Let observations are denoted by xi for 1≤i<2n . ∴Mean(x)=‌
∑
xi
2n
=‌
(a+a+a+...+a)−(a+a+a+...+a)
2n
⇒‌‌x=0 and
σx2=‌
∑x12
2n
−(x)2=‌
2
â•â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â•®
‌
a2+a2+a2+....+a2
2n
−0=a2
⇒‌‌σx2=a2⇒σx=a According to the question, adding a constant b , then new mean (y)=x+b=5 ⇒‌‌0+b=5 ⇒‌‌b=5 and new SD (σy)=σx   (no change in SD) ⇒‌σy=σx=20 ⇒‌a=20 ‌∴a2+b2=400+25=425