Calculate Incorrect Sum and Sum of Squares

Find the initial sum of observations and sum of squares using the given mean and standard deviation formulas.

$\sum x_i = 5 \times 40 = 200, \quad \sum x_i^2 = 5 \times (3^2 + 40^2) = 8045$

Find Correct Sum and Correct Mean

Adjust the total sum by subtracting the incorrect observation and adding the correct one to get the new mean.

$S_{\text{corr}} = 200 - 35 + 45 = 210, \quad \mu_{\text{corr}} = \frac{210}{5} = 42$

Find Correct Sum of Squares

Update the sum of squares by replacing the square of the incorrect value with the square of the correct value.

$\sum x_{i,\text{corr}}^2 = 8045 - 35^2 + 45^2 = 8845$

Calculate Correct Standard Deviation

Apply the variance formula with the corrected sum of squares and mean, then take the square root.

$\sigma = \sqrt{\frac{8845}{5} - 42^2} = \sqrt{1769 - 1764} = \sqrt{5} \approx 2.24$