Find the median from the following frequency distribution.
`{:("Number of students (f)",12,8,32,14,16,3),("Marks (x)",40,18,50,100,80,160):}`

Answer

To find the median from the given frequency distribution, we will follow these steps:

### Step 1: Create the Frequency Distribution Table
We have the following data:
- Marks (x): 40, 18, 50, 100, 80, 160
- Number of students (f): 12, 8, 32, 14, 16, 3

### Step 2: Calculate the Cumulative Frequency (CF)
We will calculate the cumulative frequency by adding the frequencies step by step.

1. **CF for 40**: 12
2. **CF for 18**: 12 + 8 = 20
3. **CF for 50**: 20 + 32 = 52
4. **CF for 100**: 52 + 14 = 66
5. **CF for 80**: 66 + 16 = 82
6. **CF for 160**: 82 + 3 = 85

So, the cumulative frequency table is:
- Marks (x): 40, 18, 50, 100, 80, 160
- Cumulative Frequency (CF): 12, 20, 52, 66, 82, 85

### Step 3: Determine the Total Number of Observations (n)
The total number of observations (n) is the last value in the cumulative frequency, which is 85.

### Step 4: Find the Median Position
Since n is 85 (which is odd), we use the formula for the median position:
\[ \text{Median Position} = \frac{n + 1}{2} = \frac{85 + 1}{2} = \frac{86}{2} = 43 \]

### Step 5: Locate the Median Class
Now, we need to find the cumulative frequency that is greater than or equal to 43. Looking at our cumulative frequency table:
- CF = 12 (for 40)
- CF = 20 (for 18)
- CF = 52 (for 50)

The first cumulative frequency greater than 43 is 52, which corresponds to the marks 50.

### Step 6: Conclusion
Thus, the median of the given frequency distribution is:
\[ \text{Median} = 50 \]

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