91影视

A constant c is added to each\({x_i}\) in a sample, giving equation as\({y_i} = {x_i} + c\).

A constant c is multiplied to each \({x_i}\) in a sample, giving \({y_i} = c{x_i}\).

a.

The variable\({y_i} = {x_i} + c\)where, c is the constant.

The mean of the variable\({x_i}\)is\(\bar x = \frac{{\sum {{x_i}} }}{n}\).

The mean of variable\({y_i}\)is,

\(\begin{array}{c}\bar y &=& \frac{{\sum {{y_i}} }}{n}\\ &=& \frac{{\sum {\left( {{x_i} + c} \right)} }}{n}\\ &=& \frac{{\sum {{x_i}} }}{n} + \frac{{\sum c }}{n}\\ &=& \frac{{\sum {{x_i}} }}{n} + \frac{{n \times c}}{n}\\\bar y &=& \bar x + c\end{array}\)

Thus, the relation between sample mean of \({x_i}'s\) and \({y_i}'s\) is \(\bar y = \bar x + c\).

Let the median of the variable\({x_i}\)is\(\tilde x\).

\(\tilde x\)is the middle value of the ordered sequence of\({x_i}\).

The median of the variable\({y_i}\)is\(\tilde y\).

\(\tilde y\)is the middle value of the ordered sequence of\({y_i}\).

Mathematically,

\(\begin{array}{c}\tilde y &=& median\,of\;\left( {{y_i}} \right)\\ &=& median\,of\;\left( {{x_i} + c} \right)\\ &=& median\,of\;\left( {{x_i}} \right) + median\,of\;\left( c \right)\\ &=& \tilde x + c\end{array}\)

Thus, the relation between sample median of \({x_i}\) and \({y_i}\) is \(\tilde y = \tilde x + c\).

b.

The variable\({y_i} = {x_i} + c\)where, c is the constant.

The mean of variable\({y_i}\)is,

\(\begin{array}{c}\bar y &=& \frac{{\sum {{y_i}} }}{n}\\ &=& \frac{{{y_1} + {y_2} + ... + {y_n}}}{n}\\ &=& \frac{{\left( {c{x_1}} \right) + \left( {c{x_2}} \right) + \left( {c{x_3}} \right) + ... + \left( {c{x_n}} \right)}}{n}\\ &=& \frac{{c\left( {{x_1} + {x_2} + ... + {x_n}} \right)}}{n}\\\bar y &=& c\bar x\end{array}\)

Thus, the relation between sample mean of \({x_i}'s\) and \({y_i}'s\) is \(\bar y = c\bar x\).

Let the median of the variable\({x_i}\)is\(\tilde x\).

\(\tilde x\)is the middle value of the ordered sequence of\({x_i}\).

The median of the variable\({y_i}\)is\(\tilde y\).

\(\tilde y\)is the middle value of the ordered sequence of\({y_i}\).

Mathematically,

\(\begin{array}{c}\tilde y &=& median\,of\;\left( {{y_i}} \right)\\ &=& median\,of\;\left( {c{x_i}} \right)\\ &=& c \times median\,of\;\left( {{x_i}} \right)\\ &=& c\tilde x\end{array}\)

Thus, the relation between sample median of \({x_i}\) and \({y_i}\) is \(\tilde y = c\tilde x\).