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Mathematics · Statistics · NCERT Exercises

Exercise 13.2

Complete, independently verified solutions for NCERT Exercise 13.2.

Mathematics · Chapter 13 · NCERT Exercises

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Question

Question 1

mode and mean · patient ages

The ages of patients admitted to a hospital during a year are grouped below. Find the mode and mean, then compare and interpret them.

Age (years)

Patients

5–15

6

15–25

11

25–35

21

35–45

23

45–55

14

55–65

5

Solution

Board-exam working:

Mean calculation:

Class

fᵢ

xᵢ

fᵢxᵢ

5–15

6

10

60

15–25

11

20

220

25–35

21

30

630

35–45

23

40

920

45–55

14

50

700

55–65

5

60

300

∑fi=80,∑fixi=2830\sum f_i=80,\qquad \sum f_ix_i=2830
xˉ=283080=35.375\bar{x}=\frac{2830}{80}=35.375

The highest frequency is 23, so the modal class is 35–45.

l=35,h=10,f1=23,f0=21,f2=14l=35,\quad h=10,\quad f_1=23,\quad f_0=21,\quad f_2=14
Mode=35+23−212(23)−21−14×10\text{Mode}=35+\frac{23-21}{2(23)-21-14}\times10
=35+2011=36.818…=35+\frac{20}{11}=36.818\ldots

The average admitted patient was about 35.38 years old, while the most typical age was about 36.82 years.

Final answer

Mode≈36.82 years;xˉ=35.375 years\boxed{\text{Mode}\approx36.82\text{ years};\quad \bar{x}=35.375\text{ years}}

Question

Question 2

mode lifetime · electrical components

The observed lifetimes of 225 electrical components are given below. Determine the modal lifetime.

Lifetime (hours)

Frequency

0–20

10

20–40

35

40–60

52

60–80

61

80–100

38

100–120

29

Solution

Board-exam working:

The highest frequency is 61, so the modal class is 60–80.

l=60,h=20,f1=61,f0=52,f2=38l=60,\quad h=20,\quad f_1=61,\quad f_0=52,\quad f_2=38
Mode=60+61−522(61)−52−38×20\text{Mode}=60+\frac{61-52}{2(61)-52-38}\times20
=60+932×20=65.625=60+\frac9{32}\times20=65.625

Final answer

Modal lifetime=65.625 hours\boxed{\text{Modal lifetime}=65.625\text{ hours}}

Question

Question 3

mode and mean expenditure · households

The monthly household expenditure of 200 village families is distributed below. Find the modal and mean monthly expenditure.

Expenditure (₹)

Families

1000–1500

24

1500–2000

40

2000–2500

33

2500–3000

28

3000–3500

30

3500–4000

22

4000–4500

16

4500–5000

7

Solution

Board-exam working:

Mean calculation:

Class

fᵢ

xᵢ

fᵢxᵢ

1000–1500

24

1250

30000

1500–2000

40

1750

70000

2000–2500

33

2250

74250

2500–3000

28

2750

77000

3000–3500

30

3250

97500

3500–4000

22

3750

82500

4000–4500

16

4250

68000

4500–5000

7

4750

33250

∑fi=200,∑fixi=532500\sum f_i=200,\qquad \sum f_ix_i=532500
xˉ=532500200=2662.5\bar{x}=\frac{532500}{200}=2662.5

The modal class is 1500–2000.

l=1500,h=500,f1=40,f0=24,f2=33l=1500,\quad h=500,\quad f_1=40,\quad f_0=24,\quad f_2=33
Mode=1500+40−242(40)−24−33×500\text{Mode}=1500+\frac{40-24}{2(40)-24-33}\times500
=1500+800023=1847.826…=1500+\frac{8000}{23}=1847.826\ldots

Final answer

Mode≈Rs. 1847.83;xˉ=Rs. 2662.50\boxed{\text{Mode}\approx\text{Rs. }1847.83;\quad \bar{x}=\text{Rs. }2662.50}

Question

Question 4

mode and mean · teacher student ratio

The state-wise teacher-student ratio in higher secondary schools is distributed below. Find the mode and mean, then interpret them.

Students per teacher

States/U.T.

15–20

3

20–25

8

25–30

9

30–35

10

35–40

3

40–45

0

45–50

0

50–55

2

Solution

Board-exam working:

Mean calculation:

Class

fᵢ

xᵢ

fᵢxᵢ

15–20

3

17.5

52.5

20–25

8

22.5

180

25–30

9

27.5

247.5

30–35

10

32.5

325

35–40

3

37.5

112.5

40–45

0

42.5

0

45–50

0

47.5

0

50–55

2

52.5

105

∑fi=35,∑fixi=1022.5\sum f_i=35,\qquad \sum f_ix_i=1022.5
xˉ=1022.535=29.214…\bar{x}=\frac{1022.5}{35}=29.214\ldots

The modal class is 30–35.

l=30,h=5,f1=10,f0=9,f2=3l=30,\quad h=5,\quad f_1=10,\quad f_0=9,\quad f_2=3
Mode=30+10−92(10)−9−3×5=30.625\text{Mode}=30+\frac{10-9}{2(10)-9-3}\times5=30.625

Across the states and U.T.s, the average is about 29.21 students per teacher; the most typical ratio is about 30.63 students per teacher.

Final answer

Mode=30.625;xˉ≈29.21 students per teacher\boxed{\text{Mode}=30.625;\quad \bar{x}\approx29.21\text{ students per teacher}}

Question

Question 5

mode runs · cricket

The runs scored by some leading one-day international batsmen are distributed below. Find the mode.

Runs

Batsmen

3000–4000

4

4000–5000

18

5000–6000

9

6000–7000

7

7000–8000

6

8000–9000

3

9000–10000

1

10000–11000

1

Solution

Board-exam working:

The modal class is 4000–5000.

l=4000,h=1000,f1=18,f0=4,f2=9l=4000,\quad h=1000,\quad f_1=18,\quad f_0=4,\quad f_2=9
Mode=4000+18−42(18)−4−9×1000\text{Mode}=4000+\frac{18-4}{2(18)-4-9}\times1000
=4000+1400023=4608.695…=4000+\frac{14000}{23}=4608.695\ldots

Final answer

Mode≈4608.70 runs\boxed{\text{Mode}\approx4608.70\text{ runs}}

Question

Question 6

mode cars · traffic count

The number of cars passing a road spot was recorded for 100 periods of three minutes. Find the mode.

Cars

Frequency

0–10

7

10–20

14

20–30

13

30–40

12

40–50

20

50–60

11

60–70

15

70–80

8

Solution

Board-exam working:

The modal class is 40–50.

l=40,h=10,f1=20,f0=12,f2=11l=40,\quad h=10,\quad f_1=20,\quad f_0=12,\quad f_2=11
Mode=40+20−122(20)−12−11×10\text{Mode}=40+\frac{20-12}{2(20)-12-11}\times10
=40+8017=44.7058…=40+\frac{80}{17}=44.7058\ldots

Final answer

Mode≈44.71 cars per three-minute period\boxed{\text{Mode}\approx44.71\text{ cars per three-minute period}}