Variance and standard deviation for following data: {xᵢ} 4 8 11 17 20 24 32 {fᵢ} 3 5 9 5 4 3 1

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13.Statistics

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Find the variance and standard deviation for the following data:

${x_i}$ $4$ $8$ $11$ $17$ $20$ $24$ $32$

${f_i}$ $3$ $5$ $9$ $5$ $4$ $3$ $1$

A

$6.77$

B

$6.77$

C

$6.77$

D

$6.77$

Solution

Presenting the data in tabular form (Table), we get

${x_i}$	${f_i}$	${f_i}{x_i}$	${{x_i} – \bar x}$	${\left( {{x_i} – \bar x} \right)^2}$	${f_i}{\left( {{x_i} – \bar x} \right)^2}$
$4$	$3$	$12$	$-10$	$100$	$300$
$8$	$5$	$40$	$-6$	$36$	$180$
$11$	$9$	$99$	$-3$	$9$	$81$
$17$	$5$	$85$	$3$	$9$	$45$
$20$	$4$	$80$	$6$	$36$	$144$
$24$	$3$	$72$	$10$	$100$	$300$
$32$	$1$	$32$	$18$	$324$	$324$
	$30$	$420$			$1374$

$N = 30,\sum\limits_{i = 1}^7 {{f_i}{x_i}} = 420,\sum\limits_{i = 1}^7 {{f_i}{{\left( {{x_i} – \bar x} \right)}^2} = 1374} $

Therefore $\bar x = \frac{{\sum\limits_{i = 1}^7 {{f_i}{x_i}} }}{N} = \frac{1}{{30}} \times 420 = 14$

Hence Variance $\left( {{\sigma ^2}} \right) = \frac{1}{N}\sum\limits_{i = 1}^7 {{f_i}{{\left( {{x_i} – \bar x} \right)}^2}} $

$\left( {{\sigma ^2}} \right) = \frac{1}{N}\sum\limits_{i = 1}^7 {{f_i}{{\left( {{x_i} – \bar x} \right)}^2}} $

and Standard deviation $\left( \sigma \right) = \sqrt {45.8} = 6.77$

Standard 11

Mathematics

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