Question

uppose the data set yIn has a wedia n value of M. Consider a transfaaned verson o this data sef 2h where a. Explain at a high level, in one entenenhatfu3 nd formation doar to the dataset. b)How is the mediun of 2n related p that 7 e) How i the mean abso lus te devia ti on firen the median

Answer 1

$(a)~z_i=y_i&plus;3(y_i-M)r_2=(1&plus;3r_2)y_i-Mr_2\\ If~r_2>-\frac{1}{3}~then\\ z_1<z_2<...<z_n~as~y_1<y_2<...<y_n\\ ~or~z_1>z_2>...>z_n~as~y_1>y_2>...>y_n\\ If~r_2<-\frac{1}{3}~then\\ z_1<z_2<...<z_n~as~y_1>y_2>...>y_n\\ ~or~z_1>z_2>...>z_n~as~y_1<y_2<...<y_n\\ (b)~Let~M_z~be~a~median~of~z_1,z_2,...,z_n~then\\ M_z=(1&plus;3r_2)M-Mr_2=(1&plus;2r_2)M~since~z~is~a~linear~function~of~y.\\ (c)~Mean~deviation~about~median~of~z~variable\\ MD(z)=\frac{1}{n}\sum_{i=1}^n|z_i-M_z|=\frac{1}{n}\sum_{i=1}^n|(1&plus;3r_2)y_i-Mr_2-(1&plus;2r_2)M|\\ =|1&plus;3r_2|\times \frac{1}{n}\sum_{i=1}^n|y_i-M|=|1&plus;3r_2|\times MD(y)\\ where,~MD(y)=mean~deviation~about~median~of~y~variable.$