Question

If $x_1,x_2,x_3,....$ are n values of the variable x, then mathematical averages, Arithmetic Mean (A.M.), Geometric Mean (G.M) and Harmonic Mean (H.M) are count by the following formulas's.

$A.M.=\frac{x_1+x_2+x_3+...+x_n}{n}=\frac{1}{n}\left\{\sum _{i=1}^n{x}_1\right\}\nless x_1>0$

$G.M.={(x_1\cdot x_2...x_n)}^{1/n}$ if each $x_1(i=1,2,....,n)$ is positive and

$H.M=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}=\frac{1}{\frac{1}{n}{\sum }_{i=1}^n\left(\frac{1}{x_1}\right)}$

* In case of frequency distribution $x/f_1(i=1,2,...,n)$ where $f_1$ is the frequency of the variable $x_r$ then the calculation of A.M.is counted as $A.M=\sum _{i=1}^n{f}_1{x}_1/NwhereN=f_1+f_2....+f_n$

** If $w_1,w_2,....,w_{{}_n}$ be the weight assigned to the n values

$x_1,x_2,....,x_{{}_n}$ then weighted A.M is counted by $\frac{{\sum }_{i=1}^n{w}_1{x}_1}{{\sum }_{i=1}^n{w}_1}$

*** if $G_1,G_2$ are the G. M's of two series of sizes $n_1$ & $n_2$ respectively, then the geometric mean (G M.) of the combined series is counted by $\log (G.M)=\frac{n_1\log G_1+n_2\log G_2}{n_1+n_2}$

On the basis of above information answer the following questions,If the mean of a set of observations $x_1,x_2,....,x_n$ is $\overline{x}$ then the mean of observations $x_1+4i$ $,i=1,2,3,...,n$ is

• A. $\overline{x}+2(n+1)$
• B. $\overline{x}+4(n+1)$
• C. $\overline{x}+4n$
• D. $\overline{x}+n$

Answer 1

The correct option is A $\overline{x}+2(n+1)$

It is given that $A.M.=\overline{x}=\frac{x_1+x_2+...+x_n}{n}$

$\therefore$ $n\overline{x}=x_1+x_2+x_3+...+x_n$

Let $\overline{y}$ be the mean of observations $x_i+4i$, $(i=1,2,...,n)$

Then $\overline{y}=\frac{(x_1+4.1)+(x_2+4.2)+...+(x_n+4.n)}{n}$

$=\frac{{\sum }_{i=1}^n{x}_i+4(1+2+3+...+n)}{n}$

$=\frac{1}{n}\sum _{i=1}^n{x}_i+\frac{4n(n+1)}{2n}=\overline{x}+2(n+1)$