Key Concepts and Formulas

This problem requires the application of fundamental statistical measures: the mean and the variance, along with essential algebraic identities.

1. Mean ($\bar{x}$): The arithmetic average of a data set. For $n$ observations $x_1, x_2, \ldots, x_n$, the mean is defined as: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ where $\sum x_i$ is the sum of all data points.

2. Variance ($\sigma^2$): A measure of the spread or dispersion of data points around the mean. The most convenient computational formula for variance, especially when the mean is known, is: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$ where $\sum x_i^2$ is the sum of the squares of each data point.

3. Algebraic Identities: The following identities are crucial for relating sums and squares of variables:

   • $(a+b)^2 = a^2 + b^2 + 2ab$
   • $(a-b)^2 = a^2 + b^2 - 2ab$

Step-by-Step Solution

We are given the data set: $6, 10, 7, 13, a, 12, b, 12$. The number of observations is $n=8$. The mean, $\bar{x} = 9$. The variance, $\sigma^2 = \frac{37}{4}$. Our objective is to find the value of $(a-b)^2$.

Step 1: Use the Mean to Find a Relationship between $a$ and $b$

• What we are doing: We will use the given mean and the definition of the mean to form our first equation involving $a$ and $b$.
• Why this step? The mean formula directly uses the sum of all data points. By substituting the known mean and summing the numerical values, we can isolate the term $(a+b)$, providing a crucial initial relationship.

The formula for the mean is $\bar{x} = \frac{\sum x_i}{n}$. Substitute the given values: $9 = \frac{6 + 10 + 7 + 13 + a + 12 + b + 12}{8}$

First, sum the known numerical values: $6 + 10 + 7 + 13 + 12 + 12 = 60$.

Now, substitute this sum back into the equation: $9 = \frac{60 + a + b}{8}$

Multiply both sides by 8 to clear the denominator: $9 \times 8 = 60 + a + b$ $72 = 60 + a + b$

Subtract 60 from both sides to find $(a+b)$: $a + b = 72 - 60$ $\boldsymbol{a + b = 12 \quad \ldots \text{(Equation 1)}}$

Step 2: Calculate the Sum of Squares of the Data Points ($\sum x_i^2$)

• What we are doing: We need to compute the sum of the squares of all data points, including $a^2$ and $b^2$, as this term is required for the variance formula.
• Why this step? The computational variance formula ($\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$) directly uses $\sum x_i^2$. Preparing this term now simplifies the subsequent variance calculation.

Square each known data point and sum them:

• $6^2 = 36$
• $10^2 = 100$
• $7^2 = 49$
• $13^2 = 169$
• $12^2 = 144$
• $12^2 = 144$

Sum of squares of known terms $= 36 + 100 + 49 + 169 + 144 + 144 = 642$.

The total sum of squares, including the unknown terms, is: $\sum x_i^2 = a^2 + b^2 + 642$

Step 3: Use the Variance to Find a Relationship between $a^2$ and $b^2$

• What we are doing: We will use the given variance, the mean, and the sum of squares calculated in Step 2 to form a second equation, this time for $a^2+b^2$.
• Why this step? We have already used the mean. The variance provides an independent piece of information that, when combined with the sum of squares, allows us to determine the value of $a^2+b^2$. This is a necessary component for finding $(a-b)^2$.

The computational formula for variance is $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$. Substitute the given values for $\sigma^2$, $n$, $\bar{x}$, and our expression for $\sum x_i^2$: $\frac{37}{4} = \frac{a^2 + b^2 + 642}{8} - (9)^2$

Calculate $(9)^2$: $(9)^2 = 81$

Substitute this back into the equation: $\frac{37}{4} = \frac{a^2 + b^2 + 642}{8} - 81$

Add 81 to both sides of the equation: $\frac{37}{4} + 81 = \frac{a^2 + b^2 + 642}{8}$

To combine the terms on the left side, express 81 with a denominator of 4: $81 = \frac{81 \times 4}{4} = \frac{324}{4}$. $\frac{37}{4} + \frac{324}{4} = \frac{a^2 + b^2 + 642}{8}$ $\frac{37 + 324}{4} = \frac{a^2 + b^2 + 642}{8}$ $\frac{361}{4} = \frac{a^2 + b^2 + 642}{8}$

Multiply both sides by 8 to simplify: $\frac{361}{4} \times 8 = a^2 + b^2 + 642$ $361 \times 2 = a^2 + b^2 + 642$ $722 = a^2 + b^2 + 642$

Subtract 642 from both sides to isolate $(a^2+b^2)$: $a^2 + b^2 = 722 - 642$ $\boldsymbol{a^2 + b^2 = 80 \quad \ldots \text{(Equation 2)}}$

Step 4: Find the Value of $2ab$

• What we are doing: We will use the algebraic identity $(a+b)^2 = a^2 + b^2 + 2ab$ and substitute the values from Equation 1 and Equation 2 to find $2ab$.
• Why this step? To calculate $(a-b)^2$, we need $a^2+b^2$ (which we have from Equation 2) and $2ab$. This identity provides a direct way to find $2ab$ using the sum of the variables and the sum of their squares.

Recall the identity: $(a+b)^2 = a^2 + b^2 + 2ab$

Substitute $a+b=12$ (from Equation 1) and $a^2+b^2=80$ (from Equation 2): $(12)^2 = 80 + 2ab$ $144 = 80 + 2ab$

Subtract 80 from both sides to find $2ab$: $2ab = 144 - 80$ $\boldsymbol{2ab = 64}$

Step 5: Calculate $(a-b)^2$

• What we are doing: We will use the algebraic identity $(a-b)^2 = a^2 + b^2 - 2ab$ and substitute the values we found for $a^2+b^2$ and $2ab$.
• Why this step? This is the final step that directly answers the question by combining all the relationships derived from the given statistical information.

Recall the identity: $(a-b)^2 = a^2 + b^2 - 2ab$

Substitute $a^2+b^2=80$ (from Equation 2) and $2ab=64$ (from Step 4): $(a-b)^2 = 80 - 64$ $\boldsymbol{(a-b)^2 = 16}$

Common Mistakes & Tips

1. Incorrect Variance Formula: A common mistake is using the definitional variance formula $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$ without simplifying it. While correct, the computational formula $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ is almost always more efficient, especially when $\bar{x}$ is known.
2. Arithmetic Errors: Be meticulous with sums, squares, and fraction manipulations. A small calculation error can propagate through the problem.
3. Forgetting Algebraic Identities: Problems like this often bridge statistics with algebra. Ensure you are comfortable with identities involving $(a+b)^2$, $(a-b)^2$, and their relationship.

Summary

This problem effectively tests the understanding of mean and variance definitions and their computational applications, combined with fundamental algebraic identities. By systematically using the given mean to establish a linear relationship between $a$ and $b$, and then the variance to establish a relationship for $a^2+b^2$, we were able to deduce the value of $2ab$. Finally, we used the identity for $(a-b)^2$ to arrive at the solution. This systematic approach is crucial for solving such integrated problems.

The final answer is $\boxed{\text{16}}$, which corresponds to option (D).