Find the median for the following frequency distribution :
`{:("Marks obtained"(x_(i)),29,32,34,27,42,37),("Number of students"(f_(i)),2,5,10,12,8,4):}`

Answer

To find the median for the given frequency distribution, we can follow these steps:

### Step 1: Organize the Data
We have the marks obtained (x) and the number of students (f) as follows:

| Marks obtained (x) | Number of students (f) |
|---------------------|------------------------|
| 29 | 2 |
| 32 | 5 |
| 34 | 10 |
| 27 | 12 |
| 42 | 8 |
| 37 | 4 |

### Step 2: Calculate the Total Frequency (n)
To find the median, we first need to calculate the total frequency (n), which is the sum of all frequencies (f).

\[ n = 2 + 5 + 10 + 12 + 8 + 4 = 41 \]

### Step 3: Calculate the Cumulative Frequency (CF)
Next, we will calculate the cumulative frequency for each class.

| Marks obtained (x) | Number of students (f) | Cumulative Frequency (CF) |
|---------------------|------------------------|----------------------------|
| 29 | 2 | 2 |
| 32 | 5 | 2 + 5 = 7 |
| 34 | 10 | 7 + 10 = 17 |
| 27 | 12 | 17 + 12 = 29 |
| 42 | 8 | 29 + 8 = 37 |
| 37 | 4 | 37 + 4 = 41 |

### Step 4: Find the Median Position
The formula for the median position is given by:

\[ \text{Median Position} = \frac{n + 1}{2} \]

Substituting the value of n:

\[ \text{Median Position} = \frac{41 + 1}{2} = \frac{42}{2} = 21 \]

### Step 5: Identify the Median Class
Now, we need to find the cumulative frequency that includes the 21st observation. From our cumulative frequency table:

- CF for 29 is 2
- CF for 32 is 7
- CF for 34 is 17
- CF for 27 is 29 (this is the first CF that is greater than or equal to 21)

Thus, the median class is the one corresponding to the marks obtained of 27.

### Step 6: Conclusion
The median of the frequency distribution is 27.

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