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KEAM 2019 Math Paper

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Question : 110 of 121

Marks: +1, -0

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

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60

Solution:

Let

x_{1}, x_{2}, \ldots, x_{20}

be the given observations

We have,

\frac{1}{20} \sum\limits_{i=1}^{20} (x i - \overline{x})^{2}=5

To find variance of

2 x_{1}, 2 x_{2}, 2 x_{3}, \ldots, 2 x_{20}

Let - denotes the mean of new observations.

\text{ Clearly,}\; \overline{X} = \frac{\sum\limits_{i=1}^{20} 2 x_i}{20} = \frac{2 \sum\limits_{i=1}^{20} 2 x_i}{20} = 2 \overline{x}

Now, variance of new observation

\; \; = \frac{1}{20} \sum\limits_{i=1}^{20} (2 x_i - 2 \overline{x})^{2} = \frac{1}{20} \sum\limits_{i=1}^{20} 4 (x_i - \overline{x})^{2}

\; \; = 4 \left( \frac{1}{20} \sum\limits_{i=1}^{20} (x_i - \overline{x})^{2} \right) = 4 \times 5 = 20

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