
The mid-value (x) for each class interval can be calculated using the formula:
\[ \text{Mid-value} = \frac{\text{Lower limit} + \text{Upper limit}}{2} \]

- For the interval 9-11:
\[ \text{Mid-value} = \frac{9 + 11}{2} = 10 \]

- For the interval 11-13:
\[ \text{Mid-value} = \frac{11 + 13}{2} = 12 \]

- For the interval 13-15:
\[ \text{Mid-value} = \frac{13 + 15}{2} = 14 \]

- For the interval 15-17:
\[ \text{Mid-value} = \frac{15 + 17}{2} = 16 \]

- For the interval 17-19:
\[ \text{Mid-value} = \frac{17 + 19}{2} = 18 \]

### Step 2: Create a table with mid-values and number of students.
We will summarize the data as follows:

| Marks Interval | Number of Students (f) | Mid-value (x) |
|----------------|------------------------|----------------|
| 9 - 11 | 6 | 10 |
| 11 - 13 | 5 | 12 |
| 13 - 15 | 2 | 14 |
| 15 - 17 | 2 | 16 |
| 17 - 19 | 5 | 18 |

### Step 3: Calculate the product of the number of students and mid-values (fx).
Now we will calculate \( fx \) for each interval:

- For 9-11:
\[ fx = 6 \times 10 = 60 \]

- For 11-13:
\[ fx = 5 \times 12 = 60 \]

- For 13-15:
\[ fx = 2 \times 14 = 28 \]

- For 15-17:
\[ fx = 2 \times 16 = 32 \]

- For 17-19:
\[ fx = 5 \times 18 = 90 \]

### Step 4: Calculate the summation of f and fx.
Now we will find the total number of students (summation of f) and the total of fx (summation of fx):

- Summation of f:
\[ \text{Total } f = 6 + 5 + 2 + 2 + 5 = 20 \]

- Summation of fx:
\[ \text{Total } fx = 60 + 60 + 28 + 32 + 90 = 270 \]

### Step 5: Calculate the mean.
The mean can be calculated using the formula:
\[ \text{Mean} = \frac{\Sigma fx}{\Sigma f} \]

Substituting the values we calculated:
\[ \text{Mean} = \frac{270}{20} = 13.5 \]

### Final Answer:
The mean marks per student is **13.5**.

---