If the variance of the following frequency distribution:

is 50, then x is equal to __________

We have a grouped frequency distribution with three classes: 10‒20, 20‒30 and 30‒40. Their respective frequencies are 2, $$x$$ and 2, while the variance is given as $$50$$. Our goal is to determine the unknown frequency $$x$$.

For every class we first find the class mark (mid-point). Using the formula “mid-point = (lower limit + upper limit)/2”, we get

$$m_1 = \frac{10+20}{2}=15, \qquad m_2 = \frac{20+30}{2}=25, \qquad m_3 = \frac{30+40}{2}=35.$$

Let the total frequency be $$N$$ and the arithmetic mean be $$\bar{x}$$. Clearly

$$N = 2 + x + 2 = x + 4.$$

Using “mean $$\bar{x} = \dfrac{\sum f m}{\sum f}$$”, we compute the numerator

$$\sum f m = 2(15) + x(25) + 2(35) = 30 + 25x + 70 = 25x + 100.$$

Hence the mean is

$$\bar{x} = \frac{25x + 100}{x + 4}.$$

Next, for variance we use the shortcut formula for grouped data:

$$\sigma^{2} = \frac{\sum f m^{2}}{N} - \bar{x}^{2}.$$

We therefore need $$\sum f m^{2}$$:

$$\sum f m^{2} = 2(15^{2}) + x(25^{2}) + 2(35^{2}) = 2(225) + x(625) + 2(1225) = 450 + 625x + 2450 = 625x + 2900.$$

Because the variance is given as $$\sigma^{2} = 50$$, we write

$$50 = \frac{625x + 2900}{x + 4} - \left( \frac{25x + 100}{x + 4} \right)^{2}.$$

To remove denominators we multiply every term by $$(x + 4)^{2}$$:

$$50(x + 4)^{2} = (625x + 2900)(x + 4) - (25x + 100)^{2}.$$

Now we simplify each expression separately.

First expand the left side:

$$(x + 4)^{2} = x^{2} + 8x + 16,$$

so

$$50(x + 4)^{2} = 50x^{2} + 400x + 800.$$

Next expand the first product on the right:

$$(625x + 2900)(x + 4) = 625x^{2} + 2500x + 2900x + 11600 = 625x^{2} + 5400x + 11600.$$

Then expand the square:

$$(25x + 100)^{2} = (25x)^{2} + 2(25x)(100) + 100^{2} = 625x^{2} + 5000x + 10000.$$

Subtracting that square from the previous expansion gives

$$(625x^{2} + 5400x + 11600) - (625x^{2} + 5000x + 10000) = 400x + 1600.$$

So the equation becomes

$$50x^{2} + 400x + 800 = 400x + 1600.$$

Subtract $$400x + 1600$$ from both sides:

$$50x^{2} + 400x + 800 - 400x - 1600 = 0,$$

which simplifies to

$$50x^{2} - 800 = 0.$$

Dividing every term by $$50$$ yields

$$x^{2} - 16 = 0.$$

This factors as

$$(x - 4)(x + 4) = 0,$$

leading to

$$x = 4 \quad \text{or} \quad x = -4.$$

Since a frequency cannot be negative, we discard $$x = -4$$ and keep

$$x = 4.$$

So, the answer is $$4$$.