Question:

Step 1: To find the correct standard deviation, we first need to understand the impact of the incorrect observation on the mean of the dataset. The original mean was given as 10, and there were 20 observations. The sum of the original observations can be calculated using the formula for the mean: $\text{Mean} = \frac{\sum x_i}{n}$, where $n$ is the number of observations. Thus, the original sum of the observations is $10 \times 20 = 200$. However, one observation was incorrectly recorded as 8 instead of 12, so the correct sum of the observations should be $200 - 8 + 12 = 204$. Step 2: Next, we calculate the correct mean using the corrected sum of the observations. The correct mean is $\frac{204}{20} = 10.2$. This step is essential because the mean is used in the calculation of the variance and standard deviation. Step 3: We are given that the correct $\sum x_i^2 = 2160$. This value represents the sum of the squares of the individual observations. To find the variance, we use the formula $\text{Variance} = \frac{\sum x_i^2}{n} - \text{Mean}^2$. Substituting the given values, we get $\text{Variance} = \frac{2160}{20} - (10.2)^2$. Step 4: Now, let's calculate the variance using the values from Step 3. The variance is $\frac{2160}{20} - (10.2)^2 = 108 - 104.04 = 3.96$. This step is crucial because the variance is the average of the squared differences from the Mean. Step 5: Finally, to find the standard deviation, we take the square root of the variance. The standard deviation is $\sqrt{3.96}$. This is the correct standard deviation after correcting the observation mistake. Therefore, the correct standard deviation is $\sqrt{3.96}$, which matches Option 3. The final answer is $\boxed{3}$.