The minimum value of $(x - 6)^{2} + (x + 3)^{2} + (x - 8)^{2} + (x + 4)^{2} + (x - 3)^{2}$ is

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Solution

Minimum value of $(x - 6)^{2} + (x + 3)^{2} + (x - 8)^{2} + (x + 4)^{2} + (x - 3)^{2}$ i.e., Sum of squares of deviations of observation $x$ would be minimum when it's value is taken about the mean of observations $6, - 3, 8, - 4, 3$
$∵$ Mean $= 2$
Substituting $x = 2$ in $(x - 6)^{2} + (x + 3)^{2} + (x - 8)^{2} + (x + 4)^{2} + (x - 3)^{2}$
We get minimum value $= 16 + 25 + 36 + 36 + 1 = 114$

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(x - 6)^{2} + (x + 3)^{2} + (x - 8)^{2} + (x + 4)^{2} + (x - 3)^{2}

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