If x_1, x_​2, x_3.....,x_n are the means of n groups with n₁ n₂ n₃......,n_{n number of observations respectively, then find the mean x of all the groups taken together}

Dear Student,

Please find below the solution to the asked query:

We know :  Mean  = $\frac{\mathrm{Sum} \mathrm{of} \mathrm{observations}}{\mathrm{Number} \mathrm{of} \mathrm{observations}}$ , So  We get

$\frac{{\left(\mathrm{Sum} \mathrm{of} \mathrm{observations}\right)}_1}{{\mathrm{n}}_1} = {\mathrm{x}}_1$ ,

( Sum of observations )₁ = n₁x₁  , Similarly we get 

( Sum of observations )₂ = n₂x₂ , 

( Sum of observations )₃ = n₃x₃ 
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( Sum of observations )_n = n_nx_n

Then ,

Mean of all group together = x  = $\frac{{\mathbf{n}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{ }\mathbf{+}{\mathbf{n}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{ }\mathbf{ }\mathbf{+}\mathbf{ }{\mathbf{n}}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{+}\mathbf{ }{\mathbf{n}}_{\mathbf{n}}{\mathbf{x}}_{\mathbf{n}}}{{\mathbf{n}}_{\mathbf{1}}\mathbf{ }\mathbf{+}\mathbf{ }{\mathbf{n}}_{\mathbf{2}}\mathbf{ }\mathbf{+}\mathbf{ }{\mathbf{n}}_{\mathbf{3}}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{+}\mathbf{ }{\mathbf{n}}_{\mathbf{n}}}$                                   ( Ans )
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