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

Question 1: $83, 37, 70, 29, 45, 63, 41, 70, 34, 54$

Population size: $10$

Ascending order: $29,34,37,41,45,54,63,70,70,83$

Median $= 49.5$

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Question 2: $133, 73, 89, 108, 94, 104, 94, 85, 100, 120$

Population size: $10$

Ascending order: $73,85,89,94,94,100,104,108,120,133$

Median $= 97$

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Question 3: $31, 38, 27, 28, 36, 25, 35, 40$

Population size: $8$

Ascending order: $25,27,28,31,35,36,38,40$

Median $= 33$

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Question 4: $15, 6, 16,8, 22, 21, 9, 18, 25$

Population size: $9$

Ascending order: $6,8,9,15,16,18,21,22,25$

Median $= 16$

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Question 5: $41, 43, 127, 99, 71, 92, 71, 58, 57$

Population size: $9$

Ascending order: $43,47,57,58,71,71,92,99,127$

Median $= 71$

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Question 6: $25, 34, 31, 23, 22, 26, 35, 29, 20, 32$

Population size: $10$

Ascending order: $20,22,23,25,26,29,31,32,34,35$

Median $= 27.5$

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Question 7: $12, 17, 3, 14,5, 8, 7, 15$

Population size: $8$

Ascending order: $3,5,7,8,12,14,15,17$

Median $= 10$

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Question 8:  $92, 35, 67, 85, 72, 81, 56, 51, 42, 69$

Population size: $10$

Ascending order: $35,42,51,56,67,69,72,81,85,92$

Median $= 68$

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Question 9: Numbers $50, 42, 35, 2x+10, 2x+1, 12,11, 8, 6$ are written in descending order and their median is $25$, find $x$.

Population size: $9$

Therefore, the Median is the $5^{th}$ term.

Hence $2x+1 = 25 \Rightarrow 2x = 24 \Rightarrow x = 12$

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Question 10: Find the median of the following observations: $46, 64, 87, 41, 58, 77, 35, 90, 55, 92, 33$. If $92$ is replaced by $99$ and $41$ by $43$ in the above data, find the new median?

Population size: $11$

Ascending order: $33,35,41,46,55,58,64,77,87,90,92$

Median $= 58$

If $92$ is replaced by $99$ and $41$ by $43$ Then

Population size: $11$

Ascending order: $33,35,43,46,55,58,64,77,87,90,99$

Median $= 58$

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Question 11:  Find the median of the following data $41, 43, 127, 99, 61, 92, 71, 58, 57$. If $58$ is replaced by $85$, what will be the new median

Population size: $9$

Ascending order: $41,43,57,58,61,71,92,99,127$

Median $= 61$

If $58$ is replaced by $85$ Then

Population size: $9$

Ascending order: $41,43,57,61,71,85,92,99,127$

Median $= 71$

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Question 12: The weights (in kg) of $15$ students are: $31,35,27,29,32,43,37,41,34,28,36,44, 45, 42, 30$. Find the median. If the weight $44 \ kg$ is replaced by $46 \ kg$ and $27 \ kg$ by $25 \ kg$, find the new median.

Population size: $15$

Ascending order: $27,28,29,30,31,32,34,35,36,37,41,42,43,44,45$

Median $= 35$

If $58$ is replaced by $85$ Then

Population size: $15$

Ascending order: $25,28,29,30,31,32,34,35,36,37,41,42,43,45,46$

Median $= 35$

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Question 13: The following observations have been arranged in ascending order. If the median of the data is $63$, find the value of $x: 29,32, 48,50,x, x+2,72,78, 84,95$

Population Size: $10$

Therefore $\displaystyle \frac{x + (x+2)}{2} = 63$

$\Rightarrow 2x + 2 = 126 \Rightarrow x = 62$

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Question 14: If the mean of the observations $45, 52, 60 x, 69,70,76, 81, 94$ is $68$, find its median.

$\displaystyle \text{Mean: } 68 =$ $\frac{45+52+60+x+69+70+76+81+94}{9}$

$\Rightarrow 612 = 547+x$

$\Rightarrow x = 65$

Population Size: $9$

Therefore Median is the $5^{th}$ term

Hence Median is $69$

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Question 15: The numbers $6,8,10,12,13$ and $x$ are arranged in ascending order. If the mean of these observations is equal to the median find the value of $x$.

Population size: $6$

$\displaystyle \text{Median } =$ $\frac{10+12}{2} = 11$

$\displaystyle \text{Therefore } 11 = \frac{6+8+10+12+13+x}{6}$

$\Rightarrow 66 = 49 + x$

$\Rightarrow x = 17$

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Question 16: The median of the following observations : $11, 12, 14, (x - 2), (x + 4) , (x + 9), 32, 38,47$ arranged in ascending order is $24$. Find the value of $x$ and hence find the median.

Population size: $9$

Median $= (x+4)$

Therefore $x+4 = 24 \Rightarrow x = 20$

$\displaystyle \text{Mean } = \frac{11+ 12+ 14+ (20 - 2)+(20 + 4)+(20 + 9)+32+38+47}{9} = 25$

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Question 17: If the mean of observations : $8, 10, 7, 6, 10, x, 6, 13, 10$ is $9$, find the median.

$\displaystyle \text{Mean: } 9 = \frac{8+10+7+6+10+x+6+13+10}{9}$

$\Rightarrow 81 = 70+x$

$\Rightarrow x = 11$

Population Size: $9$

Ascending order: $6,6,7,8,10,10,10,11,13$

Therefore Median is the $5^{th}$ term

Hence Median is $10$

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