1. Key Concepts and Formulas

To solve this problem efficiently, a solid understanding of the definitions of mean and standard deviation, along with their properties under linear data transformations, is essential.

• Mean ($\overline{x}$): Measure of Central Tendency For a set of $N$ observations $x_1, x_2, \dots, x_N$, the mean is given by: $\overline{x} = \frac{\sum_{i=1}^{N} x_i}{N}$

• Standard Deviation ($\sigma$): Measure of Data Spread The standard deviation quantifies the dispersion of data points around the mean. A commonly used computational formula is: $\sigma = \sqrt{\frac{\sum_{i=1}^{N} x_i^2}{N} - (\overline{x})^2}$ Note that $\sigma$ is always non-negative.

• Properties of Data Transformation (Change of Origin): If a constant $b$ is added to each observation $x_i$ to form a new set $x'_i = x_i + b$:

  • New Mean: The new mean $\overline{x}'$ is the old mean plus the constant: $\overline{x}' = \overline{x} + b$
  • New Standard Deviation: The new standard deviation $\sigma'$ remains unchanged: $\sigma' = \sigma$

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2. Step-by-Step Solution

Let's apply these fundamental principles to systematically solve the problem.

Step 2.1: Analyze the Original Series of Observations

We are given $2n$ observations. Specifically, $n$ of these observations are equal to $a$, and the remaining $n$ observations are equal to $-a$. The total number of observations, $N = 2n$.

• Calculate the Mean of the Original Series ($\overline{x}_{\text{old}}$): We use the formula $\overline{x} = \frac{\sum x_i}{N}$. First, let's find the sum of all observations: $\sum_{i=1}^{2n} x_i = \underbrace{(a + a + \dots + a)}_{n \text{ times}} + \underbrace{((-a) + (-a) + \dots + (-a))}_{n \text{ times}}$ $\sum_{i=1}^{2n} x_i = (n \cdot a) + (n \cdot (-a)) = na - na = 0$ Now, calculate the mean: $\overline{x}_{\text{old}} = \frac{0}{2n} = 0$ Why: The observations are perfectly symmetric around zero. For every positive value $a$, there is a corresponding negative value $-a$, which cancels out in the summation, resulting in a mean of zero.

• Calculate the Standard Deviation of the Original Series ($\sigma_{\text{old}}$): We use the computational formula $\sigma = \sqrt{\frac{\sum x_i^2}{N} - (\overline{x})^2}$. Since we found $\overline{x}_{\text{old}} = 0$, the formula simplifies to $\sigma_{\text{old}} = \sqrt{\frac{\sum x_i^2}{N}}$.

  First, let's find the sum of the squares of the observations, $\sum x_i^2$: $\sum_{i=1}^{2n} x_i^2 = \underbrace{(a^2 + a^2 + \dots + a^2)}_{n \text{ times}} + \underbrace{((-a)^2 + (-a)^2 + \dots + (-a)^2)}_{n \text{ times}}$ Since $(-a)^2 = a^2$, this simplifies to: $\sum_{i=1}^{2n} x_i^2 = (n \cdot a^2) + (n \cdot a^2) = 2n a^2$ Now, substitute this into the standard deviation formula: $\sigma_{\text{old}} = \sqrt{\frac{2n a^2}{2n}}$ $\sigma_{\text{old}} = \sqrt{a^2}$ $\sigma_{\text{old}} = |a|$ Why: The standard deviation must always be a non-negative value, as it measures spread. The square root of $a^2$ is always the absolute value of $a$, i.e., $|a|$.

Step 2.2: Apply Transformations and Use Given Information

A constant $b$ is added to each of the original observations. Let the new observations be $x'_i = x_i + b$. We are given that the mean of this new set ($\overline{x}_{\text{new}}$) is 5, and the standard deviation of this new set ($\sigma_{\text{new}}$) is 20.

• Determine the value of $b$ using the new mean: We use the property for the transformation of the mean: $\overline{x}_{\text{new}} = \overline{x}_{\text{old}} + b$. We are given $\overline{x}_{\text{new}} = 5$ and we calculated $\overline{x}_{\text{old}} = 0$. Substituting these values: $5 = 0 + b$ $b = 5$ Why: Adding a constant $b$ to every observation shifts the entire dataset, and consequently its mean, by exactly $b$. Since the original mean was 0, the new mean of 5 directly tells us the value of $b$.

• Determine the value of $a$ using the new standard deviation: We use the property for the transformation of the standard deviation: $\sigma_{\text{new}} = \sigma_{\text{old}}$. We are given $\sigma_{\text{new}} = 20$ and we calculated $\sigma_{\text{old}} = |a|$. Substituting these values: $20 = |a|$ This implies that $a$ can be either $20$ or $-20$. Why: Adding a constant to each data point represents a "change of origin." This operation shifts the entire distribution without changing the distances between data points or their spread around the mean. Therefore, the standard deviation remains unaffected.

Step 2.3: Calculate the Required Value

The problem asks for the value of $a^2 + b^2$. We have found $b=5$ and $a=\pm 20$.

Let's calculate $a^2$ and $b^2$: $a^2 = (\pm 20)^2 = 400$ $b^2 = (5)^2 = 25$ Now, sum these values: $a^2 + b^2 = 400 + 25 = 425$

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3. Common Mistakes & Tips

• Absolute Value: Always remember that $\sqrt{x^2} = |x|$, not just $x$. This is crucial for standard deviation, which must be non-negative.
• Transformation Rules: Be precise about how mean and standard deviation are affected by transformations. Adding a constant ($x_i \to x_i + b$) changes the mean but not the standard deviation. Multiplying by a constant ($x_i \to c x_i$) changes both the mean (by $c$) and the standard deviation (by $|c|$).
• Symmetry in Data: For symmetric data sets (like $a, -a$), quickly identifying a mean of zero can save time and simplify calculations.

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4. Summary

This problem effectively tests the understanding of fundamental statistical measures and, more importantly, the properties of mean and standard deviation under a change of origin (adding a constant). By first calculating the mean and standard deviation of the original dataset, and then applying the transformation properties, we efficiently determined the values of $a$ and $b$. The original mean was 0 and standard deviation was $|a|$. After adding $b$, the new mean became $0+b=5$, giving $b=5$. The new standard deviation remained $|a|=20$. Finally, calculating $a^2+b^2$ yielded 425.

The final answer is $\boxed{425}$, which corresponds to option (A).