The median of the data 78, 56, 22, 34, 45, 54, 39, 68, 54 and 84 is

Options

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

Explanation

To find the median of the given data set: 78, 56, 22, 34, 45, 54, 39, 68, 54, and 84, we will follow these steps:

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

**Data in ascending order:**
22, 34, 39, 45, 54, 54, 56, 68, 78, 84

### Step 2: Count the number of observations
Next, we need to determine how many numbers are in the data set.

**Number of observations (n):**
There are 10 observations.

### Step 3: Determine if n is odd or even
Since the number of observations (n = 10) is even, we will use the formula for finding the median in an even set of numbers.

### Step 4: Use the median formula for an even number of observations
The median is calculated as the average of the two middle numbers. The positions of these numbers can be found using the formula:
- The two middle positions are \( \frac{n}{2} \) and \( \frac{n}{2} + 1 \).

For our data:
- \( \frac{10}{2} = 5 \) (5th observation)
- \( \frac{10}{2} + 1 = 6 \) (6th observation)

### Step 5: Identify the 5th and 6th observations
Now, we look at our sorted data to find the 5th and 6th observations:
- 5th observation = 54
- 6th observation = 54

### Step 6: Calculate the median
Now, we calculate the median by averaging the 5th and 6th observations:
\[ \text{Median} = \frac{\text{5th observation} + \text{6th observation}}{2} = \frac{54 + 54}{2} = \frac{108}{2} = 54 \]

### Final Answer:
The median of the given data is **54**.

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