The mean of 100 observations is 40. It is found that an observation 53 was misread as 83. Find the correct mean.

Answer

To find the correct mean after identifying the misread observation, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Total Sum of Observations**:
- We know the mean of 100 observations is 40.
- The formula for mean is:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} \]
- Rearranging gives us:
\[ \text{Sum of Observations} = \text{Mean} \times \text{Number of Observations} \]
- Substituting the given values:
\[ \text{Sum of Observations} = 40 \times 100 = 4000 \]

2. **Identify the Misread Observation**:
- The observation was misread as 83 instead of the correct value of 53.

3. **Adjust the Sum of Observations**:
- To correct the sum, we need to subtract the misread value (83) and add the correct value (53):
\[ \text{Corrected Sum} = \text{Original Sum} - \text{Misread Value} + \text{Correct Value} \]
- Plugging in the numbers:
\[ \text{Corrected Sum} = 4000 - 83 + 53 \]
- Calculate:
\[ \text{Corrected Sum} = 4000 - 83 + 53 = 4000 - 30 = 3970 \]

4. **Calculate the New Mean**:
- Now that we have the corrected sum, we can find the new mean:
\[ \text{New Mean} = \frac{\text{Corrected Sum}}{\text{Number of Observations}} = \frac{3970}{100} \]
- Performing the division:
\[ \text{New Mean} = 39.7 \]

### Final Answer:
The correct mean after adjusting for the misread observation is **39.7**.