1. **State the problem:** We have a dataset of 200 items with an initial mean of 80. Two items were recorded incorrectly: 61 was recorded as 16, and 45 was recorded as 15. We need to find the corrected mean. 2. **Recall the formula for mean:** The mean $\bar{x}$ is given by $$\bar{x} = \frac{\text{sum of all items}}{N}$$ where $N$ is the number of items. 3. **Calculate the original total sum:** Given the mean $\bar{x} = 80$ and $N = 200$, the total sum of all items is $$\text{sum} = \bar{x} \times N = 80 \times 200 = 16000$$ 4. **Adjust the sum for the incorrect entries:** The incorrect sum includes 16 and 15 instead of 61 and 45. So, the sum was underestimated by $$ (61 - 16) + (45 - 15) = 45 + 30 = 75 $$ 5. **Calculate the corrected sum:** $$\text{corrected sum} = 16000 + 75 = 16075$$ 6. **Calculate the corrected mean:** $$\text{corrected mean} = \frac{\text{corrected sum}}{N} = \frac{16075}{200} = 80.375$$ **Final answer:** The corrected mean is $80.375$.