Check sibling questions

Example 18 - If each observation x1, x2, x3 is increased by a - Standard deviation and variance - Ungrouped data

Example 18 - Chapter 15 Class 11 Statistics - Part 2
Example 18 - Chapter 15 Class 11 Statistics - Part 3

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Example 15 If each observation 𝑥﷮1﷯, 𝑥﷮2﷯, 𝑥﷮3﷯, ..., 𝑥﷮𝑛﷯ is increased by a, where a is a negative or positive number, show that the variance remains unchanged. Let the mean of the observations 𝑥﷮1﷯, 𝑥﷮2﷯, 𝑥﷮3﷯, ..., 𝑥﷮𝑛﷯ be 𝑥﷯ Variance of these observations is given by Old Variance = 1﷮n﷯ ﷮﷮( 𝑥﷮𝑖﷯﷯− 𝑥﷯)﷮2﷯ If each observation is increased by a , we get new observations, Let the new observations be 𝑦﷮1﷯, 𝑦﷮2﷯, 𝑦﷮3﷯, ..., 𝑦﷮𝑛﷯ where 𝑦﷮𝑖﷯ = 𝑥﷮𝑖﷯ + a We need to find variance of the new observations i.e. New Variance = 1﷮n﷯ ﷮﷮( 𝑦﷮𝑖﷯﷯− 𝑦﷯)﷮2﷯ Now, We know 𝑦﷮𝑖﷯ = 𝑥﷮𝑖﷯ + a Calculating 𝑦﷯ in terms of 𝑥﷯, 𝑦﷯ = 1﷮𝑛﷯ ﷮﷮ 𝑦﷮𝑖﷯﷯ 𝑦﷯ = 1﷮𝑛﷯ ﷮﷮( 𝑥﷮𝑖﷯﷯ + a) 𝑦﷯ = 1﷮𝑛﷯ ﷮﷮ 𝑥﷮𝑖﷯﷯ + ﷮﷮𝑎﷯﷯ 𝑦﷯ = 1﷮𝑛﷯ ﷮﷮ 𝑥﷮𝑖﷯﷯ + 1﷮𝑛﷯ ﷮﷮𝑎﷯ 𝑦﷯ = 𝑥﷯ + 1﷮𝑛﷯ × n(a) 𝑦﷯ = 𝑥﷯ + a Calculating new variance New Variance = 1﷮n﷯ ﷮﷮( 𝑦﷮𝑖﷯﷯− 𝑦﷯)﷮2﷯ = 1﷮n﷯ ﷮﷮( 𝑦﷮𝑖﷯﷯− 𝑦﷯)﷮2﷯ = 1﷮n﷯ ﷮﷮( 𝑥﷮𝑖﷯+𝑎﷯−( 𝑥﷯+𝑎))﷮2﷯ = 1﷮n﷯ ﷮﷮( 𝑥﷮𝑖﷯+𝑎﷯− 𝑥﷯−𝑎)﷮2﷯ = 1﷮n﷯ ﷮﷮( 𝑥﷮𝑖﷯﷯− 𝑥﷯)﷮2﷯ = Old variance Thus, the variance of the new observations is same as that of the original observations.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.