Question

If the variance of $1,2,3,4,5,....,10$ is $\dfrac{99}{12}$ , then the standard deviation of $3,6,9,12,....,30$ is
1 $\dfrac{297}{4}$
2 $\left( \dfrac{3}{2} \right)\sqrt{33}$
3 $\left( \dfrac{3}{2} \right)\sqrt{99}$
4 $\dfrac{\sqrt{99}}{12}$
5 $\dfrac{3\sqrt{3}}{2}$

Answer 1

Hint : This question will become easier when you have the knowledge of the terms standard deviation and variance. When a constant term is multiplied in the series then its variance is also multiplied by the factor of the square of that constant term. By this statement you can get the answer to the problem.

Complete step-by-step answer :
To solve this question we should know the meaning of the terms variance and standard deviation. They both are important in the field of statistics. Let us discuss one by one.
The variance is used to measure how far a set of numbers is away from the average value of the given data set. Variance is the square of the standard deviation. The symbol used for the representation of variance is ${{\sigma }^{2}}$ or $Var(X)$ . It is calculated as, the variance of $X$ is equal to the subtraction of the mean of the square of $X$ and the square of the mean of $X$ . Mathematically it is represented as,
$Var(X)=E[{{X}^{2}}] -E{{[X] }^{2}}$
Where $E$ denotes the expected value or the mean.
Now let us discuss standard deviation. Standard deviation is used to measure how far each of the values is from the mean of the given data. Standard deviation is the square root of the variance. It is represented by the symbol $\sigma$ .
Now let us try to solve the question. The given series is $1,2,3,4,5,....,10$ and the variance of this particular series is also given and that is $\dfrac{99}{12}$ .
But we have to solve the standard deviation of the given series
$3,6,9,12,....,30$
By the definition of standard deviation we can easily calculate it by simply finding the variance of this series and then performing the root operation and we will get the value of standard deviation for this series.
If we take $\text{3}$ common from the series we will have
$\Rightarrow 3(1,2,3,4,....,10)$
We know that the variance of this series is $\dfrac{99}{12}$ .
And when a constant term is multiplied then variance also gets multiplied by the square of that constant term.
Variance of the series given below:
$3,6,9,12,....,30$
$Variance={{(3)}^{2}}\times \dfrac{99}{12}$
And we know that Standard deviation is the square root of variance.
$\Rightarrow \sigma =\sqrt{9\times \dfrac{99}{12}}$
$\Rightarrow \sigma =\left( \dfrac{3}{2} \right)\sqrt{33}$
Hence we can conclude that option $(2)$ is correct.
So, the correct answer is “Option 2”.

Note : The variance is used to measure the variability in the given data. Standard deviation is totally based upon the mean that means it is relative to the mean. They both are very useful in the field of finance. These two are used in making the trading strategy more effective and more creative.