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Mathematics

Assertion (A): If xi’s are the mid-points of the class intervals of a grouped data, Σfi’s are the corresponding frequencies and x̄ is the mean, then Σfi(xi − x̄) = 1.

Reason (R): The sum of the deviations from the mean is 0.

  1. A is true, R is false

  2. A is false, R is true

  3. Both A and R are true

  4. Both A and R are false

Measures of Central Tendency

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Answer

The expression ∑fi(xi−xˉ)\sum fi(xi - \bar{x}) represents the sum of the deviations of all observations from their mean, weighed by their frequencies.

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,

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

Assertion (A) is false.

The sum of the deviations from the mean is 0. This is a fundamental and correct property of the arithmetic mean.

Reason (R) is true.

Hence, option 2 is the correct option.

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