The mean of a data set consisting of 20 observation is 40. If one observation 53 was wrongly recorded as 33, then the correct mean will be

Options

A. 41
B. 49
C. `40.5`
D. `42.5`

Answer

To find the correct mean of the data set after correcting the wrongly recorded observation, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have a data set with 20 observations.
- The mean of these observations is given as 40.
- One observation was wrongly recorded as 33 instead of the correct value, which is 53.

2. **Calculate the Total Sum of the Observations**:
- The mean (average) is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
- Given that the mean is 40 and there are 20 observations, we can find the total sum:
\[ \text{Sum of observations} = \text{Mean} \times \text{Number of observations} = 40 \times 20 = 800 \]

3. **Adjust the Total Sum for the Correct Observation**:
- Since one observation was recorded as 33 instead of 53, we need to adjust the total sum:
- The incorrect sum includes 33, so we subtract 33 and add 53 to get the correct sum:
\[ \text{Correct Sum} = 800 - 33 + 53 = 800 - 33 + 53 = 820 \]

4. **Calculate the Correct Mean**:
- Now, we can find the correct mean using the corrected sum:
\[ \text{Correct Mean} = \frac{\text{Correct Sum}}{\text{Number of observations}} = \frac{820}{20} = 41 \]

5. **Conclusion**:
- The correct mean of the 20 observations is **41**.