Examples

Example 1

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 2023 Exams

Example 14 Important Deleted for CBSE Board 2023 Exams

Example 15 Important Deleted for CBSE Board 2023 Exams

Example 16

Example 17 Important

Example 18 You are here

Example 19 Important

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.