The median of certain observations 17, 18, 23, 27, x - 3, x + 5, 45, 49, 74 and 84, arranged in ascending order is 35. Later on, it was found that one observation 72 was misread as 27 by mistake. The correct median of the data is

Options

A. 36
B. 38
C. 42
D. 47

Answer

To find the correct median of the observations after correcting the misread value, we will follow these steps:

### Step 1: Determine the value of x
We know that the median of the observations is given as 35. The observations are:
17, 18, 23, 27, x - 3, x + 5, 45, 49, 74, and 84.

Since there are 10 observations (an even number), the median is calculated as the average of the 5th and 6th terms when arranged in ascending order.

The 5th term is (x - 3) and the 6th term is (x + 5).

The formula for the median is:
\[ \text{Median} = \frac{(x - 3) + (x + 5)}{2} = 35 \]
Simplifying this:
\[ \frac{2x + 2}{2} = 35 \]
\[ 2x + 2 = 70 \]
\[ 2x = 68 \]
\[ x = 34 \]

### Step 2: Substitute x back into the observations
Now that we have the value of x, we can substitute it back into the observations:
- x - 3 = 34 - 3 = 31
- x + 5 = 34 + 5 = 39

So the observations become:
17, 18, 23, 27, 31, 39, 45, 49, 74, and 84.

### Step 3: Correct the misread observation
The problem states that 27 was misread as 72. Therefore, we replace 27 with 72 in the observations.

The updated observations are:
17, 18, 23, 31, 39, 45, 49, 72, 74, and 84.

### Step 4: Arrange the observations in ascending order
The observations in ascending order are:
17, 18, 23, 31, 39, 45, 49, 72, 74, and 84.

### Step 5: Calculate the new median
Again, since we have 10 observations (even number), the median will be the average of the 5th and 6th terms.

The 5th term is 39 and the 6th term is 45.

Calculating the median:
\[ \text{Median} = \frac{39 + 45}{2} = \frac{84}{2} = 42 \]

### Final Answer
The correct median of the data is **42**.

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