1. **State the problem:** Find the range, mean, median, and mode of the fractions: $3 \frac{1}{3}$, $4 \frac{1}{2}$, $5 \frac{1}{6}$, $3 \frac{2}{3}$, $4 \frac{5}{6}$, $3 \frac{1}{2}$, $3 \frac{2}{3}$.\n\n2. **Convert mixed numbers to improper fractions or decimals for easier calculation:**\n$3 \frac{1}{3} = 3 + \frac{1}{3} = \frac{10}{3} \approx 3.333$\n$4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{9}{2} = 4.5$\n$5 \frac{1}{6} = 5 + \frac{1}{6} = \frac{31}{6} \approx 5.167$\n$3 \frac{2}{3} = 3 + \frac{2}{3} = \frac{11}{3} \approx 3.667$\n$4 \frac{5}{6} = 4 + \frac{5}{6} = \frac{29}{6} \approx 4.833$\n$3 \frac{1}{2} = 3 + \frac{1}{2} = \frac{7}{2} = 3.5$\n$3 \frac{2}{3} = 3.667$ (repeated)\n\n3. **Arrange the numbers in ascending order:**\n$3.333, 3.5, 3.667, 3.667, 4.5, 4.833, 5.167$\n\n4. **Find the range:**\nRange = Maximum - Minimum = $5.167 - 3.333 = 1.834$\n\n5. **Find the mean (average):**\nMean = $\frac{3.333 + 4.5 + 5.167 + 3.667 + 4.833 + 3.5 + 3.667}{7}$\nCalculate numerator stepwise:\n$3.333 + 4.5 = 7.833$\n$7.833 + 5.167 = 13$\n$13 + 3.667 = 16.667$\n$16.667 + 4.833 = 21.5$\n$21.5 + 3.5 = 25$\n$25 + 3.667 = 28.667$\nMean = $\frac{28.667}{7} = 4.095$\n\n6. **Find the median:**\nSince there are 7 numbers (odd count), median is the middle number in the ordered list.\nThe 4th number is $3.667$\nMedian = $3.667$\n\n7. **Find the mode:**\nMode is the number that appears most frequently.\n$3.667$ appears twice, others appear once.\nMode = $3 \frac{2}{3}$ or $3.667$\n\n**Final answers:**\nRange = $1.834$\nMean = $4.095$\nMedian = $3.667$\nMode = $3.667$