The mean of $25$ observations was found to be $78.4$. But later on it was found that $96$ was misread as $69$. The correct mean is

### Concept & Formula When a value is incorrectly read while calculating a mean, we can find the correct mean by adjusting the total sum. Instead of recalculating the entire sum from scratch, we can simply find the net difference caused by the error and distribute it evenly across all observations. $$\text{Correct Mean} = \text{Old Mean} + \frac{\text{Correct Value} - \text{Incorrect Value}}{\text{Total Observations}}$$ ### Step-by-Step Solution * **Given:** * Total observations ($n$) = $25$ * Incorrect mean = $78.4$ * Correct value = $96$ * Incorrect value = $69$ * **Calculation:** 1. Determine the net error in the total sum. The sum was recorded lower than it should have been because $96$ was read as $69$. 2. Net increase required in sum = $96 - 69 = 27$. 3. This increase of $27$ needs to be distributed equally among the $25$ observations. 4. Increase in average = $\frac{27}{25} = \frac{108}{100} = 1.08$. 5. Correct Mean = $78.4 + 1.08 = 79.48$. ### Exam Strategy & Shortcut **The Deviation Method:** Never multiply $25 \times 78.4$ in an exam! Calculate the difference between the correct and incorrect values ($+27$). Divide this difference by the total number of items ($25$) to find the change in the average ($+1.08$). Add this to the old average. ### Common Pitfall Students often waste precious time calculating the total sum ($25 \times 78.4 = 1960$), subtracting $69$, adding $96$, and then dividing by $25$ again. While this works, it is highly prone to arithmetic errors and slows you down significantly. ### Final Answer **Therefore, the correct answer is 79.48.**