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Let $\bar{x}$ be the mean of n observations $x_{1}, x_{2}, \ldots . ., x_{n}$. If $(a-b)$ is added to each observation, then what is the mean of new set of observations?
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The correct answer is:
$\overline{\mathrm{x}}+(\mathrm{a}-\mathrm{b}) \quad\mathrm{}$
Let $\bar{x}$ is the mean of $n$ observation $x_{1}, x_{2}, \ldots ., x_{n}$
$\Rightarrow \bar{x}=\frac{x_{1}+x_{2}+x_{3}+\ldots .+x_{n}}{n}$
Now, $(a-b)$ is added to each term. New mean $=\frac{\mathrm{x}_{1}+(\mathrm{a}-\mathrm{b})+\mathrm{x}_{2}+(\mathrm{a}-\mathrm{b})+\ldots . .+\mathrm{x}_{\mathrm{n}}+(\mathrm{a}-\mathrm{b})}{\mathrm{n}}$
$=\frac{\mathrm{x}_{1}+\mathrm{x}_{2}+\ldots .+\mathrm{x}_{\mathrm{n}}}{\mathrm{n}}+\frac{\mathrm{n}(\mathrm{a}-\mathrm{b})}{\mathrm{n}}$
$=\overline{\mathrm{x}}+(\mathrm{a}-\mathrm{b})$
$\Rightarrow \bar{x}=\frac{x_{1}+x_{2}+x_{3}+\ldots .+x_{n}}{n}$
Now, $(a-b)$ is added to each term. New mean $=\frac{\mathrm{x}_{1}+(\mathrm{a}-\mathrm{b})+\mathrm{x}_{2}+(\mathrm{a}-\mathrm{b})+\ldots . .+\mathrm{x}_{\mathrm{n}}+(\mathrm{a}-\mathrm{b})}{\mathrm{n}}$
$=\frac{\mathrm{x}_{1}+\mathrm{x}_{2}+\ldots .+\mathrm{x}_{\mathrm{n}}}{\mathrm{n}}+\frac{\mathrm{n}(\mathrm{a}-\mathrm{b})}{\mathrm{n}}$
$=\overline{\mathrm{x}}+(\mathrm{a}-\mathrm{b})$
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