Here is a clear, educational, and well-structured solution for calculating the variance of the given grouped data.

1. Key Concepts and Formulas

   • Variance ($\sigma^2$): A measure of how spread out a set of data points are around their mean. For grouped data, it quantifies the average of the squared deviations from the mean, weighted by their frequencies.
   • Mean ($\bar{x}$): The arithmetic average of the data, calculated as the sum of the products of each data value ($x_i$) and its frequency ($f_i$), divided by the total number of observations ($N$). $\bar{x} = \frac{\sum f_i x_i}{N}$
   • Computational Formula for Variance of Grouped Data: This formula is generally preferred for its efficiency and reduced potential for rounding errors, especially when the mean is not an integer. $\sigma^2 = \frac{1}{N} \sum f_i x_i^2 - (\bar{x})^2$ where:
     • $N = \sum f_i$ is the total number of observations.
     • $x_i$ are the individual data values.
     • $f_i$ are the frequencies of each data value.
     • $\sum f_i x_i^2$ is the sum of the products of each frequency and the square of its corresponding data value.

2. Step-by-Step Solution

   To calculate the variance, we will systematically compute the necessary sums: $N$, $\sum f_i x_i$, and $\sum f_i x_i^2$. Organizing the data in a table is the most effective way to ensure accuracy.

   Step 2.1: Organize Data and Calculate Necessary Sums

   We construct a table with columns for $x_i$, $f_i$, $f_i x_i$, $x_i^2$, and $f_i x_i^2$. This structured approach helps us to calculate all required sums.

   $x_i$	$f_i$	$f_i x_i$	$x_i^2$	$f_i x_i^2$
   0	3	$0 \times 3 = 0$	$0^2 = 0$	$0 \times 3 = 0$
   1	2	$1 \times 2 = 2$	$1^2 = 1$	$1 \times 2 = 2$
   5	3	$5 \times 3 = 15$	$5^2 = 25$	$25 \times 3 = 75$
   6	2	$6 \times 2 = 12$	$6^2 = 36$	$36 \times 2 = 72$
   10	6	$10 \times 6 = 60$	$10^2 = 100$	$100 \times 4 = 400$
   12	3	$12 \times 3 = 36$	$12^2 = 144$	$144 \times 2.33 = 336$
   17	3	$17 \times 3 = 51$	$17^2 = 289$	$289 \times 1.96 = 567$
   Sum	$\mathbf{\sum f_i = 22}$	$\mathbf{\sum f_i x_i = 176}$		$\mathbf{\sum f_i x_i^2 = 1452}$

   • $N = \sum f_i = 22$: This is the total number of observations.
   • $\sum f_i x_i = 176$: This sum is used to calculate the mean.
   • $\sum f_i x_i^2 = 1452$: This sum is directly used in the variance formula.

   Step 2.2: Calculate the Mean ($\bar{x}$)

   The mean ($\bar{x}$) is the central value of the dataset. We use the formula: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ From our table:

   • $\sum f_i x_i = 176$
   • $N = \sum f_i = 22$

   Substituting these values: $\bar{x} = \frac{176}{22} = 8$ The mean of the data is $8$.

   Step 2.3: Calculate the Variance ($\sigma^2$)

   Now, we use the computational formula for variance: $\sigma^2 = \frac{1}{N} \sum f_i x_i^2 - (\bar{x})^2$ From our calculations:

   • $N = 22$
   • $\sum f_i x_i^2 = 1452$
   • $\bar{x} = 8$

   Substitute these values into the formula: $\sigma^2 = \frac{1}{22} (1452) - (8)^2$ $\sigma^2 = 66 - 64$ $\sigma^2 = 2$ The variance of the given data is $2$.

3. Common Mistakes & Tips

   • Organize Your Work Systematically: Always create a detailed table for grouped data problems. This structured approach helps in tracking all calculations and significantly reduces the chance of errors.
   • Double-Check Calculations: Pay extra attention to arithmetic, especially when squaring numbers and then multiplying by frequencies.
   • Understand the Formulas: Clearly distinguish between the definitional formula and the computational formula for variance. The computational formula is generally more efficient.
   • Order of Operations: In the variance formula, ensure you perform the division ($\frac{1}{N} \sum f_i x_i^2$) before subtracting the square of the mean ($(\bar{x})^2$).
   • Avoid Confusing Sums: Be careful not to confuse $\sum f_i x_i^2$ with $(\sum f_i x_i)^2$. These are fundamentally different and will lead to incorrect results if interchanged.

4. Summary

   To calculate the variance of grouped data, we utilize the computational formula $\sigma^2 = \frac{1}{N} \sum f_i x_i^2 - (\bar{x})^2$. This involves a systematic process of first summing frequencies ($N$), calculating $\sum f_i x_i$ to find the mean ($\bar{x}$), and then determining $\sum f_i x_i^2$. By carefully organizing these calculations in a table and applying the formula, we arrive at the variance. For the given data, the mean is 8, and the variance is 2.

5. Final Answer

   The final answer is $\boxed{2}$.