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  • Fill in the blanks If X bar is the mean of n values of x, then is always equal to _______. If a has any value other than , then is _________ than .

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If \bar{X}is the mean of n values of x, then  \sum _{i=1}^{n} \left( x_{i}-\overline{x} \right) is always equal to _______.
If a has any value other than \bar{X} , then \sum _{i=1}^{n} \left( x_{i}-\overline{x} \right) ^{2} is _________ than \sum \left (X_i-a \right )^2

Answers (1)

Zero, less than: Given x is mean of n values, the sum of all n terms is denoted by  \sum _{i=1}^{n}x_{i}, \\

So, the difference between both of these is always equal to zero, i.e. } \sum _{i=1}^{n} \left( x_{i}-\overline{x} \right) =0 \\

And square of the above equation is also equal to zero, so } \sum _{i=1}^{n} \left( x_{i}-\overline{x} \right) ^{2}=0 \\

  Now if "a"  has a value other than x, then  ~ \sum _{i=1}^{n} \left( x_{i}-\overline{x} \right) ^{2}>0~ \\

\\ So,~ \sum _{i=1}^{n} \left( x_{i}-\overline{x} \right) ^{2}<~ \sum _{i=1}^{n} \left( x_{i}-a \right) ^{2}~~ \\ \\ \text{So, if }\mathrm{a}\text{~ has any value other than }\overline{x},~then~~ \sum _{i=1}^{n} \left( x_{i}-\overline{x} \right) ^{2}\text{~is less than } \sum _{i=1}^{n} \left( x_{i}-a \right) ^{2} \\

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