Show that the two formulae for the standard deviation of ungrouped data- -1- n - x i - X 2 and -1- n - x i 2- X 2 are equivalent- where X -1- n - x i

The correct option is σ = √(1/n∑ (x_i X)^2) = √(1/n∑ (x^2_i 2x_i X+ X)^2) = √(1/n∑ x^2_i 1/n∑ 2x_i X+ 1/n∑X^2) =√(1/n∑ x^2_i 1/n× 2X∑ x_i+1/n×X^2 ∑ 1 ) = ...

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Standard XII

Mathematics

Standard Deviation

Show that the...

Question

Show that the two formulae for the standard deviation of ungrouped data .

$σ = \sqrt{\frac{1}{n} \sum (x_{i} - ¯ ¯¯¯ ¯ X)^{2}} a n d σ^{'} = \sqrt{\frac{1}{n} \sum x_{i}^{2} - {¯ ¯¯¯ ¯ X}^{2}}$ are equivalent , where $¯ ¯¯¯ ¯ X = \frac{1}{n} \sum x_{i}$

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Solution

$σ = \sqrt{\frac{1}{n} \sum (x_{i} - ¯ ¯¯¯ ¯ X)^{2}}$

$= \sqrt{\frac{1}{n} \sum (x_{i}^{2} - 2 x_{i} ¯ ¯¯¯ ¯ X + ¯ ¯¯¯ ¯ X)^{2}} = \sqrt{\frac{1}{n} \sum x_{i}^{2} - \frac{1}{n} \sum 2 x_{i} ¯ ¯¯¯ ¯ X + \frac{1}{n} \sum {¯ ¯¯¯ ¯ X}^{2}}$

$= \sqrt{\frac{1}{n} \sum x_{i}^{2} - \frac{1}{n} \times 2 ¯ ¯¯¯ ¯ X \sum x_{i} + \frac{1}{n} \times {¯ ¯¯¯ ¯ X}^{2} \sum 1}$

$= \sqrt{\frac{1}{n} \sum x_{i}^{2} - \frac{1}{n} \times 2 ¯ ¯¯¯ ¯ X \times n ¯ ¯¯¯ ¯ X + \frac{1}{n} \times {¯ ¯¯¯ ¯ X}^{2} \times n}$ $(¯ ¯¯¯ ¯ X = \frac{1}{n} \sum x_{i})$

$= \sqrt{\frac{1}{n} \sum x_{i}^{2} - 2 ¯ ¯¯¯ ¯ X + {¯ ¯¯¯ ¯ X}^{2}} = \sqrt{\frac{1}{n} \sum x_{i}^{2} - {¯ ¯¯¯ ¯ X}^{2}} = σ^{1}$

Hence, the formula $σ = \sqrt{\frac{1}{n} \sum (x_{i} - ¯ ¯¯¯ ¯ X)^{2}} a n d σ = \sqrt{\frac{1}{n} \sum x_{i}^{2} - {¯ ¯¯¯ ¯ X}^{2}}$ are equivalent, where $¯ ¯¯¯ ¯ X = \frac{1}{n} \sum x_{i}$

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Similar questions

Q. Show that the two formulae for the standard deviation of ungrouped data

σ = \sqrt{\frac{1}{n} \underset{}{\sum {(x_{i} - X)}^{2}}}

and

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are equivalent, where

X = \frac{1}{n} \sum_{} x_{i}

PQ. Show that the two formulae for the standard deviation of ungrouped data $δ = \sqrt{\frac{1}{n} \sum (x_{i} - ¯ X)^{2}}$ and $δ = \sqrt{\frac{1}{n} \sum (x_{i} - ¯ X)^{2}}$ are equivalent, wehere $¯ X = \frac{1}{n} \sum x_{i}$ .

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For the given distinct values $x_{1}, x_{2}, x_{3}, . . . . . x_{n}$ occurring with frequencies $f_{1}, f_{2}, f_{3}, . . . f_{n}$ respectively. The mean deviation about mean where $¯ x$ is the mean and N is total number of observations would be

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Show that the two formulae for the standard deviation of ungrouped data . σ=√1n∑(xi−¯¯¯¯¯X)2 and σ′=√1n∑x2i−¯¯¯¯¯X2 are equivalent , where ¯¯¯¯¯X=1n∑xi

Show that the two formulae for the standard deviation of ungrouped data .

$σ = \sqrt{\frac{1}{n} \sum (x_{i} - ¯ ¯¯¯ ¯ X)^{2}} a n d σ^{'} = \sqrt{\frac{1}{n} \sum x_{i}^{2} - {¯ ¯¯¯ ¯ X}^{2}}$ are equivalent , where $¯ ¯¯¯ ¯ X = \frac{1}{n} \sum x_{i}$