Question 1: Find the mean of:

(i) $11, 13, 17, 19, 23$

(ii) $22, 24, 26, 28, 30, 32, 34, 36$

(iii) $\frac{1}{4}, 3\frac{1}{4}, 4\frac{3}{4}, 5\frac{1}{4}, 7\frac{1}{2}$

(iv) $6.5, 8.2, 9.4, 4.6, 7.8, 4.9$

(i) $Mean =$ $\frac{\Sigma x_i}{n}$

$Mean =$ $\frac{16+ 13+7+19+23}{5}$

$Mean =$ $\frac{83}{5}$ $= 16.6$

(ii) $Mean =$ $\frac{\Sigma x_i}{n}$

$Mean =$ $\frac{22+ 24+ 26+ 28+ 30+ 32+ 34+ 36}{8}$

$Mean =$ $\frac{232}{8}$ $= 29$

(iii) $Mean =$ $\frac{\Sigma x_i}{n}$

$Mean =$ $\frac{\frac{1}{4}+ 3\frac{1}{4}+ 4\frac{3}{4}+ 5\frac{1}{4}, 7\frac{1}{2}}{5}$

$Mean =$ $\frac{20\frac{1}{2}}{5}$ $= 4.2$

(iv) $Mean =$ $\frac{\Sigma x_i}{n}$

$Mean =$ $\frac{6.5+ 8.2+ 9.4+ 4.6+ 7.8+ 4.9}{6}$

$Mean =$ $\frac{41.4}{6}$ $= 6.9$

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Question 2:

(i) Find the mean of all prime numbers between $20$ and $40$

(ii) Find the mean of first seven whole numbers

(iii) Find the mean of first five multiples of $6$

(i) Prime numbers between $20$ and $40$ are $23, 29, 31, 37$

$Mean =$ $\frac{\Sigma x_i}{n}$

$Mean =$ $\frac{23+29+31+37}{4}$

$Mean =$ $\frac{120}{4}$ $= 30$

(ii) First seven whole numbers $= 0, 1, 2, 3, 4, 5, 6$

$Mean =$ $\frac{\Sigma x_i}{n}$

$Mean =$ $\frac{0+1+2+3+4+5+6}{7}$

$Mean =$ $\frac{21}{7}$ $= 3$

(iii) First five multiple of $6 = 6, 12, 18, 24, 30$

$Mean =$ $\frac{\Sigma x_i}{n}$

$Mean =$ $\frac{6+12+18+24+30}{5}$

$Mean =$ $\frac{90}{5}$ $= 18$

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Question 3: The mean of $9, 14, x, 16, 7$ and $18$ is $11.5$. Find the value of $x$.

$Mean =$ $\frac{\Sigma x_i}{n}$

$11.5 =$ $\frac{9+14+x+16+7+18}{6}$

$11.5 =$ $\frac{64+x}{6}$

$69 = 64+x \Rightarrow x = 5$

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Question 4: The mean of $7, 9, x+3, 12, 2x-1$ and $3$ is $9$. Find the value of $x$.

$Mean =$ $\frac{\Sigma x_i}{n}$

$9 =$ $\frac{7+9+x+3+12+2x-1+3}{6}$

$9 =$ $\frac{33+3x}{6}$

$54 = 33+x \Rightarrow x = 7$

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Question 5: Find the mean of the following distribution:

 Marks $12$ $15$ $20$ $23$ $25$ $27$ $30$ Number of students $8$ $6$ $11$ $7$ $4$ $1$ $3$

Mean of a distribution $=$ $\frac{\Sigma(f_ix_i)}{\Sigma f_i}$

 $x_i$ $f_i$ $x_if_i$ $12$ $8$ $96$ $15$ $6$ $90$ $20$ $11$ $220$ $23$ $7$ $161$ $25$ $4$ $100$ $27$ $1$ $27$ $30$ $3$ $90$ $\Sigma(f_i)= 40$ $\Sigma(f_ix_i)= 784$

Mean of a distribution $=$ $\frac{784}{40}$ $= 19.6$

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Question 6: The following data gives the number of boys of a particular age in a class of 40 students:

 Age in Years $15$ $16$ $17$ $18$ $19$ $20$ Number of Boys $3$ $8$ $9$ $11$ $6$ $3$

Calculate the mean age of the students.

Mean of a distribution $=$ $\frac{\Sigma(f_ix_i)}{\Sigma f_i}$

 $x_i$ $f_i$ $x_if_i$ $15$ $3$ $45$ $16$ $8$ $128$ $17$ $9$ $153$ $18$ $11$ $198$ $19$ $6$ $114$ $20$ $3$ $60$ $\Sigma(f_i)= 40$ $\Sigma(f_ix_i)= 698$

Mean of a distribution $=$ $\frac{698}{40}$ $= 17.45 \ years$

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Question 7: The weight of 40 students in a class are given below:

 Weight in Kg $30$ $32$ $33$ $35$ $36$ $37$ $38$ Number of students $5$ $6$ $3$ $8$ $4$ $9$ $5$

Find the mean weight.

Mean of a distribution $=$ $\frac{\Sigma(f_ix_i)}{\Sigma f_i}$

 $x_i$ $f_i$ $x_if_i$ $30$ $5$ $150$ $32$ $6$ $192$ $33$ $3$ $99$ $35$ $8$ $280$ $36$ $4$ $144$ $37$ $9$ $333$ $38$ $5$ $190$ $\Sigma(f_i)= 40$ $\Sigma(f_ix_i)= 1388$

Mean of a distribution $=$ $\frac{1388}{40}$ $= 34.7 \ kg$

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Question 8: Find the mean of the following frequency distribution:

 $x_i$ $7$ $8$ $9$ $10$ $11$ $f_i$ $19$ $23$ $31$ $27$ $20$

Mean of a distribution $=$ $\frac{\Sigma(f_ix_i)}{\Sigma f_i}$

 $x_i$ $f_i$ $x_if_i$ $7$ $19$ $133$ $8$ $23$ $184$ $9$ $31$ $279$ $10$ $27$ $270$ $11$ $20$ $220$ $\Sigma(f_i)= 120$ $\Sigma(f_ix_i)= 1086$

Mean of a distribution $=$ $\frac{1086}{120}$ $= 9.05$

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Question 9: If the mean of the following frequency distribution is 50, find the value of p.

 $x_i$ $10$ $30$ $50$ $70$ $90$ $f_i$ $17$ $28$ $32$ $p$ $19$

Mean of a distribution $=$ $\frac{\Sigma(f_ix_i)}{\Sigma f_i}$

 $x_i$ $f_i$ $x_if_i$ $10$ $17$ $170$ $30$ $28$ $840$ $50$ $32$ $1600$ $70$ $p$ $70p$ $90$ $19$ $1710$ $\Sigma(f_i)= 96+p$ $\Sigma(f_ix_i) = 4320+70p$

Mean of a distribution:  $50 =$ $\frac{4320+70p}{96+p}$

$4800 +50p = 4320+70p \Rightarrow p = 24$

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Question 10: Find the mean of the following grouped frequency distribution:

(i)

 Class – Interval $0-10$ $10-20$ $20-30$ $30-40$ $40-50$ Frequency $11$ $7$ $9$ $5$ $8$

(ii)

 Class – Interval $10-16$ $16-22$ $22-28$ $28-34$ $34-40$ Frequency $12$ $8$ $5$ $9$ $6$

(i) Mean of a distribution $=$ $\frac{\Sigma(f_ix_i)}{\Sigma f_i}$

 Class – Interval $f_i$ $x_i$ $x_if_i$ $0-10$ $11$ $5$ $55$ $10-20$ $7$ $15$ $105$ $20-30$ $9$ $25$ $225$ $30-40$ $5$ $35$ $175$ $40-50$ $8$ $45$ $360$ $40$ $920$

Mean of a distribution $=$ $\frac{920}{40}$ $= 23$

(ii) Mean of a distribution $=$ $\frac{\Sigma(f_ix_i)}{\Sigma f_i}$

 Class – Interval $f_i$ $x_i$ $x_if_i$ $10-16$ $12$ $13$ $156$ $16-22$ $8$ $19$ $152$ $22-28$ $5$ $25$ $125$ $28-34$ $9$ $31$ $279$ $34-40$ $6$ $37$ $222$ $40$ $934$

Mean of a distribution $=$ $\frac{934}{40}$ $= 23.35$

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Question 11: The daily wages of 60 workers in a factory are given below:

 Daily wages in Rs. $100-120$ $120-140$ $140-160$ $160-180$ $180-200$ Number of workers $24$ $12$ $8$ $11$ $5$

Find the mean daily wages.

Mean of a distribution $=$ $\frac{\Sigma(f_ix_i)}{\Sigma f_i}$

 Daily wages in Rs. $f_i$ $x_i$ $x_if_i$ $100-120$ $24$ $110$ $2640$ $120-140$ $12$ $130$ $1560$ $140-160$ $8$ $150$ $1200$ $160-180$ $11$ $170$ $1870$ $180-200$ $5$ $190$ $950$ $60$ $8220$

Mean of a distribution $=$ $\frac{8220}{60}$ $= 137$

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Question 12: Find the mode of:

(i) $11, 7, 3, 7, 0, 7, 8, 10, 8, 9, 11, 7$

(ii) $16, 23, 18, 20, 23, 18, 30, 25, 18, 16$

(i) In the given data, $7$ occurs the maximum number of times. Hence, the mode of the given data $= 7$

(ii) In the given data, $18$ occurs the maximum number of times. Hence, the mode of the given data $= 18$

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Question 13: The ages (in years) of $10$ good players of a class are given below: $13, 15, 13, 14, 16, 15, 13, 16, 13$. Find the mode age.

In the given data, $13$ occurs the maximum number of times. Hence, the mode of the given data $= 13$

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Question 14: Daily wages of \$latex 40 workers in a factory are given below.

 Daily wages in Rs. $100$ $125$ $150$ $175$ $200$ Number of workers $8$ $14$ $6$ $9$ $3$

Find the mode of the data.

Since the frequency of $125$ is the highest, mode of the data $= 125$.

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Question 15: The height of plants (in cm) in a nursery are given below.

 Height in cm $28$ $30$ $32$ $34$ $36$ Number of plants $36$ $47$ $80$ $58$ $72$

Find the mode of the data.

Since the frequency of $32$ is the highest, mode of the data $= 32$.

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Question 16: Find the median of each of the following data:

(i) $7, 11, 20, 6, 3, 16, 15, 23, 12$

(ii) $9, 25, 32, 51, 7, 16, 37, 50, 0, 13, 19$

(iii) $5.6, 7.2, 1.8, 4.3, 9.1, 2.6, 3.4$

(iv) $122, 127, 109, 118, 125, 108$

(i) On arranging the data in ascending order we get:

$3, 6, 7, 11, 12, 15, 16, 20, 23$

Number of terms $= 9$

Therefore middle term $=$ $\frac{1}{2}$ $(9+1)^{th} term = 5^{th} term = 12$

Therefore Median $= 12$

(ii) On arranging the data in ascending order we get:

$0, 7, 9, 13, 16, 19, 25, 32, 37, 50, 51$

Number of terms $= 11$

Therefore middle term $=$ $\frac{1}{2}$ $(11+1)^{th} term = 6^{th} term = 19$

Therefore Median $= 19$

(iii) On arranging the data in ascending order we get:

$1.8, 2.6, 3.4, 4.3, 5.6, 7.2, 9.1$

Number of terms $= 7$

Therefore middle term $=$ $\frac{1}{2}$ $(7+1)^{th} term = 4^{th} term = 4.3$

Therefore Median $= 4.3$

(iv) On arranging the data in ascending order we get:

$108, 109, 118, 122, 125, 127$

Number of terms $= 6$

Therefore middle term $=$ $\frac{1}{2}$ $(6)^{th} term = 3^{th} term = 118$

Therefore Median $= 118$

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