$${{{n^2} - 1} \over {12}} = 14 \Rightarrow n = 13$$

$$5 = {{3 + 7 + x + y} \over 4} \Rightarrow x + y = 10$$

Var(x) = $$10 = {{{3^2} + {7^2} + {x^2} + {y^2}} \over 4} - 25$$

$$140 = 49 + 9 + {x^2} + {y^2}$$

$${x^2} + {y^2} = 82$$

x + y = 10

$$\Rightarrow$$ (x, y) = (9, 1)

Four numbers are 21, 9, 10, 8

Mean = $${{48} \over 4}$$ = 12

$$\because$$ Sum of frequencies = 584

$$\Rightarrow$$ $$\alpha$$ + $$\beta$$ = 390

Now, median is at $${{584} \over 2}$$ = 292^th

$$\because$$ Median = 45 (lies in class 40 - 50)

$$\Rightarrow$$ $$\alpha$$ + 110 + 54 + 15 = 292

$$\Rightarrow$$ $$\alpha$$ = 113, $$\beta$$ = 277

$$\Rightarrow$$ | $$\alpha$$ $$-$$ $$\beta$$ | = 164

Class	Frequency	$${x_i}$$	$${f_i}{x_i}$$
0-6	a	3	3a
6-12	b	9	9b
12-18	12	15	180
18-24	9	21	189
24-30	5	27	135
	$$N = (26 + a + b)$$		$$(504 + 3a + 9b)$$

Mean = $${{3a + 9b + 180 + 189 + 135} \over {a + b + 26}} = {{309} \over {22}}$$

$$ \Rightarrow 66a + 198b + 11088 = 309a + 309b + 8034$$

$$ \Rightarrow 243a + 111b = 3054$$

$$ \Rightarrow 81a + 37b = 1018$$ $$\to$$ (1)

Now, Median $$ = 12 + {{{{a + b + c} \over 2} - (a + b)} \over {12}} \times 6 = 14$$

$$ \Rightarrow {{13} \over 2} - \left( {{{a + b} \over 4}} \right) = 2$$

$$ \Rightarrow {{a + b} \over 4} = {9 \over 2}$$

$$ \Rightarrow a + b = 18$$ $$\to$$ (2)

From equation (1) $ (2)

a = 8, b = 10

$$\therefore$$ $${(a - b)^2} = {(8 - 10)^2}$$