Let the correct observation be x and incorrect observation be x'.
∴ \(x+ x' = 70 \) .....(1) (Given)
Incorrect sum of obs. = Mean x 50 = 15 x 50 = 750
∴ Correct sum = Incorrect sum - x' + x
50 x Correct mean = 750 - x' + x
⇒ \(750 - x' + x = 50 \times 16\)
⇒ \(x - x' = 800 - 750 = 50\) .....(2)
Adding (1) & (2), we get
\(2x = 120\)
⇒ \(x =60\) & \(x' = 10\)
Incorrect standard deviation = 2 ⇒ incorrect variance = 4
⇒ \(\frac{(\sum{x_i}^2)_{incorect}}{n} - (Incorrect\, mean)^2 = 4\)
⇒ \((\sum {x_1}^2)_{incorrect }= 50 (4 + 15^2) = 50 \times 229 = 11450\)
\(\therefore(\sum {x_i}^2)_{correct} = (\sum {x_i}^2)_{incorrect} - x'^2 + x^2\)
\(= 11450 - 100 + 3600\)
\(= 14950\)
∴ Correct variance = \(\frac{(\sum {x_i}^2)_{correct}}n - (correct\, mean)^2\)
\(=\frac{ 14950 }{50} - 16^2\)
\(= 299 - 256\)
\(= 43\)