1. Key Concepts and Formulas

To solve this problem, we need to understand and apply the fundamental formulas for the mean and standard deviation of a dataset.

• Mean ($\bar{x}$): The average of all observations. For a set of $N$ observations $x_1, x_2, \ldots, x_N$: $\bar{x} = \frac{\sum_{i=1}^{N} x_i}{N}$
• Standard Deviation ($\sigma$): A measure of the typical spread or dispersion of data points around the mean. It is the square root of the variance. $\sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N}}$ Here, $(x_i - \bar{x})$ represents the deviation of each observation from the mean, and $(x_i - \bar{x})^2$ is the squared deviation.

2. Step-by-Step Solution

Step 1: Understand the Given Observations and Parameters

We are given a series of $2n$ observations.

• Exactly half of these observations, which is $n$ observations, are equal to $a$.
• The remaining half, also $n$ observations, are equal to $-a$.

So, our dataset can be represented as: $\underbrace{a, a, \ldots, a}_{n \text{ times}}, \underbrace{-a, -a, \ldots, -a}_{n \text{ times}}$

• The total number of observations is $N = 2n$.
• The standard deviation of these observations is given as $\sigma = 2$.
• Our goal is to find the value of $|a|$.

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

We first need to calculate the mean because the standard deviation formula measures deviations from the mean.

• Why this step? The standard deviation formula requires the mean ($\bar{x}$) to calculate the deviations $(x_i - \bar{x})$.
• Calculation: The sum of all observations, $\sum x_i$, is the sum of $n$ values of $a$ and $n$ values of $-a$: $\sum x_i = (a \times n) + (-a \times n)$ $\sum x_i = na - na$ $\sum x_i = 0$ Now, substitute this sum and the total number of observations $N=2n$ into the mean formula: $\bar{x} = \frac{\sum x_i}{N} = \frac{0}{2n}$ $\bar{x} = 0$ The mean of the observations is 0. This is expected, as the dataset is perfectly symmetric around zero.

Step 3: Calculate the Sum of Squared Deviations ($\sum (x_i - \bar{x})^2$)

Next, we calculate the sum of the squared deviations of each observation from the mean. Since $\bar{x} = 0$, this simplifies to calculating $\sum x_i^2$.

• Why this step? This sum forms the numerator of the variance (and thus standard deviation) formula. We need to calculate it for all $2n$ observations.

• Calculation: For the $n$ observations that are equal to $a$: Each squared deviation is $(a - \bar{x})^2 = (a - 0)^2 = a^2$. The sum of squared deviations for these $n$ observations is $n \times a^2 = na^2$.

  For the $n$ observations that are equal to $-a$: Each squared deviation is $(-a - \bar{x})^2 = (-a - 0)^2 = (-a)^2 = a^2$. The sum of squared deviations for these $n$ observations is $n \times a^2 = na^2$.

  The total sum of squared deviations for all $2n$ observations is: $\sum (x_i - \bar{x})^2 = na^2 + na^2$ $\sum (x_i - \bar{x})^2 = 2na^2$ Notice that squaring the deviations ensures that both positive and negative deviations contribute positively to the measure of spread.

Step 4: Apply the Standard Deviation Formula and Solve for $|a|$

Now we have all the components to use the standard deviation formula and solve for $|a|$.

• Why this step? We use the given standard deviation ($\sigma=2$) and our calculated values for $\bar{x}$ and $\sum (x_i - \bar{x})^2$ to form an equation and solve for the unknown $|a|$.
• Substitution and Solution: Recall the standard deviation formula: $\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N}}$ Substitute the known values: $\sigma = 2$, $\sum (x_i - \bar{x})^2 = 2na^2$, and $N = 2n$: $2 = \sqrt{\frac{2na^2}{2n}}$ Simplify the expression under the square root: $2 = \sqrt{a^2}$ To solve for $a$, we must remember that the square root of $a^2$ is always the absolute value of $a$, denoted as $|a|$, because the square root symbol represents the principal (non-negative) square root. $|a| = 2$

Thus, the value of $|a|$ is 2.

3. Common Mistakes & Tips

• Incorrectly Calculating the Mean: Always calculate the mean explicitly. Forgetting that negative values exist or incorrectly summing can lead to errors.
• Error in Squaring Deviations: A common mistake is not correctly handling $(-a)^2$. Remember that $(-a)^2 = a^2$, which means both $a$ and $-a$ contribute the same amount to the sum of squared deviations.
• Misinterpreting $\sqrt{a^2}$: The most critical mistake is writing $\sqrt{a^2} = a$. This is only true if $a \ge 0$. The correct mathematical simplification is $\sqrt{a^2} = |a|$. This ensures the result is non-negative, consistent with the definition of standard deviation.
• Algebraic Errors: Be careful with cancellations and simplifications, especially when dealing with variables like $n$.

4. Summary

This problem required a direct application of the formulas for mean and standard deviation. We began by calculating the mean, which simplified to 0 due to the symmetric nature of the observations. Next, we calculated the sum of the squared deviations from this mean, finding it to be $2na^2$. Finally, we substituted these values into the standard deviation formula, set it equal to the given standard deviation of 2, and solved for $|a|$, correctly recognizing that $\sqrt{a^2} = |a|$. The symmetric distribution of the data around zero significantly simplified the calculations.

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