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The mean and standard deviation of a set of n_(1)observation are barx_(1)and s_(1) respectively while the mean and standard deviation of another set of n_(2) observations are barx_(2) and s_(2) respectively . Show that the standard deviation of the combined set of (n_(1)+n_(2)) observations is given bySD=sqrt((n_(1)(s_(1))^(2)+n_(2)(s_(2))^(2))/(n_(1)+n_(2))+(n_(1)n_(2)(barx_(1)-barx_(2))^(2))/((n_(1)-n_(2))^(2))) |
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Answer» `thereforebarx_(1)=(1)/(n_(1))overset(n_(1))underset(i=1)Sigmax_(i) and barx_(2)=(1)/(n_(2))overset(n_(2))underset(i=1)Sigmay_(i)` `rArrsigma_(1)^(2)=(1)/(n_(1))overset(n_(1))underset(i=1)(x_(i)-barx_(1))^(2)` and`sigma_(2)^(2)=(1)/(n_(2))overset(n_(1))underset(i=1)(y_(i)-barx_(2))^(2)` Now , mean `barx`of the COMBINED series is given by `barx=(1)/(n_(1)+n_(2))[overset(n_(1))underset(i=1)Sigmax_(i)+overset(n_(2))underset(i=1)Sigmay_(i)]=(n_(1)barx_(1)+n_(2)barx_(2))/(n_(1)+n_(2))` The variance `sigma^(2)` of the combined series is given by `sigma ^(2)=(1)/(n_(2)+n_(2))[overset(n_(1))underset(i=1)Sigmax_(i(x_(1)-barx)^(2))+overset(n_(2))underset(i=1)Sigmay_(i(y_(1)-barx)^(2))]` Now`overset(n_(1))underset(i=1)Sigmax_(i(x_(1)-barx)^(2))=overset(n_(1))underset(i=1)Sigmax_(i(x_(1)-barx_(j)+barx_(j)-barx)^(2)` `=overset(n_(1))underset(i=1)Sigma(x_(i)-barx_(j))^(2)+n_(1)(barx_(j)-barx)^(2)+2(barx_(j)-barx)underset(i=1)overset(n_(1))Sigma(x_(i)-barx)^2` But`overset(n_(1))underset(i=1)Sigma(x_(i)-barx_(i))=0` [algbraic SUM of the deviation of VALUES of first series from their mean is zero] Also`overset(n_(1))underset(i=1)Sigma(x_(i)-barx_(i))^(2)=n_(1)s_(1)^2+n_(1)(barx_(1)-barx)^2=n_1s_1^2+n_1d_1^2` Where,`d_1=(barx_1-barx)` Similarly, `overset(n_2)underset(j=1)Sigma(y_1-barx)^2=overset(n_2)underset(j=1)Sigma(y_1-barx_i+barx_i-barx)^2=n_2s_2^2+n_2d_2^2` Where, `d_2=barx_2-barx` `sigma=sqrt(([n_1(s_1^2+d_1^2)+n_2(s_2^2+d_2^2)])/(n_1+n_2))` where, `d_1=barx_1-barx=barx_1-((n_1barx_1+n_2barx_2)/(n_1+n_2))=((n_2barx_1-barx_2))/(n_1+n_2)` `d_2=barx_2-barx=barx_2-(n_1barx_1+n_2barx_2)/(n_1+n_2)=(n_1(barx_2-barx_1))/(n_1+n_2)` `thereforesigma^2=(1)/(n_1+n_2)[n_1s_1^2+n_2s_2^2+(n_1n_2(barx_1-barx_2)^2)/(n_1+n_2)^2+(n_2n_1(barx_2-x_1)^2)/((n_1+n_2)^2)]` Also`sigma=sqrt((n_1s_1^2+n_2s_2^2)/(n_1+n_2)+(n_1n_2(barx_1-barx_2)^2)/((n_1+n_2)^2))` |
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