Let in a series of $$2n$$ observations, half of them are equal to $$a$$ and remaining half are equal to $$-a$$. Also by adding a constant $$b$$ in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of $$a^2 + b^2$$ is equal to :

In the original series of $$2n$$ observations, $$n$$ values equal $$a$$ and $$n$$ values equal $$-a$$. The original mean is $$\frac{n \cdot a + n \cdot (-a)}{2n} = 0$$.

After adding constant $$b$$ to each observation, the new values are $$a + b$$ ($$n$$ times) and $$-a + b$$ ($$n$$ times). The new mean is $$0 + b = b = 5$$, so $$b = 5$$.

Adding a constant does not change the standard deviation, so the standard deviation of the new set equals the standard deviation of the original set, which is 20. For the original data, the variance is $$\frac{n \cdot a^2 + n \cdot a^2}{2n} - 0^2 = a^2$$. Therefore the standard deviation is $$|a| = 20$$, giving $$a^2 = 400$$.

Hence $$a^2 + b^2 = 400 + 25 = 425$$.