Standard deviation $\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n} x_i^2 - \left(\frac{1}{n}\sum_{i=1}^{n} x_i\right)^2}$
Step-by-Step Solution
Key Concept: Standard deviation measures spread using the formula involving the mean of squares minus the square of the mean; corrections must account for both incorrect and correct data points.
Using the standard deviation formula with the given data, we have $\sigma = \sqrt{\frac{1}{100} \times \text{Incorrect} \sum_{i=1}^{n} x_i^2 - \bar{x}^2}$. The incorrect value of $\sum x_i^2 = 1600$. Correcting this: Incorrect $\sum x_i^2 = 100(26.01 + 1600) = 162601$. Correct mean $\bar{x} = \frac{3900}{100} = 39.0$. The correct $\sum x_i^2 = 162601 - 2500 + 1600 = 161701$. Therefore, correct standard deviation $= \sqrt{\frac{161701}{100} - (39.9)^2} = \sqrt{1617.01 - 1592.01} = \sqrt{25} = 5$.
Correct Answer: 5