If the mean of n observation ax1, ax2, ax3,....,axn is a`bar"X"`, show that `(ax_1 - abar"X") + (ax_2 - abar"X") + ...(ax_"n" - abar"X")` = 0.

English

If the Mean of N Observation Ax1, Ax2, Ax3,....,Axn is a ¯¯¯ X , Show that ( a X 1 − a ¯¯¯ X ) + ( a X 2 − a ¯¯¯ X ) + ... ( a X N − a ¯¯¯ X ) = 0. - Mathematics

Question

If the mean of n observation ax₁, ax₂, ax₃,....,ax_n is a`bar"X"`, show that `(ax_1 - abar"X") + (ax_2 - abar"X") + ...(ax_"n" - abar"X")` = 0.

Sum

Solution

We have
a`bar"X" = (ax_1 + ax_2 + ... + ax_"n")/"n"`
⇒ `ax_1 + ax_2 + ... + ax_"n" = "n"(abar"X")` ...(i)
Now `(ax_1 - abar"X") + (ax_2 - abar"X") + ... (ax_"n" - abar"X")`
= `(ax_1 + ax_2 + ... + axn) - (abar"X" + abar"X" + ... + abar"X" "n - times")`
= `"n"(abar"X" - "n"(abar"X")` = 0. ...[Using (i)]
Hence proved.

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