Consider the observation `x_(1)=1, x_(2), x_(3)=3, .............x_(100)=100`
`x_(101), x_(102), x_(103), x_(104)=104` Median of the given data is

Options

A. 51
B. 52.5
C. 51.5
D. 53

Answer

To find the median of the given data set, we will follow these steps:

### Step 1: Identify the Data Set
The observations are given as:
- \( x_1 = 1 \)
- \( x_2 = 2 \)
- \( x_3 = 3 \)
- ...
- \( x_{100} = 100 \)
- \( x_{101} = 104 \)
- \( x_{102} = 104 \)
- \( x_{103} = 104 \)
- \( x_{104} = 104 \)

This means the data set consists of the numbers 1 through 100, followed by the number 104 repeated four times.

### Step 2: Count the Total Number of Observations
The total number of observations is:
- From \( x_1 \) to \( x_{100} \): 100 observations
- From \( x_{101} \) to \( x_{104} \): 4 observations

Thus, the total number of observations is:
\[ 100 + 4 = 104 \]

### Step 3: Determine the Position of the Median
Since there are 104 observations (an even number), the median will be the average of the values at the positions \( \frac{n}{2} \) and \( \frac{n}{2} + 1 \), where \( n \) is the total number of observations.

Calculating these positions:
- \( \frac{104}{2} = 52 \)
- \( \frac{104}{2} + 1 = 53 \)

### Step 4: Find the Values at the Median Positions
Now we need to find the values at the 52nd and 53rd positions in the ordered data set.

The ordered data set is:
- \( 1, 2, 3, \ldots, 100, 104, 104, 104, 104 \)

The 52nd observation is 52, and the 53rd observation is 53.

### Step 5: Calculate the Median
The median is the average of the 52nd and 53rd observations:
\[ \text{Median} = \frac{x_{52} + x_{53}}{2} = \frac{52 + 53}{2} = \frac{105}{2} = 52.5 \]

### Final Answer
The median of the given data set is \( 52.5 \).

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