1. Key Concepts and Formulas

To accurately determine the standard deviation of a data set, we rely on the following fundamental statistical definitions and computational formulas:

• Standard Deviation ($\sigma$): A measure of the average amount of variability or dispersion of data points around the mean. The most common computational formula for standard deviation is: $\sigma = \sqrt{\frac{\sum_{i=1}^n x_i^2}{n} - \left(\bar{x}\right)^2}$ where $n$ is the number of observations, $\sum x_i^2$ is the sum of the squares of the observations, and $\bar{x}$ is the mean.

• Arithmetic Mean ($\bar{x}$): The average of all observations in the data set. $\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$ where $\sum x_i$ is the sum of all observations.

• Properties of Summation: These properties are crucial for expanding and simplifying the given equations:

  • $\sum_{i=1}^n (A_i \pm B_i) = \sum_{i=1}^n A_i \pm \sum_{i=1}^n B_i$ (Distributivity over addition/subtraction)
  • $\sum_{i=1}^n c = nc$ (Sum of a constant $c$ for $n$ terms)
  • $\sum_{i=1}^n c \cdot A_i = c \cdot \sum_{i=1}^n A_i$ (Constant multiple rule)

2. Step-by-Step Solution

Our goal is to find $\sum x_i$ (to get $\bar{x}$) and $\sum x_i^2$ from the given equations. We will expand the summations and form a system of linear equations.

Step 1: Expand and Simplify the First Given Equation We are given the equation: \sum_{i=1}^n {{\left( {{x_i} + 1} \right)}^2}} = 8n

• Why this step? We need to transform the given summation into a form that explicitly involves $\sum x_i^2$ and $\sum x_i$, which are the components needed for the standard deviation formula.

• Action: First, expand the squared term $(x_i + 1)^2$ using the identity $(a+b)^2 = a^2 + 2ab + b^2$: $(x_i + 1)^2 = x_i^2 + 2x_i + 1$ Substitute this expansion back into the summation: $\sum_{i=1}^n {\left( {x_i^2 + 2x_i + 1} \right)} = 8n$

• Action: Apply the summation properties: distribute the summation, pull out the constant factor, and sum the constant term. $\sum_{i=1}^n {x_i^2} + \sum_{i=1}^n {2x_i} + \sum_{i=1}^n {1} = 8n$ $\sum_{i=1}^n {x_i^2} + 2\sum_{i=1}^n {x_i} + n = 8n$

• Action: Isolate the terms involving $x_i$ by subtracting $n$ from both sides: $\sum_{i=1}^n {x_i^2} + 2\sum_{i=1}^n {x_i} = 8n - n$ $\sum_{i=1}^n {x_i^2} + 2\sum_{i=1}^n {x_i} = 7n \quad \dots(1)$ This is our first simplified equation.

Step 2: Expand and Simplify the Second Given Equation We are given the second equation: \sum_{i=1}^n {{\left( {{x_i} - 1} \right)}^2}} = 4n

• Why this step? Similar to Step 1, we need to expand this summation to get another equation in terms of $\sum x_i^2$ and $\sum x_i$, forming a system of equations.

• Action: Expand the squared term $(x_i - 1)^2$ using the identity $(a-b)^2 = a^2 - 2ab + b^2$: $(x_i - 1)^2 = x_i^2 - 2x_i + 1$ Substitute this expansion into the summation: $\sum_{i=1}^n {\left( {x_i^2 - 2x_i + 1} \right)} = 4n$

• Action: Apply the summation properties: $\sum_{i=1}^n {x_i^2} - \sum_{i=1}^n {2x_i} + \sum_{i=1}^n {1} = 4n$ $\sum_{i=1}^n {x_i^2} - 2\sum_{i=1}^n {x_i} + n = 4n$

• Action: Isolate the terms involving $x_i$: $\sum_{i=1}^n {x_i^2} - 2\sum_{i=1}^n {x_i} = 4n - n$ $\sum_{i=1}^n {x_i^2} - 2\sum_{i=1}^n {x_i} = 3n \quad \dots(2)$ This is our second simplified equation.

Step 3: Solve the System of Equations for $\sum x_i^2$ and $\sum x_i$ We now have a system of two linear equations: (1) $\sum x_i^2 + 2\sum x_i = 7n$ (2) $\sum x_i^2 - 2\sum x_i = 3n$

• Why this step? We need the exact values of $\sum x_i^2$ and $\sum x_i$ to calculate the mean and standard deviation. Solving this system will provide those values.

• Action (Finding $\sum x_i^2$): Add Equation (1) and Equation (2) to eliminate the $\sum x_i$ term. $(\sum x_i^2 + 2\sum x_i) + (\sum x_i^2 - 2\sum x_i) = 7n + 3n$ $2\sum x_i^2 = 10n$ Divide by 2: $\sum x_i^2 = 5n$

• Action (Finding $\sum x_i$): Subtract Equation (2) from Equation (1) to eliminate the $\sum x_i^2$ term. $(\sum x_i^2 + 2\sum x_i) - (\sum x_i^2 - 2\sum x_i) = 7n - 3n$ $4\sum x_i = 4n$ Divide by 4: $\sum x_i = n$

Step 4: Calculate the Mean ($\bar{x}$) Now that we have $\sum x_i$, we can calculate the mean.

• Why this step? The mean is a critical component required for the standard deviation formula.

• Action: Use the formula $\bar{x} = \frac{\sum x_i}{n}$ and substitute $\sum x_i = n$. $\bar{x} = \frac{n}{n}$ $\bar{x} = 1$

Step 5: Calculate the Standard Deviation ($\sigma$) We have all the necessary components:

• $\sum x_i^2 = 5n$

• $\bar{x} = 1$

• $n$ (number of observations)

• Why this step? This is the final calculation to answer the question.

• Action: Substitute these values into the standard deviation formula: $\sigma = \sqrt{\frac{\sum x_i^2}{n} - \left(\bar{x}\right)^2}$ $\sigma = \sqrt{\frac{5n}{n} - (1)^2}$ $\sigma = \sqrt{5 - 1}$ $\sigma = \sqrt{4}$ $\sigma = 2$

3. Common Mistakes & Tips

• Algebraic Expansion: Be meticulous when expanding terms like $(x_i \pm c)^2$. A common error is omitting the middle term ($2cx_i$) or getting its sign incorrect.
• Summation Properties: Remember that $\sum_{i=1}^n c = nc$, not just $c$. Also, ensure constant factors are correctly pulled outside the summation.
• System of Equations: Treat $\sum x_i^2$ and $\sum x_i$ as distinct variables and solve the system carefully. Double-check arithmetic, especially during subtraction.
• Variance vs. Standard Deviation: Remember to take the square root at the end. The quantity $\frac{\sum x_i^2}{n} - (\bar{x})^2$ represents the variance ($\sigma^2$), not the standard deviation itself.

4. Summary

This problem effectively tests your ability to apply algebraic manipulation alongside core statistical definitions. The solution involved:

1. Expanding the given summations using algebraic identities and summation properties.
2. Forming and solving a system of two linear equations to find $\sum x_i^2$ and $\sum x_i$.
3. Calculating the mean ($\bar{x}$) from $\sum x_i$.
4. Finally, substituting these derived values into the standard deviation formula to obtain the answer. The careful handling of summations and simultaneous equations is key to solving such problems efficiently.

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