The mean of the series $${x_1},\,{x_2},......,\,{x_n}$$    is $$\overline X $$. If $${x_2}$$ is replaced by $$\lambda $$, then what is the new mean ?

Solution :
$$\eqalign{ & {\text{Mean of series}}\left( {{x_1},\,{x_2},\,{x_3},......,\,{x_n}} \right) \cr & \overline x = \frac{{{x_1} + {x_2} + {x_3}, +......,\,{x_n}}}{n} \cr & \Rightarrow {x_1} + {x_2} + {x_3} + ...... + \,{x_n} = n\overline x \cr & {\text{Now we will replace }}{x_2}{\text{ by }}\lambda \cr & {\text{So number of elements in series will not change}}{\text{.}} \cr & {\text{New series will include }}\lambda {\text{ and exclude }}{x_2} \cr & {\text{Hence new series sum :}} \cr & \left( {{x_1} + {x_2} + ...... \,{x_n}} \right) - {x_2} + \lambda = n\overline x + \lambda - {x_2} \cr & {\text{Now new mean}} \cr & = \frac{{n\overline x + \lambda - {x_2}}}{n} = \frac{{n\overline x - {x_2} + \lambda }}{n} \cr} $$