If average of 20 observations x<sub>1</sub>, x<sub>2</sub>, . . . . . x<sub>20</sub> is y, then the average of x<sub>1</sub> - 101, x<sub>2</sub> - 101, x<sub>3</sub> - 101, . . . . . x<sub>20</sub> - 101 is :
Correct Answer: y - 101
According to the question,
$$\eqalign{ & \Rightarrow \frac{{{x_1} + {x_2} + {x_3} + {x_4} + .... + {x_{20}}}}{{20}} = y \cr & \Rightarrow {x_1} + {x_2} + {x_3} + {x_4} + .... + {x_{20}} = 20y \cr} $$
$$ = \frac{{{x_1} - 101 + {x_2} - 101 + {x_3} - 101 + {x_4} - 101 + .... + {x_{20}} - 101}}{{20}}$$
$$ = \frac{{\left( {{x_1} + {x_2} + {x_3} + {x_4} + .... + {x_{20}}} \right) - 20 \times 101}}{{20}}$$
$$\eqalign{ & = \frac{{20y - 20 \times 101}}{{20}} \cr & = y - 101 \cr} $$
$$\eqalign{ & \Rightarrow \frac{{{x_1} + {x_2} + {x_3} + {x_4} + .... + {x_{20}}}}{{20}} = y \cr & \Rightarrow {x_1} + {x_2} + {x_3} + {x_4} + .... + {x_{20}} = 20y \cr} $$
$$ = \frac{{{x_1} - 101 + {x_2} - 101 + {x_3} - 101 + {x_4} - 101 + .... + {x_{20}} - 101}}{{20}}$$
$$ = \frac{{\left( {{x_1} + {x_2} + {x_3} + {x_4} + .... + {x_{20}}} \right) - 20 \times 101}}{{20}}$$
$$\eqalign{ & = \frac{{20y - 20 \times 101}}{{20}} \cr & = y - 101 \cr} $$