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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

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.

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