1. Key Concepts and Formulas

• Mean ($\bar{x}$): For a grouped dataset with distinct outcomes $x_i$ and their corresponding frequencies $f_i$, the mean is the average of all observations. $\bar{x} = \frac{\sum_{i=1}^k f_i x_i}{N}$ where $N = \sum_{i=1}^k f_i$ is the total number of observations.
• Variance ($\sigma^2$): A measure of how spread out the data points are from the mean. For grouped data, the variance is calculated as the mean of the squared deviations from the mean. $\sigma^2 = \frac{\sum_{i=1}^k f_i (x_i - \bar{x})^2}{N}$ An alternative computational formula is $\sigma^2 = \frac{\sum_{i=1}^k f_i x_i^2}{N} - (\bar{x})^2$. For this particular problem, the first definition simplifies calculations due to the symmetric nature of the data.

2. Step-by-Step Solution

Given Data:

• Total number of items, $N = 30$.
• The outcomes ($x_i$) and their frequencies ($f_i$):
  • $x_1 = {1 \over 2} - d$, $f_1 = 10$
  • $x_2 = {1 \over 2}$, $f_2 = 10$
  • $x_3 = {1 \over 2} + d$, $f_3 = 10$
• The variance, $\sigma^2 = {4 \over 3}$.
• Our objective is to find the value of $|d|$.

Step 1: Calculate the Mean ($\bar{x}$)

• What we are doing: We are calculating the average value of the given outcomes, which is a necessary component for computing the variance.
• Why we are doing it: The variance formula fundamentally relies on the mean to measure the spread of data points around it.
• Math and Reasoning: Using the formula for the mean of grouped data: $\bar{x} = \frac{f_1 x_1 + f_2 x_2 + f_3 x_3}{f_1 + f_2 + f_3}$ Substitute the given values: $\bar{x} = \frac{10 \left({1 \over 2} - d\right) + 10 \left({1 \over 2}\right) + 10 \left({1 \over 2} + d\right)}{10 + 10 + 10}$ Factor out 10 from the numerator and sum the frequencies in the denominator: $\bar{x} = \frac{10 \left[\left({1 \over 2} - d\right) + \left({1 \over 2}\right) + \left({1 \over 2} + d\right)\right]}{30}$ Inside the brackets, the 'd' terms cancel out ($-d + d = 0$): $\bar{x} = \frac{10 \left({1 \over 2} + {1 \over 2} + {1 \over 2}\right)}{30}$ $\bar{x} = \frac{10 \left({3 \over 2}\right)}{30}$ $\bar{x} = \frac{15}{30}$ $\bar{x} = {1 \over 2}$ The mean of the outcome data is $1/2$. This is expected due to the symmetric distribution of the outcomes around $1/2$.

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

• What we are doing: We are calculating the sum of the squared differences between each outcome and the mean, weighted by their frequencies. This sum forms the numerator of our chosen variance formula.

• Why we are doing it: This term is a direct component of the fundamental variance formula, which we will use to solve for 'd'.

• Math and Reasoning: First, let's find the deviation $(x_i - \bar{x})$ for each outcome:

  • For $x_1 = {1 \over 2} - d$: $x_1 - \bar{x} = \left({1 \over 2} - d\right) - {1 \over 2} = -d$
  • For $x_2 = {1 \over 2}$: $x_2 - \bar{x} = {1 \over 2} - {1 \over 2} = 0$
  • For $x_3 = {1 \over 2} + d$: $x_3 - \bar{x} = \left({1 \over 2} + d\right) - {1 \over 2} = d$

  Next, square each deviation:

  • $(x_1 - \bar{x})^2 = (-d)^2 = d^2$
  • $(x_2 - \bar{x})^2 = (0)^2 = 0$
  • $(x_3 - \bar{x})^2 = (d)^2 = d^2$

  Now, multiply each squared deviation by its corresponding frequency $f_i$ and sum these products: $\sum f_i (x_i - \bar{x})^2 = f_1 (x_1 - \bar{x})^2 + f_2 (x_2 - \bar{x})^2 + f_3 (x_3 - \bar{x})^2$ $\sum f_i (x_i - \bar{x})^2 = 10(d^2) + 10(0) + 10(d^2)$ $\sum f_i (x_i - \bar{x})^2 = 10d^2 + 0 + 10d^2 = 20d^2$

Step 3: Apply the Variance Formula and Solve for $d^2$

• What we are doing: We substitute the calculated sum of squared deviations and the total number of observations, along with the given variance, into the variance formula to form an equation for $d^2$.
• Why we are doing it: This step allows us to use all the given information to solve for the unknown variable 'd'.
• Math and Reasoning: The variance formula is: $\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{N}$ We are given $\sigma^2 = {4 \over 3}$. From previous steps, we found $\sum f_i (x_i - \bar{x})^2 = 20d^2$ and $N=30$. Substitute these values into the formula: ${4 \over 3} = \frac{20d^2}{30}$ Simplify the fraction on the right side by dividing the numerator and denominator by 10: ${4 \over 3} = \frac{2d^2}{3}$ To solve for $d^2$, multiply both sides of the equation by 3: $4 = 2d^2$ Now, divide by 2: $d^2 = {4 \over 2}$ $d^2 = 2$

Step 4: Find $|d|$

• What we are doing: We are determining the absolute value of 'd' from the calculated value of $d^2$.
• Why we are doing it: The question specifically asks for $|d|$, not just 'd' or $d^2$.
• Math and Reasoning: We have found that $d^2 = 2$. To find 'd', we take the square root of both sides: $d = \pm \sqrt{2}$ The absolute value of $d$ is the non-negative value of $d$: $|d| = |\pm \sqrt{2}|$ $|d| = \sqrt{2}$

3. Common Mistakes & Tips

• Choosing the Right Variance Formula: While both variance formulas are equivalent, for problems with symmetric data and a simple mean, using $\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{N}$ can often lead to fewer algebraic complexities and cancellations, reducing the chance of error.
• Algebraic Accuracy: Pay close attention to basic arithmetic, expanding squared terms, and manipulating fractions. Simple calculation errors are frequent pitfalls.
• Leveraging Symmetry: Always look for symmetry in the data. In this problem, the outcomes $(1/2-d, 1/2, 1/2+d)$ are symmetric around $1/2$, which immediately tells you the mean is $1/2$ and simplifies the deviation calculations.
• Understanding the Question: Double-check what the question asks for (e.g., $d$, $d^2$, or $|d|$) to ensure you provide the correct final form of the answer.

4. Summary

This problem is a straightforward application of statistical measures for grouped data. The solution involved systematically calculating the mean, then the sum of the squared deviations from that mean, and finally using the variance formula to set up and solve an equation for $d^2$. The symmetric arrangement of the data points significantly simplified the algebraic steps, particularly in finding the mean and the sum of squared deviations. The final step was to find the absolute value of 'd'.

5. Final Answer

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