If the variance of the frequency distribution x _ i 2 3 4 5 6 7 8 Frequency f _ i 3 6 16 α 9 5 6 is 3 , then α is equal to

Correct answer: 0 We have, x _ i f _ i d _ i = x _ i - 5 f _ i d _ i ^ 2 f _ i d _ i 2 3 - 3 27 - 9 3 6 - 2 24 - 12 4 16 - 1 16 - 16 5 α 0 0 0 6 9 1 9 9 7 5 2 20 10 8 6 3 54 18 σ _ x ^ 2 = σ _ d ^ 2 = ∑ f _ i d _ i ^ 2 ∑ f _ i - ( ∑ f _ i d _ i ∑ f _ i ) ^ 2 ⇒ 3 = (150)/(45 + α) - 0 ⇒ 150 = 135 + 3α ⇒ 3α = 15 ⇒ α = 5

JEE Main 202331 JanuaryMorning Shift

Exam: JEE Main
Year: 2023
Date: 31 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Statistics

JEE Main 2023 · 31 January · Morning Shift · Mathematics · Statistics

If the variance of the frequency distribution x _ i 2 3 4 5 6 7 8...

If the variance of the frequency distribution

${x}_{i}$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
Frequency ${f}_{i}$	$3$	$6$	$16$	$\alpha$	$9$	$5$	$6$

is $3$ , then $\alpha$ is equal to

Correct answer

Step-by-step solution

We have,

${x}_{i}$	${f}_{i}$	${d}_{i} = {x}_{i} - 5$	${f}_{i}{d}_{i}^{2}$	${f}_{i}{d}_{i}$
$2$	$3$	$- 3$	$27$	$- 9$
$3$	$6$	$- 2$	$24$	$- 12$
$4$	$16$	$- 1$	$16$	$- 16$
$5$	$\alpha$	$0$	$0$	$0$
$6$	$9$	$1$	$9$	$9$
$7$	$5$	$2$	$20$	$10$
$8$	$6$	$3$	$54$	$18$

${\sigma}_{x}^{2} = {\sigma}_{d}^{2} = \dfrac{∑ {f}_{i}{d}_{i}^{2}}{∑ {f}_{i}} - {\left(\dfrac{∑ {f}_{i}{d}_{i}}{∑ {f}_{i}}\right)}^{2}$

$⇒ 3 = \dfrac{150}{45 + \alpha} - 0$

$⇒ 150 = 135 + 3\alpha$

$⇒ 3\alpha = 15 ⇒ \alpha = 5$

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