


Examples
Example 2
Example 3 Important
Example 4
Example 5 Important
Example 6 (Normal Method)
Example 6 (Shortcut Method) Important
Example 7 Important
Example 8
Example 9 Important
Example 10 Important
Example 11
Example 12 Important
Example 13 Deleted for CBSE Board 2022 Exams
Example 14 Important
Example 15 Important Deleted for CBSE Board 2022 Exams
Example 16
Example 17 Important
Example 18 You are here
Example 19 Important
Examples
Last updated at May 29, 2018 by Teachoo
Example 18 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.