A statistics problem involves finding the value of X in a frequency distribution, determining the modal class, and calculating the mean.

The total number of students is 100. The frequencies are 4, 6, 10, 14, X, 24, 14, 6.
Sum of known frequencies $= 4 + 6 + 10 + 14 + 24 + 14 + 6 = 78$.

The sum of all frequencies must equal the total number of students.

The value of X is $\boxed{22}$.

The frequencies are 4, 6, 10, 14, 22 (for X), 24, 14, 6.
The highest frequency is 24.

The class interval corresponding to the frequency 24 is 55-59.
The modal class is $\boxed{55-59}$.

• 30-34: $x = \frac{30+34}{2} = 32$
• 35-39: $x = \frac{35+39}{2} = 37$
• 40-44: $x = \frac{40+44}{2} = 42$
• 45-49: $x = \frac{45+49}{2} = 47$
• 50-54: $x = \frac{50+54}{2} = 52$
• 55-59: $x = \frac{55+59}{2} = 57$
• 60-64: $x = \frac{60+64}{2} = 62$
• 65-69: $x = \frac{65+69}{2} = 67$

• $4 \times 32 = 128$
• $6 \times 37 = 222$
• $10 \times 42 = 420$
• $14 \times 47 = 658$
• $22 \times 52 = 1144$
• $24 \times 57 = 1368$
• $14 \times 62 = 868$
• $6 \times 67 = 402$

The total frequency $\sum f = 100$.

The mean is $\boxed{52.1}$.