The median of 100 observations grouped in classes of equal width is 25. If the median class interval is 20-30 and the number of observations less than 20 is 45, then the frequency of median class is

Options

A. 20
B. 12
C. 10
D. 15

Explanation

To find the frequency of the median class, we can use the formula for the median in a grouped frequency distribution. Let's break down the solution step-by-step.

### Step 1: Identify the given values
- Total number of observations (n) = 100
- Number of observations less than 20 (f) = 45
- Median class interval = 20-30
- Lower boundary of the median class (l) = 20
- Width of the class interval (h) = 10 (since the class is 20-30)

### Step 2: Calculate the cumulative frequency less than the median class
The cumulative frequency less than the median class (20-30) is given as:
- Cumulative frequency less than 20 = 45

Let the frequency of the median class (20-30) be denoted as f. Therefore, the cumulative frequency less than 30 will be:
- Cumulative frequency less than 30 = Cumulative frequency less than 20 + Frequency of the median class
- Cumulative frequency less than 30 = 45 + f

### Step 3: Use the median formula
The formula for the median in a grouped frequency distribution is given by:
\[ \text{Median} = l + \frac{\frac{n}{2} - F}{f} \times h \]
Where:
- Median = 25
- l = 20 (lower boundary of the median class)
- n = 100 (total number of observations)
- F = cumulative frequency less than the median class = 45
- f = frequency of the median class
- h = width of the class = 10

### Step 4: Substitute the known values into the median formula
Substituting the values into the formula:
\[ 25 = 20 + \frac{\frac{100}{2} - 45}{f} \times 10 \]
\[ 25 = 20 + \frac{50 - 45}{f} \times 10 \]
\[ 25 = 20 + \frac{5}{f} \times 10 \]
\[ 25 = 20 + \frac{50}{f} \]

### Step 5: Rearrange the equation to solve for f
Subtract 20 from both sides:
\[ 5 = \frac{50}{f} \]
Now, cross-multiply to solve for f:
\[ 5f = 50 \]
\[ f = \frac{50}{5} \]
\[ f = 10 \]

### Conclusion
The frequency of the median class (20-30) is **10**.