The mean of 25 observations is 72. It was detected that one observation 53 was wrongly copied as 78. Find the correct mean.

Options

A. 72
B. 71
C. 78
D. 76

Answer

To find the correct mean after detecting an error in one of the observations, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Original Sum of Observations:**
The mean of the 25 observations is given as 72. The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} \]
Therefore, we can calculate the sum of the observations (let's denote it as \( S \)):
\[ S = \text{Mean} \times \text{Number of Observations} = 72 \times 25 = 1800 \]

2. **Identify the Error in Observations:**
It was detected that one observation was wrongly copied. The incorrect observation was 78, and it should have been 53.

3. **Adjust the Sum of Observations:**
To find the correct sum, we need to subtract the incorrect observation and add the correct observation:
\[ \text{Correct Sum} = S - \text{Incorrect Observation} + \text{Correct Observation} \]
Plugging in the values:
\[ \text{Correct Sum} = 1800 - 78 + 53 \]
Simplifying this:
\[ \text{Correct Sum} = 1800 - 78 + 53 = 1800 - 25 = 1775 \]

4. **Calculate the Correct Mean:**
Now, we can find the correct mean using the corrected sum:
\[ \text{Correct Mean} = \frac{\text{Correct Sum}}{\text{Number of Observations}} = \frac{1775}{25} \]
Performing the division:
\[ \text{Correct Mean} = 71 \]

### Final Answer:
The correct mean of the observations is **71**.