Here is a clear, educational, and well-structured solution to the problem:

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1. Key Concepts and Formulas

To determine the standard deviation of a dataset, we primarily rely on the following statistical measures and formulas:

• Mean ($\bar{x}$): The average of all observations in a dataset. For a dataset $x_1, x_2, \ldots, x_n$, the mean is calculated as: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
• Variance ($\sigma^2$): A measure of how much the data points deviate from the mean. While there's a definitional formula involving squared differences from the mean, for computational efficiency, especially when the sum of squares of observations ($\sum x_i^2$) is given, we use the following formula: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$
• Standard Deviation ($\sigma$): The positive square root of the variance. It's preferred for interpretation as it is in the same units as the original data. $\sigma = \sqrt{\sigma^2}$

Our goal is to find the standard deviation ($\sigma$) of the entire dataset. This requires us to first calculate the overall mean ($\bar{x}$) and then use the given sum of squares ($\sum x_i^2$) to find the variance ($\sigma^2$), before taking its square root.

2. Step-by-Step Solution

We are given a dataset of 10 observations: $x_1, x_2, \ldots, x_{10}$. Therefore, the total number of observations, $n = 10$.

The problem provides the following information:

• Mean of the first four observations ($n_1=4$) is $\bar{x}_1 = 11$.
• Mean of the remaining six observations ($n_2=6$) is $\bar{x}_2 = 16$.
• Sum of squares of all 10 observations is $\sum_{i=1}^{10} x_i^2 = 2000$.

Let's proceed with the calculations:

Step 1: Calculate the Total Sum of All Observations ($\sum_{i=1}^{10} x_i$)

• Why this step? To find the overall mean of the entire dataset, we first need the sum of all its observations. The problem provides means for two subsets, which allows us to find the sum of observations for each subset and then combine them. Recall that $\text{Sum of observations} = \text{Mean} \times \text{Number of observations}$.

  • For the first four observations ($x_1, x_2, x_3, x_4$): Number of observations = 4 Mean ($\bar{x}_1$) = 11 Sum of these observations ($S_1$) = $11 \times 4 = 44$.

  • For the remaining six observations ($x_5, x_6, \ldots, x_{10}$): Number of observations = 6 Mean ($\bar{x}_2$) = 16 Sum of these observations ($S_2$) = $16 \times 6 = 96$.

• Total Sum of all 10 observations: The total sum is the sum of the sums of the two subsets: $\sum_{i=1}^{10} x_i = S_1 + S_2 = 44 + 96 = 140$

Step 2: Calculate the Mean of the Entire Dataset ($\bar{x}$)

• Why this step? The overall mean ($\bar{x}$) is a crucial component for calculating the variance using the computational formula: $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$. We now have the total sum of observations and the total number of observations.

• The mean of the entire dataset of 10 observations is: $\bar{x} = \frac{\sum_{i=1}^{10} x_i}{n} = \frac{140}{10} = 14$

Step 3: Identify the Sum of Squares of All Observations ($\sum_{i=1}^{10} x_i^2$)

• Why this step? This value is directly provided in the problem statement and is the other essential component for the variance formula.

• From the problem statement: $\sum_{i=1}^{10} x_i^2 = 2000$

Step 4: Calculate the Variance ($\sigma^2$)

• Why this step? Now that we have all the necessary components ($\sum x_i^2 = 2000$, $\bar{x} = 14$, and $n = 10$), we can directly apply the computational formula for variance.

• Using the formula: $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ Substitute the values we found: $\sigma^2 = \frac{2000}{10} - (14)^2$ Perform the calculations: $\sigma^2 = 200 - 196$ $\sigma^2 = 4$

Step 5: Calculate the Standard Deviation ($\sigma$)

• Why this step? The standard deviation is the positive square root of the variance. This is the final quantity the question asks for.

• Taking the positive square root of the variance: $\sigma = \sqrt{\sigma^2}$ $\sigma = \sqrt{4}$ $\sigma = 2$

3. Common Mistakes & Tips

• Computational Formula is Your Friend: Always remember the computational formula for variance ($\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$) as it often simplifies calculations significantly, especially in competitive exams.
• Don't Confuse Variance and Standard Deviation: A frequent error is to calculate the variance and present it as the final answer. Remember to take the square root to get the standard deviation.
• Accuracy in Arithmetic: Even with the correct formulas, small calculation errors can lead to incorrect answers. Double-check your arithmetic, especially when dealing with sums and squares.
• Overall Mean vs. Subset Means: When dealing with combined data, ensure you calculate the overall mean for the entire dataset, not just average the means of the subsets.

4. Summary

In this problem, we systematically determined the standard deviation of a dataset of 10 observations. We began by using the given means of two subsets (first four and remaining six observations) to calculate the total sum of all observations. This total sum, combined with the total number of observations, allowed us to find the overall mean of the entire dataset. Finally, using the given sum of squares of all observations and the calculated overall mean, we applied the efficient computational formula for variance. Taking the square root of the variance yielded the standard deviation.

The final answer is $\boxed{2}$, which corresponds to option (B).