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Chapter 13 Class 11 Statistics

Serial order wise

Last updated at May 29, 2023 by Teachoo

Example 15 If each observation 𝑥1, 𝑥2, 𝑥3, ..., 𝑥𝑛 is increased by a, where a is a negative or positive number, show that the variance remains unchanged. Let the mean of the observations 𝑥1, 𝑥2, 𝑥3, ..., 𝑥𝑛 be 𝑥 Variance of these observations is given by Old Variance = 1n ( 𝑥𝑖− 𝑥)2 If each observation is increased by a , we get new observations, Let the new observations be 𝑦1, 𝑦2, 𝑦3, ..., 𝑦𝑛 where 𝑦𝑖 = 𝑥𝑖 + a We need to find variance of the new observations i.e. New Variance = 1n ( 𝑦𝑖− 𝑦)2 Now, We know 𝑦𝑖 = 𝑥𝑖 + a Calculating 𝑦 in terms of 𝑥, 𝑦 = 1𝑛 𝑦𝑖 𝑦 = 1𝑛 ( 𝑥𝑖 + a) 𝑦 = 1𝑛 𝑥𝑖 + 𝑎 𝑦 = 1𝑛 𝑥𝑖 + 1𝑛 𝑎 𝑦 = 𝑥 + 1𝑛 × n(a) 𝑦 = 𝑥 + a Calculating new variance New Variance = 1n ( 𝑦𝑖− 𝑦)2 = 1n ( 𝑦𝑖− 𝑦)2 = 1n ( 𝑥𝑖+𝑎−( 𝑥+𝑎))2 = 1n ( 𝑥𝑖+𝑎− 𝑥−𝑎)2 = 1n ( 𝑥𝑖− 𝑥)2 = Old variance Thus, the variance of the new observations is same as that of the original observations.