Question
Let $$\overline x $$ be the mean of $$n$$ observations $${x_1},\,{x_2},\,......,\,{x_n}.$$ If $$\left( {a - b} \right)$$ is added to each observation, then what is the mean of new set of observations ?
A.
$$0$$
B.
$$\overline x $$
C.
$$\overline x - \left( {a - b} \right)$$
D.
$$\overline x + \left( {a - b} \right)$$
Answer :
$$\overline x + \left( {a - b} \right)$$
Solution :
Let $$\overline x $$ is the mean of $$n$$ observation $${x_1},\,{x_2},\,......,\,{x_n}$$
$$\eqalign{
& \Rightarrow \overline x = \frac{{{x_1} + {x_2} + {x_3} + ..... + {x_n}}}{n} \cr
& {\text{Now, }}\left( {a - b} \right){\text{ is added to each term}}{\text{.}} \cr
& \therefore \,{\text{New mean}} \cr
& = \frac{{{x_1} + \left( {a - b} \right) + {x_2} + \left( {a - b} \right) + ...... + {x_n} + \left( {a - b} \right)}}{n} \cr
& = \frac{{{x_1} + {x_2} + ..... + {x_n}}}{n} + \frac{{n\left( {a - b} \right)}}{n} \cr
& = \overline x + \left( {a - b} \right) \cr} $$