The mean of 20 numbers is 18. If 3 is added to each of the first ten numbers

Solution:

Let the numbers be x1, x2,...x20.
We know

Mean $=\frac{\text { Sum of observa tions }}{\text { Number of observations }}$

Thus, we have:

$18=\frac{x_{1}+x_{2}+\ldots \ldots+x_{20}}{20}$

$\Rightarrow x_{1}+x_{2}+\ldots \ldots+x_{20}=360 \quad \ldots \ldots$ (i)

New numbers are:

$\left(x_{1}+3\right),\left(x_{2}+3\right), \ldots\left(x_{10}+3\right), x_{11}, \ldots x_{20}$

New Mean:

$=\frac{\left(x_{1}+3\right)+\left(x_{2}+3\right)+\ldots \ldots . .+\left(x_{10}+3\right)+x_{11}+\ldots \ldots . .+x_{20}}{20}$

$=\frac{\left(x_{1}+x_{2}+\ldots . .+x_{10}\right)+(3 \times 10)+x_{11}+\ldots \ldots . .+x_{20}}{20}$

 

$=\frac{\left(x_{1}+x_{2}+\ldots \ldots+x_{20}\right)+30}{20}$

$=\frac{360+30}{20} \quad[$ From (i) $]$

$=19,5$