Let in a series of 2n observations, half of them are equal to a and remaining half are equ…

Question

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

Accepted Answer

Mean of original (before adding b): $\bar x = 0$ (equal numbers of $+a$ and $-a$).
$$\sigma_x^2 = \frac{a^2\times2n}{2n} = a^2 \implies \sigma_x=a$$

After adding constant b: new mean $=0+b=5 \implies b=5$; new S.D. $=\sigma_x=20 \implies a=20$.

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

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