For a group of 100 students, the mean x ¯ 1 and the standard deviation σ 1 of their marks were found to be 40 and 15 respectively. Later it was observed that the scores 40 and 50 were misread as 30 and 60 respectively. If the mean and the standard deviation with the corrected observations of the scores are x ¯ 2 and σ 2 respectively, then

Step-by-step solution

It is given that. x ¯ 1 = 40 σ = 15 Now, the mean is, x ¯ 1 = ∑ x i n ⇒ 40 = ∑ x i 100 ⇒ ∑ x i = 4000 Corrected summation, ∑ x i ' = ∑ x i - 30 - 60 + 40 + 50 = 4000 The corrected mean, x ¯ 2 = ∑ x i ' 100 ⇒ x ¯ 2 = 4000 100 = 40 σ 1 2 = ∑ x i 2 n - ∑ x i n 2 ⇒ 225 = ∑ x i 2 100 - 4000 100 2 ⇒ ∑ x i 2 = 182500 Corrected summation, ∑ x i 2 = 182500 - 30 2 - 60 2 + 40 2 + 50 2 ⇒ ∑