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Question
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.
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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