Question

The mean and variance of a data set comprising $15$ observations are $15$ and $5$ respectively. If one of the observation $15$ is deleted and two new observations $6$ and $8$ are added to the data, then the new variance of resulting data is

• A. $\text{10}\text{.3715}$
• B. $\text{11}\text{.8125}$
• C. $\text{13}\text{.25}$
• D. $\text{5}\text{.7516}$

Answer 1

Answer: (B)

Let $15$ observations are $x_1,x_2\dots \dots x_{15}$
$\frac{x_1+x_2+x_3\dots \dots x_{15}}{15}=15$
If $x_{15}=15$
${\bar{x}}_{new}=\frac{x_1+x_2+x_3\dots x_{15}-15+6+8}{16}=\frac{224}{16}=14$
$\frac{\Sigma x_i^2}{15}-{\left(\bar{x}\right)}^2=5$ $\Rightarrow x_1^2+x_2^2\dots \dots x_{14}^2+{\left(15\right)}^2=230\times 15$
$\Rightarrow x_1^2+x_2^2\dots \dots x_{14}^2=3225$
New variance = $\frac{x_1^2+x_2^2\dots \dots x_{14}^2+6^2+8^2}{16}-{\left({\left(\bar{x}\right)}_{new}\right)}^2$
$=\frac{3325}{16}-196$
$=11.8125$