The median of the data 78,56,22,34,45,54,39,68,54,84 is
(i) 45
(ii) 49.5
(iii) 54
(iv) 56

Options

A. 45
B. 49.5
C. 54
D. 56

Answer

To find the median of the data set \( 78, 56, 22, 34, 45, 54, 39, 68, 54, 84 \), we can follow these steps:

### Step 1: Arrange the data in ascending order
First, we need to sort the numbers from the smallest to the largest.

**Sorted data:**
\[ 22, 34, 39, 45, 54, 54, 56, 68, 78, 84 \]

### Step 2: Determine the number of data points
Next, we count the number of data points in the sorted list.

**Number of data points (n):**
\[ n = 10 \]

### Step 3: Identify if n is even or odd
Since \( n = 10 \) is even, we will use the formula for finding the median in an even dataset.

### Step 4: Use the median formula
The median for an even number of observations is calculated using the formula:
\[ \text{Median} = \frac{\text{n/2 th value} + \text{(n/2 + 1) th value}}{2} \]

### Step 5: Calculate the positions
We need to find the \( \frac{n}{2} \)th and \( \left(\frac{n}{2} + 1\right) \)th values.

\[ \frac{n}{2} = \frac{10}{2} = 5 \]
\[ \frac{n}{2} + 1 = 5 + 1 = 6 \]

### Step 6: Find the 5th and 6th values in the sorted data
From our sorted data:
- The 5th value is \( 54 \)
- The 6th value is also \( 54 \)

### Step 7: Calculate the median
Now we can substitute these values into the median formula:

\[ \text{Median} = \frac{54 + 54}{2} = \frac{108}{2} = 54 \]

### Final Answer
Thus, the median of the data set is \( 54 \).

### Conclusion
The correct option is (iii) 54.

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