the mean and standard deviation of 100 observations were calculated as 40 and 5.1 resp. by a student who took by mistake 50 instead of 40 for one observation. what are correct mean and standard deviation?
Given, = 40 , = 5.1
According to the formula
2 = ∑ x x 2 n - x ¯ 2 ⇒ 5 . 1 2 = ∑ x x 2 100 - 40 2 ⇒ ∑ x x 2 100 = 5 . 1 2 + 40 2 = 1626 . 01 S o , ∑ x x 2 = 1626 . 01 × 100 = 162601 B u t c a l c u l a t e d ∑ x x 2 i s i n c o r r e c t , s o f o r c o r r e c t s u m o f s q u a r e w e s u b t r a c t s q u a r e o f 50 a n d a s s s q u a r e o f 40 S o , n e w ∑ x x 2 = 162601 - 50 2 + 40 2 = 161701 S o , c o r r e c t σ 2 = c o r r e c t ∑ x x 2 n - c o r r e c t x ¯ 2 c o r r e c t σ 2 = 161701 100 - 39 . 9 2 = 1617 . 01 - 1592 . 01 = 25 S o , σ = 25 = 5 H e n c e n e w m e a n i s 39 . 9 a n d s tan d a r d d e v i a t i o n i s 5
According to the formula