Question

Fill In the Blanks

If the mean of the data x₁,x₂,...,x_n is $\overline{\mathrm{X}}$, then the mean of ax₁ + b,ax₂+b,...,ax_n + b is ____________ .

Answer 1

It is given that, the mean of x₁, x₂,..., x_n is $\overline{X}$.

$\therefore \overline{X}=\frac{x_1+x_2+...+x_n}{n}$

$\Rightarrow x_1+x_2+...+x_n=n\overline{X}$ .....(1)

Let $\overline{X\text{'}}$ be the mean of data ax₁ + b, ax₂ + b,..., ax_n + b.

$\therefore \overline{X\text{'}}=\frac{\left(a{x}_1+b\right)+\left(a{x}_2+b\right)+...+\left(a{x}_n+b\right)}{n}$

$\Rightarrow \overline{X\text{'}}=\frac{\left(a{x}_1+a{x}_2+...+a{x}_n\right)+\left(\underset{n\mathrm{times}}{\underbrace{b+b+...+b}}\right)}{n}$

$\Rightarrow \overline{X\text{'}}=\frac{a\left(x_1+x_2+...+x_n\right)+nb}{n}$

$\Rightarrow \overline{X\text{'}}=\frac{an\overline{X}+nb}{n}$ [Using (1)]

$\Rightarrow \overline{X\text{'}}=\frac{n\left(a\overline{X}+b\right)}{n}$

$\Rightarrow \overline{X\text{'}}=a\overline{X}+b$

Thus, the mean of data ax₁ + b, ax₂ + b,..., ax_n + b is $a\overline{X}+b$.

If the mean of the data x₁, x₂,..., x_n is $\overline{X}$, then the mean of ax₁ + b, ax₂ + b,..., ax_n + b is $\underline{a\overline{X}+b}$ .