The following observations have been arranged in ascending order. If the median of the da ta is 54. Find the value of x.
29, 32, 48, 50, X, X + 2, 72, 78, 84, 95

Answer

To find the value of \( x \) such that the median of the given data set is 54, we can follow these steps:

### Step 1: Identify the number of observations
The given data set is:
\[ 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95 \]
There are 10 observations in total.

### Step 2: Determine the position of the median
Since there are 10 observations (an even number), the median is calculated as the average of the 5th and 6th terms in the ordered list.

### Step 3: Identify the 5th and 6th terms
From the data set, the 5th term is \( x \) and the 6th term is \( x + 2 \).

### Step 4: Set up the median equation
The median is given as 54. Therefore, we can set up the equation:
\[ \text{Median} = \frac{\text{5th term} + \text{6th term}}{2} = \frac{x + (x + 2)}{2} = 54 \]

### Step 5: Simplify the equation
Now, simplify the equation:
\[ \frac{2x + 2}{2} = 54 \]
This simplifies to:
\[ x + 1 = 54 \]

### Step 6: Solve for \( x \)
Now, subtract 1 from both sides:
\[ x = 54 - 1 \]
\[ x = 53 \]

### Conclusion
The value of \( x \) is \( 53 \).

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