In a set of $$2n$$ observations, half of them are equal to 'a' and the remaining half are equal to '-a'. If the standard deviation of all the observations is 2; then the value of |a| is :

We are given a set of $$2n$$ observations. Half of them, which is $$n$$ observations, are equal to $$a$$, and the other half, also $$n$$ observations, are equal to $$-a$$. The standard deviation of all observations is given as 2. We need to find the value of $$|a|$$.

Recall that the standard deviation ($$\sigma$$) is the square root of the variance ($$\sigma^2$$). Given $$\sigma = 2$$, we have:

$$\sigma^2 = 2^2 = 4$$

So, the variance is 4.

To find the variance, we first need the mean of the observations. Let the mean be denoted by $$\bar{x}$$. The sum of all observations is:

$$\text{Sum} = (a + a + \cdots + a) + (-a + (-a) + \cdots + (-a)) = n \times a + n \times (-a) = na - na = 0$$

Since there are $$2n$$ observations, the mean is:

$$\bar{x} = \frac{\text{Sum}}{2n} = \frac{0}{2n} = 0$$

The variance is defined as the average of the squared differences from the mean. Since the mean is 0, the variance simplifies to:

$$\sigma^2 = \frac{1}{2n} \sum_{i=1}^{2n} (x_i - \bar{x})^2 = \frac{1}{2n} \sum_{i=1}^{2n} x_i^2$$

Now, we compute the sum of the squares of all observations. The first $$n$$ observations are each $$a$$, so their squares sum to $$n \times a^2$$. The next $$n$$ observations are each $$-a$$, and since $$(-a)^2 = a^2$$, their squares also sum to $$n \times a^2$$. Therefore, the total sum of squares is:

$$\sum_{i=1}^{2n} x_i^2 = n a^2 + n a^2 = 2n a^2$$

Substituting this into the variance formula:

$$\sigma^2 = \frac{1}{2n} \times 2n a^2 = \frac{2n a^2}{2n} = a^2$$

We know that $$\sigma^2 = 4$$, so:

$$a^2 = 4$$

Solving for $$a$$:

$$a = \pm 2$$

The question asks for $$|a|$$, the absolute value of $$a$$:

$$|a| = |\pm 2| = 2$$

Therefore, the value of $$|a|$$ is 2.

Looking at the options:

A. 2

B. $$\sqrt{2}$$

C. 4

D. $$2\sqrt{2}$$

Hence, the correct answer is Option A.