Class XI Chapter 15 – Statistics Maths ______________________________________________________________________________ Question 4: Given that x is the mean and Ïƒ 2 is the variance of n observations xi, x2 â€¦ xn. Prove that the mean and variance of the observations ax1, ax2, ax3 â€¦axn are and a2 Ïƒ 2 , respectively (a ≠ 0). Solution 4: The given n observations are x1, x2 â€¦ xn. Mean = x Variance = Ïƒ 2 2 1 n 2 = y x x ... 1 1 i n i1 If each observation is multiplied by a and the new observations are yi, then y 1 = ax i i.e., x i = 1 y a 1 1 n 1 n a n 1 n y y1 ax1 x1 ax x x1 n i1 n i1 n i1 n i1 Therefore, mean of the observations, ax1, ax2 â€¦ axn, is ax Substituting the values of xi and x in (1), we obtain 2 2 1 n 1 1 y1 y n i 1 a a 2 2 1 n 2 a y1 y n i1 Thus, the variance of the observations, ax1, ax2 â€¦ axn, is a 2 Ïƒ 2 . Question 5: The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases: (i) If wrong item is omitted. (ii) If it is replaced by 12. Solution 5: (i) Number of observations (n) = 20 Incorrect mean = 10 Incorrect standard deviation = 2 1 20 x xi n i1 1 20 10 x 20 i i1 20 x 200 i i1 That is, incorrect sum of observations = 200 Correct sum of observations = 200 â€“ 8 = 192 Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

Class XI Chapter 15 – Statistics Maths ______________________________________________________________________________ ∴ Correct mean = correct sum 192 10.1 19 19 Standard deviation = 1 20 1 20 n 2 2 Incorrect xi - 10 i1 1 n 2 1 n xi x 2 i n i1 n i1 = 1 2 n x x n i1 i 2 2 n 2 4 Incorrect xi -100 i1 n 2 Incorrect xi 2080 i1 n n 2 2 xi xi i1 i1 2 Correct Incorrect 8 2080 64 2016 2 Correct x Correct standard deviation= i Correct mean n 2016 10.1 2 19 1061.1 102.1 4.09 2.02 (ii) When 8 is replaced by 12, Incorrect sum of observations = 200 ∴ Correct sum of observations = 200 â€“ 8 + 12 = 204 Correct sum 204 Correct mean= 10.2 20 20 Standard deviation = 1 2 Incorrect -(10) 20 1 n 2 4 Incorrect xi -100 20 i1 n 2 Incorrect xi 2080 i1 n n 2 n 2 2 i 2 i i i1 i1 i1 1 1 1 x x x x n n n n 2 2 xi i1 n 2 2 2 2 Correct xi Incorrect xi 8 12 i1 =2080-64+144 =2160 2 2 2 Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

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