the following observation are arranged in the ascending order. if the median of these observations is 51, then find the value of x: 26, 33,38,x+1,x+3, 53,57,62,67
Answers
x+3
given median =51
x+3=51
x=51-3
x=48
Given observation is arranged in ascending order.
》36, 51, 52, x, x + 4, 68, 82, 96.
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\begin{gathered}\frak{We\:have} \begin{cases} \sf Median\:of\:data\: = \frak{62} & \\ \\ \sf Number\: of\: observations,\:(n)\: = \frak{8}& \end{cases}\\\\\end{gathered}
Wehave
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Medianofdata=62
Numberofobservations,(n)=8
¤ Formula to find Median for an ( n = even number ) is,
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\begin{gathered}\star\:{\underline{\boxed{\frak{\purple{Median = \dfrac{\bigg( \frac{n}{2} \bigg)^{th} + \bigg( \frac{n}{2} + 1 \bigg)^{th}\:observation}{2}}}}}}\\\\\end{gathered}
⋆
Median=
2
(
2
n
)
th
+(
2
n
+1)
th
observation
\begin{gathered}\bf{\dag}\:{\underline{\frak{Now,\:Putting\: Given\: values\: in\: formula,}}}\\\end{gathered}
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Now,PuttingGivenvaluesinformula,
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\begin{gathered}:\implies\sf 62 = \dfrac{\bigg( \cancel{\frac{8}{2}} \bigg)^{th} + \bigg( \cancel{\frac{8}{2}} + 1 \bigg)^{th}\:observation}{2}\\\\\\ :\implies\sf 62 = \dfrac{4^{th} + 5^{th}\:observation}{2}\\\\\\ :\implies\sf 62 = \dfrac{(x) + (x + 4)}{2}\\\\\\ :\implies\sf 62 = \dfrac{2x + 4}{2}\\\\\\ :\implies\sf 62 \times 2 = 2x + 4\\\\\\\ :\implies\sf 124 = 2x + 4\\\\\\\ :\implies\sf 124 - 4 = 2x\\\\\\\ :\implies\sf 120 = 2x\\\\\\ :\implies\sf x = \cancel{\dfrac{120}{2}}\\\\\\ :\implies{\underline{\boxed{\frak{\pink{x = 60}}}}}\:\bigstar\\\\\end{gathered}
:⟹62=
2
(
2
8
)
th
+(
2
8
+1)
th
observation
:⟹62=
2
4
th
+5
th
observation
:⟹62=
2
(x)+(x+4)
:⟹62=
2
2x+4
:⟹62×2=2x+4
:⟹124=2x+4
:⟹124−4=2x
:⟹120=2x
:⟹x=
2
120
:⟹
x=60
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\qquad\qquad\qquad\therefore\:{\underline{\sf{The\:value\:of\:x\:is\:{\textsf{\textbf{60}}}.}}}∴