1. Given frequency distribution

The class marks are: $5, 15, 25, 35, 45$

2. Use the mean formula

Mean of grouped data is $\bar{x} = \frac{\sum f_i m_i}{\sum f_i}$ where $m_i$ are class marks.

Given mean $=28$.

So, $\sum f_i = 2+3+x+5+4 = x+14$

and $\sum f_i m_i = 2\cdot 5 + 3\cdot 15 + x\cdot 25 + 5\cdot 35 + 4\cdot 45$ $=10+45+25x+175+180$ $=25x+410$

Now, $\frac{25x+410}{x+14}=28$

3. Find $x$

$25x+410 = 28(x+14)$ $25x+410 = 28x+392$ $18 = 3x$ $x=6$

4. Write the completed frequencies

Now frequencies are: $2, 3, 6, 5, 4$

Hence, $N = 2+3+6+5+4 = 20$

5. Compute variance

Variance of grouped data: $\sigma^2 = \frac{\sum f_i m_i^2}{N} - \bar{x}^2$

First calculate $\sum f_i m_i^2$:

$\sum f_i m_i^2 = 2(5^2)+3(15^2)+6(25^2)+5(35^2)+4(45^2)$

$=2(25)+3(225)+6(625)+5(1225)+4(2025)$

$=50+675+3750+6125+8100$

$=18700$

Therefore, $\frac{\sum f_i m_i^2}{N} = \frac{18700}{20} = 935$

Given mean $\bar{x}=28$, so $\sigma^2 = 935 - 28^2$ $=935-784$ $=151$

6. Final answer

The variance is $\boxed{151}$