The mean of a set of 100 observations were found to be 50. On checking, it was found that two observations were wrongly taken as 32 and 15 instead of 82 and 51. Calculate the correct mean

Given:
Number of observations ($n$) = 100
Incorrect mean ($\bar{X}_{\text{incorrect}}$) = 50

The formula for the mean is $\bar{X} = \frac{\sum X}{n}$.
So, the incorrect sum of observations ($\sum X_{\text{incorrect}}$) can be calculated as:

$$\sum X_{\text{incorrect}} = \bar{X}_{\text{incorrect}} \times n$$ $$\sum X_{\text{incorrect}} = 50 \times 100$$ $$\sum X_{\text{incorrect}} = 5000$$

The two observations wrongly taken were 32 and 15.
The correct observations should have been 82 and 51.
To find the correct sum ($\sum X_{\text{correct}}$), subtract the incorrect values and add the correct values to the incorrect sum:

$$\sum X_{\text{correct}} = \sum X_{\text{incorrect}} - (\text{wrongly taken values}) + (\text{correct values})$$ $$\sum X_{\text{correct}} = 5000 - (32 + 15) + (82 + 51)$$ $$\sum X_{\text{correct}} = 5000 - 47 + 133$$ $$\sum X_{\text{correct}} = 4953 + 133$$ $$\sum X_{\text{correct}} = 5086$$

Using the correct sum and the number of observations:

$$\bar{X}_{\text{correct}} = \frac{\sum X_{\text{correct}}}{n}$$ $$\bar{X}_{\text{correct}} = \frac{5086}{100}$$ $$\bar{X}_{\text{correct}} = 50.86$$

The correct mean is $\boxed{50.86}$.