Class | 500-600 | 600-700 | 700-800 | 800-900 | 900-1000| Frequency| 36. | 32 | 32. | 20. | 30.

Question

Class | 500-600 | 600-700 | 700-800 | 800-900 | 900-1000|
Frequency| 36. | 32 | 32. | 20. | 30. |

Find the median of given distribution.​

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Iris 1 month 2021-08-13T06:43:39+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-08-13T06:44:42+00:00

    Question :

    Find the median of given distribution :

    \underbrace{\overbrace{\begin{tabular}{|c|c|}\cline{1-2}Class &  Frequency  \\\cline{1-2} 500 - 600 & 36  \\ 600 - 700 & 32  \\ 700 - 800 & 32  \\800 - 900 & 20  \\ 900 - 1000 & 30  \\\cline{1-2}\end{tabular}}}}

    Theory :

    Steps to find Median of a grouped distribution

    1) Find the cumulative frequency ( c.f)

    2) Find \sf\dfrac{N}{2} where , \sf\:N=\sum\:F

    3) See the c.f just greater than N/2 and determine the corresponding class

    4) Then , use Formula of median;

    {\purple{\boxed{\large{\bold{Median=l+(\dfrac{\frac{N}{2}-F}{f})\times\:h}}}}}

    Here ,l = lower limit of the median class

    f=frequency of the median class

    h= size of the median class

    F= c.f of thr class of the class preceding the median class \sf\:N=\sum\:F

    Solution :

    We have to find the median of the given distribution.

    Let’s solve the problem :

    First , we have to find c.f to compute the median :

    \boxed{\begin{array}{cccc}\sf Class\: interval&\sf Frequency\: (f) &\sf C.F\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 500 - 600&\sf 36&\sf 36\\\\\sf 600 - 700 &\sf 32&\sf 68\\\\\sf 700-800 &\sf 32&\sf 100\\\\\sf 800 - 900&\sf 20&\sf 120\\\\\sf 900-1000&\sf 30&\sf 150\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf & \sf \sum f = 150& \end{array}}

    We have ,

    \sf\sum\:F=N=150

    \sf\implies\dfrac{N}{2}=75

    The cumulative frequency just greater than N/2 is 100 and the corresponding corresponding class is 700-800 . Therefore , 700-800 is the median class :

    Thus , l = 700

    h = 100

    F = 68

    f= 32

    \tt\:Median=l+[\dfrac{\frac{N}{2}-F}{f}]\times\:h

    \sf=700+\dfrac{75-68}{32}\times100

    \sf=700+\dfrac{7}{32}\times100

    \sf=700+0.218\times100

    \sf=700+21.8

    \sf=721.8

    Therefore, Median of given distribution is 721.8

    0
    2021-08-13T06:44:55+00:00

    \begin{gathered}\begin{tabular}{|c|c|c|c|c|c|}\cline{1-6} \tt Class & \tt 500-600 & \tt 600-700 & \tt 700-800 & \tt 800-900 & \tt 900-100 \\\cline{1-6}\tt Frequency &\tt 36 & \tt 32& \tt 32 & \tt 20 & \tt 30 \\\cline{1-6}\end{tabular}\end{gathered}

    ⠀⠀⠀⠀⠀⠀⠀

    ☯ We have to find, Median of given distribution.

    ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

    \boxed{\begin{array}{cccc}\bf Class\: interval&\bf Frequency\: (f)& \bf C.F\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 500 - 600&\sf 36&\sf 36\\\\\sf 600 - 700 &\sf 32&\sf 68\\\\\sf 700-800 &\sf 32&\sf 100\\\\\sf 800 - 900&\sf 20&\sf 120\\\\\sf 900-1000&\sf 30&\sf 150\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\bf & \bf \sum f = 150& \end{array}}

    ⠀⠀⠀⠀⠀⠀⠀

    \dag\;{\underline{\frak{Formula\;to\:find\;Median,}}}

    ⠀⠀⠀⠀⠀⠀⠀

    \star\;{\boxed{\sf{\pink{l = \dfrac{ \frac{n}{2} - C.F.}{f} \times h}}}}

    ⠀⠀⠀⠀⠀⠀⠀

    Firstly we have to calculate \sf \dfrac{n}{2}, (where N = \sf \sum F) = \sf \dfrac{150}{2} = \bf{75}.

    ⠀⠀⠀⠀⠀⠀⠀

    So, The value of comulative frequency just greater than or equal to 75 is 100.

    ⠀⠀⠀⠀⠀⠀⠀

    \dag\;{\underline{\frak{We\;know\;that,}}}

    ⠀⠀⠀⠀⠀⠀⠀

    \boxed{\begin{minipage}{6cm}$\bigstar$\:\:\sf Median = l + $\sf\dfrac{\frac{n}{2}-C.f.}{f}\times h\\\\Here: \\1)\:n = \sum f =150\\2)\:l=Lower\:limit\:of\:median\:class=700\\3)\:C.f.=Cumulative\:frequency\:of\:class\\preceeding\:the\:median\:class=68\\4)\:f= frequency\:of\:median\:class=32\\5)\:h= Class\:interval =700-800 = 100\end{minipage}}

    ⠀⠀⠀⠀⠀⠀⠀

    {\underline{\sf{\bigstar\;Putting\;values\;in\;formula\;:}}}

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies\sf 700 + \dfrac{ \frac{150}{2} - 68}{32} \times 100

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies\sf 700 + \dfrac{ 75 - 68}{32} \times 100

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies\sf 700 + \dfrac{7}{32} \times 100

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies\sf 700 + 21.872

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies{\underline{\boxed{\frak{\purple{721.875}}}}}\;\bigstar

    ⠀⠀⠀⠀⠀⠀⠀

    \therefore\;{\underline{\sf{Median\;of\;given\; distribution\;is\; \textbf{721.875}}}}

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