Find the median of the following data (1-8)

\displaystyle \textbf{Question 1: }\text{Find the median of the following data: }83,37,70,29,45,63,41,70,34,54.
\displaystyle \text{Answer:}
\displaystyle \text{Arranging the data in ascending order:}
\displaystyle 29,34,37,41,45,54,63,70,70,83
\displaystyle n=10,\text{ which is even.}
\displaystyle \text{Median}=\frac{1}{2}\left\{\left(\frac{n}{2}\right)^{\text{th}}\text{ term}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}\left\{5^{\text{th}}\text{ term}+6^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}(45+54)=49.5
\displaystyle \therefore \text{The median is }49.5.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Find the median of the following data: }133,73,89,108,94,104,94,85,100,120.
\displaystyle \text{Answer:}
\displaystyle \text{Arranging the data in ascending order:}
\displaystyle 73,85,89,94,94,100,104,108,120,133
\displaystyle n=10,\text{ which is even.}
\displaystyle \text{Median}=\frac{1}{2}\left\{\left(\frac{n}{2}\right)^{\text{th}}\text{ term}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}\left\{5^{\text{th}}\text{ term}+6^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}(94+100)=97
\displaystyle \therefore \text{The median is }97.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Find the median of the following data: }31,38,27,28,36,25,35,40.
\displaystyle \text{Answer:}
\displaystyle \text{Arranging the data in ascending order:}
\displaystyle 25,27,28,31,35,36,38,40
\displaystyle n=8,\text{ which is even.}
\displaystyle \text{Median}=\frac{1}{2}\left\{\left(\frac{n}{2}\right)^{\text{th}}\text{ term}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}\left\{4^{\text{th}}\text{ term}+5^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}(31+35)=33
\displaystyle \therefore \text{The median is }33.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Find the median of the following data: }15,6,16,8,22,21,9,18,25.
\displaystyle \text{Answer:}
\displaystyle \text{Arranging the data in ascending order:}
\displaystyle 6,8,9,15,16,18,21,22,25
\displaystyle n=9,\text{ which is odd.}
\displaystyle \text{Median}=\left(\frac{n+1}{2}\right)^{\text{th}}\text{ term}
\displaystyle =\left(\frac{9+1}{2}\right)^{\text{th}}\text{ term}=5^{\text{th}}\text{ term}
\displaystyle =16
\displaystyle \therefore \text{The median is }16.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Find the median of the following data: }41,43,127,99,71,92,71,58,57.
\displaystyle \text{Answer:}
\displaystyle \text{Arranging the data in ascending order:}
\displaystyle 41,43,57,58,71,71,92,99,127
\displaystyle n=9,\text{ which is odd.}
\displaystyle \text{Median}=\left(\frac{n+1}{2}\right)^{\text{th}}\text{ term}
\displaystyle =\left(\frac{9+1}{2}\right)^{\text{th}}\text{ term}=5^{\text{th}}\text{ term}
\displaystyle =71
\displaystyle \therefore \text{The median is }71.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Find the median of the following data: }25,34,31,23,22,26,35,29,20,32.
\displaystyle \text{Answer:}
\displaystyle \text{Arranging the data in ascending order:}
\displaystyle 20,22,23,25,26,29,31,32,34,35
\displaystyle n=10,\text{ which is even.}
\displaystyle \text{Median}=\frac{1}{2}\left\{\left(\frac{n}{2}\right)^{\text{th}}\text{ term}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}\left\{5^{\text{th}}\text{ term}+6^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}(26+29)=27.5
\displaystyle \therefore \text{The median is }27.5.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Find the median of the following data: }12,17,3,14,5,8,7,15.
\displaystyle \text{Answer:}
\displaystyle \text{Arranging the data in ascending order:}
\displaystyle 3,5,7,8,12,14,15,17
\displaystyle n=8,\text{ which is even.}
\displaystyle \text{Median}=\frac{1}{2}\left\{\left(\frac{n}{2}\right)^{\text{th}}\text{ term}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}\left\{4^{\text{th}}\text{ term}+5^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}(8+12)=10
\displaystyle \therefore \text{The median is }10.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Find the median of the following data: }92,35,67,85,72,81,56,51,42,69.
\displaystyle \text{Answer:}
\displaystyle \text{Arranging the data in ascending order:}
\displaystyle 35,42,51,56,67,69,72,81,85,92
\displaystyle n=10,\text{ which is even.}
\displaystyle \text{Median}=\frac{1}{2}\left\{\left(\frac{n}{2}\right)^{\text{th}}\text{ term}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}\left\{5^{\text{th}}\text{ term}+6^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}(67+69)=68
\displaystyle \therefore \text{The median is }68.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{The numbers }50,42,35,2x+10,2x+1,12,11,8,6\text{ are written in}
\displaystyle \text{descending order. If their median is }25,\text{ find }x.
\displaystyle \text{Answer:}
\displaystyle n=9,\text{ which is odd.}
\displaystyle \text{Median}=\left(\frac{n+1}{2}\right)^{\text{th}}\text{ term}=5^{\text{th}}\text{ term}
\displaystyle \therefore 2x+1=25
\displaystyle 2x=24
\displaystyle x=12
\displaystyle \therefore x=12.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Find the median of the following observations: }46,64,87,41,58,77,35,90,55,92,33.
\displaystyle \text{If }92\text{ is replaced by }99\text{ and }41\text{ by }43,\text{ find the new median.}
\displaystyle \text{Answer:}
\displaystyle \text{Arranging the data in ascending order:}
\displaystyle 33,35,41,46,55,58,64,77,87,90,92
\displaystyle n=11,\text{ which is odd.}
\displaystyle \text{Median}=\left(\frac{n+1}{2}\right)^{\text{th}}\text{ term}
\displaystyle =\left(\frac{11+1}{2}\right)^{\text{th}}\text{ term}=6^{\text{th}}\text{ term}
\displaystyle =58
\displaystyle \text{After replacing }92\text{ by }99\text{ and }41\text{ by }43,
\displaystyle \text{the data in ascending order are }33,35,43,46,55,58,64,77,87,90,99.
\displaystyle n=11,\text{ which is odd.}
\displaystyle \text{Median}=\left(\frac{11+1}{2}\right)^{\text{th}}\text{ term}=6^{\text{th}}\text{ term}
\displaystyle =58
\displaystyle \therefore \text{The new median is }58.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{Find the median of the following data: }41,43,127,99,61,92,71,58,57.
\displaystyle \text{If }58\text{ is replaced by }85,\text{ what will be the new median?}
\displaystyle \text{Answer:}
\displaystyle \text{Arranging the data in ascending order:}
\displaystyle 41,43,57,58,61,71,92,99,127
\displaystyle n=9,\text{ which is odd.}
\displaystyle \text{Median}=\left(\frac{n+1}{2}\right)^{\text{th}}\text{ term}
\displaystyle =\left(\frac{9+1}{2}\right)^{\text{th}}\text{ term}=5^{\text{th}}\text{ term}
\displaystyle =61
\displaystyle \text{After replacing }58\text{ by }85,\text{ the data in ascending order are}
\displaystyle 41,43,57,61,71,85,92,99,127.
\displaystyle n=9,\text{ which is odd.}
\displaystyle \text{Median}=\left(\frac{9+1}{2}\right)^{\text{th}}\text{ term}=5^{\text{th}}\text{ term}
\displaystyle =71
\displaystyle \therefore \text{The new median is }71.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{The weights (in kg) of }15\text{ students are }31,35,27,29,32,43,37,41,34,28,
\displaystyle 36,44,45,42,30.\text{ Find the median. If }44\text{ kg is replaced by }46\text{ kg and }27\text{ kg}
\displaystyle \text{by }25\text{ kg, find the new median.}
\displaystyle \text{Answer:}
\displaystyle \text{Arranging the data in ascending order:}
\displaystyle 27,28,29,30,31,32,34,35,36,37,41,42,43,44,45
\displaystyle n=15,\text{ which is odd.}
\displaystyle \text{Median}=\left(\frac{n+1}{2}\right)^{\text{th}}\text{ term}
\displaystyle =\left(\frac{15+1}{2}\right)^{\text{th}}\text{ term}=8^{\text{th}}\text{ term}
\displaystyle =35\text{ kg}
\displaystyle \text{After replacing }44\text{ kg by }46\text{ kg and }27\text{ kg by }25\text{ kg,}
\displaystyle \text{the data in ascending order are }25,28,29,30,31,32,34,35,36,37,41,42,43,45,46.
\displaystyle n=15,\text{ which is odd.}
\displaystyle \text{Median}=8^{\text{th}}\text{ term}=35\text{ kg}
\displaystyle \therefore \text{The original median and the new median are both }35\text{ kg.}
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{The following observations are arranged in ascending order. If the median is }63,
\displaystyle \text{find }x: \ 29,32,48,50,x,x+2,72,78,84,95.
\displaystyle \text{Answer:}
\displaystyle n=10,\text{ which is even.}
\displaystyle \text{Median}=\frac{1}{2}\left\{\left(\frac{n}{2}\right)^{\text{th}}\text{ term}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{ term}\right\}
\displaystyle 63=\frac{1}{2}\left\{5^{\text{th}}\text{ term}+6^{\text{th}}\text{ term}\right\}
\displaystyle 63=\frac{x+(x+2)}{2}
\displaystyle 126=2x+2
\displaystyle 2x=124
\displaystyle x=62
\displaystyle \therefore x=62.
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{If the mean of the observations }45,52,60,x,69,70,76,81,94\text{ is }68,
\displaystyle \text{find its median.}
\displaystyle \text{Answer:}
\displaystyle n=9,\qquad \text{Mean}=68
\displaystyle 68=\frac{45+52+60+x+69+70+76+81+94}{9}
\displaystyle 612=547+x
\displaystyle x=65
\displaystyle \text{Arranging the observations in ascending order:}
\displaystyle 45,52,60,65,69,70,76,81,94
\displaystyle n=9,\text{ which is odd.}
\displaystyle \text{Median}=\left(\frac{n+1}{2}\right)^{\text{th}}\text{ term}
\displaystyle =\left(\frac{9+1}{2}\right)^{\text{th}}\text{ term}=5^{\text{th}}\text{ term}
\displaystyle =69
\displaystyle \therefore \text{The median is }69.
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{The numbers }6,8,10,12,13\text{ and }x\text{ are arranged in ascending order.}
\displaystyle \text{If the mean of these observations is equal to the median, find the value of }x.
\displaystyle \text{Answer:}
\displaystyle n=6,\text{ which is even.}
\displaystyle \text{Median}=\frac{1}{2}\left\{3^{\text{rd}}\text{ term}+4^{\text{th}}\text{ term}\right\}
\displaystyle =\frac{1}{2}(10+12)=11
\displaystyle \text{Since mean}=\text{median},
\displaystyle 11=\frac{6+8+10+12+13+x}{6}
\displaystyle 66=49+x
\displaystyle x=17
\displaystyle \therefore x=17.
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{The median of the observations }11,12,14,(x-2),(x+4),(x+9),32,38,47,
\displaystyle \text{arranged in ascending order is }24.\text{ Find }x\text{ and hence find the mean.}
\displaystyle \text{Answer:}
\displaystyle n=9,\text{ which is odd.}
\displaystyle \text{Median}=\left(\frac{n+1}{2}\right)^{\text{th}}\text{ term}=5^{\text{th}}\text{ term}
\displaystyle \therefore x+4=24
\displaystyle x=20
\displaystyle \text{Substituting }x=20,\text{ the observations are}
\displaystyle 11,12,14,18,24,29,32,38,47.
\displaystyle \text{Mean}=\frac{11+12+14+18+24+29+32+38+47}{9}
\displaystyle =\frac{225}{9}=25
\displaystyle \therefore x=20\text{ and the mean is }25.
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{If the mean of the observations }8,10,7,6,10,x,6,13,10\text{ is }9,
\displaystyle \text{find the median.}
\displaystyle \text{Answer:}
\displaystyle n=9,\qquad \text{Mean}=9
\displaystyle 9=\frac{8+10+7+6+10+x+6+13+10}{9}
\displaystyle 81=70+x
\displaystyle x=11
\displaystyle \text{Arranging the observations in ascending order:}
\displaystyle 6,6,7,8,10,10,10,11,13
\displaystyle n=9,\text{ which is odd.}
\displaystyle \text{Median}=\left(\frac{n+1}{2}\right)^{\text{th}}\text{ term}
\displaystyle =\left(\frac{9+1}{2}\right)^{\text{th}}\text{ term}=5^{\text{th}}\text{ term}
\displaystyle =10
\displaystyle \therefore \text{The median is }10.
\displaystyle \\

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