Question 1: Find the mean of:

i) $\displaystyle 11, 13, 17, 19, 23$   ii) $\displaystyle 22, 24, 26, 28, 30, 32, 34, 36$

iii) $\displaystyle \frac{1}{4}, 3\frac{1}{4}, 4\frac{3}{4}, 5\frac{1}{4}, 7\frac{1}{2}$   iv) $\displaystyle 6.5, 8.2, 9.4, 4.6, 7.8, 4.9$

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

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

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

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

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

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

iii) $\displaystyle \text{Mean} = \frac{\Sigma x_i}{n}$

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

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

iv) $\displaystyle \text{Mean} = \frac{\Sigma x_i}{n}$

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

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

$\displaystyle \\$

Question 2:

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

ii) Find the mean of first seven whole numbers

iii) Find the mean of first five multiples of $\displaystyle 6$

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

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

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

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

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

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

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

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

$\displaystyle \\$

Question 5: Find the mean of the following distribution:

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

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

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

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

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

Calculate the mean age of the students.

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

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

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

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

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

Find the mean weight.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

i)

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

ii)

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

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

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

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

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

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

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

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

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

Find the mean daily wages.

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

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

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

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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 $40$ workers in a factory are given below.

 Daily wages in Rs. $100$$100$ $125$$125$ $150$$150$ $175$$175$ $200$$200$ Number of workers $8$$8$ $14$$14$ $6$$6$ $9$$9$ $3$$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$$28$ $30$$30$ $32$$32$ $34$$34$ $36$$36$ Number of plants $36$$36$ $47$$47$ $80$$80$ $58$$58$ $72$$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 $= 118$