Question

Find mean median and mode of following data class interval 0-10 10-20 20-30 30-40 40-50 frequency 5,7,8,6,4​

Answer 1

Answer:

☯ Find mean, median and mode of following data:

$\boxed{\begin{array}{c|c|c|c|c|c}\bf Class\: interval&\sf 0-10&\sf 10-20&\sf 20-30 &\sf 30-40&\sf 40-50\\\frac{\qquad \quad \qquad}{}&\frac{\quad \qquad \qquad}{}&\frac{\qquad \quad\qquad}{}&\frac{\qquad \quad \qquad}{}&\frac{\quad \qquad \qquad}{}&\frac{\qquad \quad\qquad}{}\\\bf Frequency&\sf 5&\sf 7&\sf 8&\sf 6&\sf 4\end{array}}$

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I. Finding Mean

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$\boxed{\begin{array}{ccccc}\sf Class\: interval&\sf F_i&\sf Mid\:Value&\sf u_i = \dfrac{x_i + 25}{10}&\sf f_i u_i\\\frac{\quad \qquad\qquad}{}&\frac{\quad \qquad \qquad}{}&\frac{\qquad \qquad\qquad}{}&\frac{ \qquad \qquad \qquad}{}&\frac{\qquad \qquad\qquad}{}\\\sf 0-10&\sf 5&\sf 5&\sf -2&\sf -10\\\\\sf 10-20 &\sf 7&\sf 15&\sf -1 &\sf -7 \\\\\sf 20-30 &\sf 8&\sf A = 25&\sf 0&\sf 0\\\\\sf 30-40&\sf 6&\sf 35&\sf 1 &\sf 6\\\\\sf 40-50 &\sf 4 &\sf 45&\sf 2&\sf 8\\\frac{\qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad}{}&\frac{\qquad\qquad\qquad}{}&\frac{\qquad \qquad\qquad}{}&\frac{\qquad \qquad\qquad}{}\\\sf Total& \sf \sum f = 30&&& \sf -3\end{array}}$

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$\sf We\:have \begin{cases} & \sf{A = \bf{25}} \\ & \sf{ \sum f_i = \bf{30}} \\ & \sf{\sum f_i u_i = \bf{ - 3}} \\ & \sf{h = \bf{10}} \end{cases}$

Now,

$\dag\;{\underline{\frak{Formula\:of\:mean\:is\:given\:by,}}}$

$\star\;{\boxed{\sf{\pink{ \bar{x} = A + \dfrac{ \sum f_i u_i}{\sum f_i} \times h}}}}$

$:\implies\sf 25 + \bigg(\cancel{\dfrac{-3}{30}} \bigg)\times 10\\ \\ \\ :\implies\sf 25 - 1 \times 10\\ \\ \\ :\implies\sf 25 - 10\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{\bar{x} = 24}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Mean\:of\:given\:data\:is\: {\textsf{\textbf{24}}}.}}}$

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II. Finding Median

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$\boxed{\begin{array}{cccc}\sf Class\: interval&\sf Frequency&\sf C.F\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 0-10&\sf 5&\sf 5 \\\\\sf 10-20 &\sf 7&\sf 12 \\\\\sf 20-30 &\sf 8&\sf 20 \\\\\sf 30-40&\sf 6&\sf 26\\\\\sf 40-50 &\sf 4 &\sf 30\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\ &\sf \sum F_i = 30&\end{array}}$

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$\sf We\:have \begin{cases} & \sf{Lower\:limit,\:l = \bf{20}} \\ & \sf{Comulative\: frequency,\:C.F. = \bf{12}} \\ & \sf{N = \sum f_i = \bf{30}} \\ & \sf{Class\:interval,\:h = 30 - 20 = \bf{10}} \\ & \sf{Frequency,\:f = \bf{8}} \end{cases}$

Now,

$\dag\;{\underline{\frak{Formula\:of\:Median\:is\:given\:by,}}}$

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

$:\implies\sf 20 + \dfrac{ \frac{30}{2} - 12}{8} \times 10\\ \\ \\ :\implies\sf 20 - \dfrac{15 - 12}{8} \times 10\\ \\ \\ :\implies\sf 20 + \dfrac{3}{ \cancel{8}} \times \cancel{10}\\ \\ \\ :\implies\sf 20 + \dfrac{3}{4} \times 5\\ \\ \\ :\implies\sf 20 + \dfrac{15}{4}\\ \\ \\ :\implies\sf 20 + 3.75\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{23.75}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Median\:of\:given\:data\:is\: {\textsf{\textbf{23.75}}}.}}}$

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III. Finding Mode

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$\boxed{\begin{array}{cccc}\sf Class\: interval&\sf Frequency\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}\\\sf 0-10&\sf 5\\\\\sf 10-20 &\sf 7\\\\\sf 20-30 &\sf 8\\\\\sf 30-40&\sf 6\\\\\sf 40-50 &\sf 4\end{array}}$

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Here,

$\sf l = Lower\:limit\:of\;class,\:l = \bf{20}$

$\sf Class\:interval,\:h = 30 - 20 = \bf{10}$

$\sf Frequency\:of\:Median\;class\:,f_1 = \bf{8}$

$\sf Frequency\:of\:preceding\;class\:,f_0 = \bf{7}$

$\sf Frequency\:of\: succeeding\;class\:,f_2 = \bf{6}$

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$\dag\;{\underline{\frak{Formula\:of\:Mode\:is\:given\:by,}}}$

$\star\;{\boxed{\sf{\pink{Mode = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h}}}}$

$:\implies\sf 20 + \dfrac{8 - 7}{2 \times 8 - 7 - 6} \times 10\\ \\ \\:\implies\sf 20 + \dfrac{1}{16 - 7 - 6} \times 10\\ \\ \\:\implies\sf 20 + \dfrac{1}{16 - 7 - 6} \times 10\\ \\ \\:\implies\sf 20 + \dfrac{1}{9 - 6} \times 10\\ \\ \\:\implies\sf 20 + \dfrac{1}{3} \times 10\\ \\ \\ :\implies\sf 20 + \dfrac{10}{3}\\ \\ \\:\implies\sf 20 + 3.33\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{23.33}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Median\:of\:given\:data\:is\: {\textsf{\textbf{23.33}}}.}}}$

Answer 2

☯ Find mean, median and mode of following data:

$\boxed{\begin{array}{c|c|c|c|c|c}\bf Class\: interval&\sf 0-10&\sf 10-20&\sf 20-30 &\sf 30-40&\sf 40-50\\\frac{\qquad \quad \qquad}{}&\frac{\quad \qquad \qquad}{}&\frac{\qquad \quad\qquad}{}&\frac{\qquad \quad \qquad}{}&\frac{\quad \qquad \qquad}{}&\frac{\qquad \quad\qquad}{}\\\bf Frequency&\sf 5&\sf 7&\sf 8&\sf 6&\sf 4\end{array}}$

━━━━━━━━━━━━━━━━━━━━

I. Finding Mean

⠀⠀⠀⠀

$\boxed{\begin{array}{ccccc}\sf Class\: interval&\sf F_i&\sf Mid\:Value&\sf u_i = \dfrac{x_i + 25}{10}&\sf f_i u_i\\\frac{\quad \qquad\qquad}{}&\frac{\quad \qquad \qquad}{}&\frac{\qquad \qquad\qquad}{}&\frac{ \qquad \qquad \qquad}{}&\frac{\qquad \qquad\qquad}{}\\\sf 0-10&\sf 5&\sf 5&\sf -2&\sf -10\\\\\sf 10-20 &\sf 7&\sf 15&\sf -1 &\sf -7 \\\\\sf 20-30 &\sf 8&\sf A = 25&\sf 0&\sf 0\\\\\sf 30-40&\sf 6&\sf 35&\sf 1 &\sf 6\\\\\sf 40-50 &\sf 4 &\sf 45&\sf 2&\sf 8\\\frac{\qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad}{}&\frac{\qquad\qquad\qquad}{}&\frac{\qquad \qquad\qquad}{}&\frac{\qquad \qquad\qquad}{}\\\sf Total& \sf \sum f = 30&&& \sf -3\end{array}}$

⠀⠀⠀⠀

$\sf We\:have \begin{cases} & \sf{A = \bf{25}} \\ & \sf{ \sum f_i = \bf{30}} \\ & \sf{\sum f_i u_i = \bf{ - 3}} \\ & \sf{h = \bf{10}} \end{cases}$

Now,

$\dag\;{\underline{\frak{Formula\:of\:mean\:is\:given\:by,}}}$

$\star\;{\boxed{\sf{\pink{ \bar{x} = A + \dfrac{ \sum f_i u_i}{\sum f_i} \times h}}}}$

$:\implies\sf 25 + \bigg(\cancel{\dfrac{-3}{30}} \bigg)\times 10\\ \\ \\ :\implies\sf 25 - 1 \times 10\\ \\ \\ :\implies\sf 25 - 10\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{\bar{x} = 24}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Mean\:of\:given\:data\:is\: {\textsf{\textbf{24}}}.}}}$

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II. Finding Median

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$\boxed{\begin{array}{cccc}\sf Class\: interval&\sf Frequency&\sf C.F\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 0-10&\sf 5&\sf 5 \\\\\sf 10-20 &\sf 7&\sf 12 \\\\\sf 20-30 &\sf 8&\sf 20 \\\\\sf 30-40&\sf 6&\sf 26\\\\\sf 40-50 &\sf 4 &\sf 30\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\ &\sf \sum F_i = 30&\end{array}}$

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$\sf We\:have \begin{cases} & \sf{Lower\:limit,\:l = \bf{20}} \\ & \sf{Comulative\: frequency,\:C.F. = \bf{12}} \\ & \sf{N = \sum f_i = \bf{30}} \\ & \sf{Class\:interval,\:h = 30 - 20 = \bf{10}} \\ & \sf{Frequency,\:f = \bf{8}} \end{cases}$

Now,

$\dag\;{\underline{\frak{Formula\:of\:Median\:is\:given\:by,}}}$

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

$:\implies\sf 20 + \dfrac{ \frac{30}{2} - 12}{8} \times 10\\ \\ \\ :\implies\sf 20 - \dfrac{15 - 12}{8} \times 10\\ \\ \\ :\implies\sf 20 + \dfrac{3}{ \cancel{8}} \times \cancel{10}\\ \\ \\ :\implies\sf 20 + \dfrac{3}{4} \times 5\\ \\ \\ :\implies\sf 20 + \dfrac{15}{4}\\ \\ \\ :\implies\sf 20 + 3.75\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{23.75}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Median\:of\:given\:data\:is\: {\textsf{\textbf{23.75}}}.}}}$

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III. Finding Mode

$\boxed{\begin{array}{cccc}\sf Class\: interval&\sf Frequency\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}\\\sf 0-10&\sf 5\\\\\sf 10-20 &\sf 7\\\\\sf 20-30 &\sf 8\\\\\sf 30-40&\sf 6\\\\\sf 40-50 &\sf 4\end{array}}$

Here,

$\sf l = Lower\:limit\:of\;class,\:l = \bf{20}$

$\sf Class\:interval,\:h = 30 - 20 = \bf{10}$

$\sf Frequency\:of\:Median\;class\:,f_1 = \bf{8}$

$\sf Frequency\:of\:preceding\;class\:,f_0 = \bf{7}$

$\sf Frequency\:of\: succeeding\;class\:,f_2 = \bf{6}$

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$\dag\;{\underline{\frak{Formula\:of\:Mode\:is\:given\:by,}}}$

$\star\;{\boxed{\sf{\pink{Mode = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h}}}}$

$:\implies\sf 20 + \dfrac{8 - 7}{2 \times 8 - 7 - 6} \times 10\\ \\ \\:\implies\sf 20 + \dfrac{1}{16 - 7 - 6} \times 10\\ \\ \\:\implies\sf 20 + \dfrac{1}{16 - 7 - 6} \times 10\\ \\ \\:\implies\sf 20 + \dfrac{1}{9 - 6} \times 10\\ \\ \\:\implies\sf 20 + \dfrac{1}{3} \times 10\\ \\ \\ :\implies\sf 20 + \dfrac{10}{3}\\ \\ \\:\implies\sf 20 + 3.33\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{23.33}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Median\:of\:given\:data\:is\: {\textsf{\textbf{23.33}}}.}}}$