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Mathematics

If xi's are the class marks of the class-intervals of grouped data, fi's are the corresponding frequencies and x̄ is the mean, then Σ fi(xi − x̄) is equal to :

  1. −1

  2. 0

  3. 1

  4. 2

Measures of Central Tendency

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Answer

Given,

⇒∑fi(xi−xˉ)=∑fixi−fixˉ⇒∑fi(xi−xˉ)=∑fixi−∑fixˉ⇒∑fi(xi−xˉ)=∑fixi−xˉ∑fi …..(1)\Rightarrow \sum fi(xi - \bar x) = \sum fixi - fi \bar x \\[1em] \Rightarrow \sum fi(xi - \bar x) = \sum fixi - \sum fi \bar x \\[1em] \Rightarrow \sum fi(xi - \bar x) = \sum fixi - \bar x \sum f_i \text{ …..(1)}

We know that,

xˉ=∑fixi∑fixˉ∑fi=∑fixi\bar x = \dfrac{\sum fi xi}{\sum fi} \\[1em] \bar x \sum fi = \sum fi xi

Substituting the values in equation (1), we get :

∴∑fi(xi−xˉ)=xˉ∑fi−xˉ∑fi=0\therefore \sum fi(xi - \bar x) =\bar x \sum fi - \bar x \sum fi = 0

Hence, option 2 is the correct option.

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