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The variance of 20 observations is 5. If each observation is multiplied by 2, then find the new variance of the resulting observations.


The variance of 20 observations is 5. If each observation is multiplied by 2, then find the new
variance of the resulting observations.

Grade:upto college level

2 Answers

Latika Leekha
askIITians Faculty 165 Points
6 years ago
Hello student,
Let the given observations be x1, x2, …. x20. and xM be their mean.
Given that n = 20 and variance is 5.
So, \frac{1}{20}\sum_{i =1}^{20}(x-x_{M})^{2} = 5
\sum_{i=1}^{20}(x-x_{M})^{2}=100 ..... (1)
If each observation is multiplied by 2, then the new observations are yi = 2xi, i = 1, 2, … 20.
So, xi = yi/2.
Hence, y_{M}= \frac{1}{20}\sum_{i=1}^{20}y_{i}
= \frac{1}{20}\sum_{i=1}^{20}2x_{i}
= 2. 1/20 . xM
This gives us xM = ½ yM
Substituting the values of xi and xM in (1) we have
\sum_{i=1}^{20}(\frac{1}{2}y_{i}-\frac{1}{2}y_{M})^{2}=100
\sum_{i=1}^{20}\frac{1}{4}(y_{i}-y_{M})^{2}=100
\sum_{i=1}^{20}\(y_{i}-y_{M})^{2}=400
Hence, the variance of the resulting observations is = 1/20 . 400
= 20.
= 22.5
ankit singh
askIITians Faculty 614 Points
8 months ago

Answer

If each observation is multiplied by 2, then mean(μ) will get doubled.

Now, variance =σ2=n(Xμ)2
Now here both X and μ will be doubled, so the variance will become four times.
Thus, the new variance will be 4 times the older variance which is 4×5=20.

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