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Last updated at March 9, 2017 by Teachoo

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Misc 3 The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations. Let the observations be 𝑥1, 𝑥2, 𝑥3, ..., 𝑥6 and 𝑥 be their mean. Given that Mean = 𝑥 = 8, Standard deviation = 4 If each observation is multiplied by 3 , we get new observations, Let the new observations be 𝑦1, 𝑦2, 𝑦3, ..., 𝑦6 where 𝑦𝑖 = 3( 𝑥𝑖) Calculating new mean New mean = 1𝑛 𝑦𝑖 𝑦 = 16 3𝑥𝑖 𝑦 = 3 × 16 𝑥𝑖 𝑦 = 3 𝑥 𝑦 = 3 × 8 𝑦 = 24 So, New Mean = 24 Calculating new standard deviation First we find variance of the new observations i.e. New Variance = 1n ( 𝑦𝑖− 𝑦)2 Given Old Standard deviation = 4 So, Old Variance = 42 = 16 Now, Old Variance = 1𝑛 ( 𝑥𝑖− 𝑥)2 16 = 16 ( 𝑥𝑖− 𝑥)2 16 × 6 = ( 𝑥𝑖− 𝑥)2 96 = ( 𝑥𝑖− 𝑥)2 ( 𝑥𝑖− 𝑥)2 = 96 ( 13 𝑦𝑖− 13 𝑦)2 = 96 ( 13 (𝑦𝑖− 𝑦))2 = 96 132 ( 𝑦𝑖− 𝑦)2 = 96 19 ( 𝑦𝑖− 𝑦)2 = 96 ( 𝑦𝑖− 𝑦)2 = 96 × 9 ( 𝑦𝑖− 𝑦)2 = 864 So, New Variance = 1𝑛 ( 𝑦𝑖− 𝑦)2 = 16 × 864 = 144 Hence, New standard deviation = 𝑁𝑒𝑤 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 144 = 12

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.