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Question

Show that the two formulae for the standard deviation of ungrouped data .

σ=1n(xi¯¯¯¯¯X)2 and σ=1nx2i¯¯¯¯¯X2 are equivalent , where ¯¯¯¯¯X=1nxi



    Solution

    The correct option is

    σ=1n(xi¯¯¯¯¯X)2

    =1n(x2i2xi¯¯¯¯¯X+¯¯¯¯¯X)2=1nx2i1n2xi¯¯¯¯¯X+1n¯¯¯¯¯X2

    =1nx2i1n×2¯¯¯¯¯Xxi+1nׯ¯¯¯¯X21

    =1nx2i1n×2¯¯¯¯¯X×n¯¯¯¯¯X+1nׯ¯¯¯¯X2×n                      (¯¯¯¯¯X=1nxi)   

    =1nx2i2¯¯¯¯¯X+¯¯¯¯¯X2=1nx2i¯¯¯¯¯X2=σ1

    Hence, the formula σ=1n(xi¯¯¯¯¯X)2 and σ=1nx2i¯¯¯¯¯X2      are equivalent, where   ¯¯¯¯¯X=1nxi         


    Mathematics
    RD Sharma
    Standard XI

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