Class 10 NCERT Solutions- Chapter 14 Statistics – Exercise 14.2

Last Updated : 03 Mar, 2021

Question 1. The following table shows the ages of the patients admitted in a hospital during a year:

Age (in years)	5-15	15-25	25-35	35-45	45-55	55-65
Number of patients	6	11	21	23	14	5

Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Solution:

The greatest frequency in the given table is 23, so the modal class = 35 – 45,

l = 35,

Class width = 10, and the frequencies are

f_m = 23, f₁ = 21 and f₂ = 14

Now, we find the mode using the given formula

Mode = $l+ \left[\frac{(f_m-f_1)}{(2f_m-f_1-f_2)}\right]×h$

On substituting the values in the formula, we get

Mode = $35+\left[\frac{(23-21)}{(46-21-14)}\right]×10$

= 35 + (20/11) = 35 + 1.8

= 36.8

Hence, the mode of the given data is 36.8 year

Now, we find the mean. So for that first we need to find the midpoint.

x_i= (upper limit + lower limit)/2

Class Interval Frequency (f_i) Mid-point (x_i) f_ix_i
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
Sum f_i = 80 Sum f_ix_i = 2830

Mean = $\bar{x}$ = ∑f_ix_i /∑f_i

= 2830/80

= 35.37 years

Question 2. The following data gives the information on the observed lifetimes (in hours) of 225 electrical components:

Lifetime (in hours)	0-20	20-40	40-60	60-80	80-100	100-120
Frequency	10	35	52	61	38	29

Determine the modal lifetimes of the components.

Solution:

According to the given question

The modal class is 60 – 80

l = 60, and the frequencies are

f_m = 61, f₁ = 52, f₂ = 38 and h = 20

Now, we find the mode using the given formula

Mode = $l+ \left[\frac{(f_m-f_1)}{(2f_m-f_1-f_2)}\right]×h$

On substituting the values in the formula, we get

Mode = $60+\left[\frac{(61-52)}{(122-52-38)}\right]×20$

= $60+\frac{(9 \times 20)}{32}$

= 60 + 45/8 = 60 + 5.625

Hence, the modal lifetime of the components is 65.625 hours.

Question 3. The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure:

Expenditure	Number of 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:

According to the question

Modal class = 1500-2000,

l = 1500,and the frequencies are

f_m = 40 f₁ = 24, f₂ = 33 and

h = 500

Now, we find the mode using the given formula

Mode = $l+ \left[\frac{(f_m-f_1)}{(2f_m-f_1-f_2)}\right]×h$

On substituting the values in the formula, we get

Mode = $1500+\left[\frac{(40-24)}{(80-24-33)}\right]×500$

= $1500+\frac{16×500}{23}$

= 1500 + 8000/23 = 1500 + 347.83

So, the modal monthly expenditure of the families is 1847.83 Rupees

Now, we find the mean. So for that first we need to find the midpoint.

x_i= (upper limit + lower limit)/2

Let us considered a mean, A be 2750

Class Interval f_i x_i d_i = x_i – a u_i = d_i/h f_iu_i
1000-1500 24 1250 -1500 -3 -72
1500-2000 40 1750 -1000 -2 -80
2000-2500 33 2250 -500 -1 -33
2500-3000 28 2750 0 0 0
3000-3500 30 3250 500 1 30
3500-4000 22 3750 1000 2 44
4000-4500 16 4250 1500 3 48
4500-5000 7 4750 2000 4 28
f_i = 200 f_iu_i = -35

Mean = $\overline{x} = a +\frac{∑f_iu_i}{∑f_i}×h$

On substituting the values in the given formula

= $2750+\frac{-35}{200}×500$

= 2750 – 87.50

= 2662.50

Hence, the mean monthly expenditure of the families is 2662.50 Rupees

Question 4. The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures

No of Students per teacher	Number of 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:

According to the question

Modal class = 30 – 35,

l = 30,

Class width (h) = 5, and the frequencies are

f_m = 10, f₁ = 9 and f₂ = 3

Now, we find the mode using the given formula

Mode = $l+ \left[\frac{(f_m-f_1)}{(2f_m-f_1-f_2)}\right]×h$

On substituting the values in the formula, we get

Mode = $30+\frac{(10-9)}{(20-9-3)}×5$

= 30 + 5/8 = 30 + 0.625

= 30.625

Hence, the mode of the given data is 30.625

Now, we find the mean. So for that first we need to find the midpoint.

x_i= (upper limit + lower limit)/2

Class Interval Frequency (f_i) Mid-point (x_i) f_ix_i
15-20 3 17.5 52.5
20-25 8 22.5 180.0
25-30 9 27.5 247.5
30-35 10 32.5 325.0
35-40 3 37.5 112.5
40-45 0 42.5 0
45-50 0 47.5 0
50-55 2 52.5 105.5
Sum f_i = 35 Sum f_ix_i = 1022.5

Mean = $\bar{x} = \frac{∑f_ix_i }{∑f_i}$

= 1022.5/35

= 29.2

Hence, the mean is 29.2

Question 5. The given distribution shows the number of runs scored by some top batsmen of the world in one- day international cricket matches.

Run Scored	Number of Batsman
3000-4000	4
4000-5000	18
5000-6000	9
6000-7000	7
7000-8000	6
8000-9000	3
9000-10000	1
10000-11000	1

Find the mode of the data.

Solution:

According to the question

Modal class = 4000 – 5000,

l = 4000,

class width (h) = 1000, and the frequencies are

f_m = 18, f₁ = 4 and f₂ = 9

Now, we find the mode using the given formula

Mode = $l+ \left[\frac{(f_m-f_1)}{(2f_m-f_1-f_2)}\right]×h$

On substituting the values in the formula, we get

Mode = $4000+\frac{(18-4)}{(36-4-9)}×1000$

Mode = 4000 + 14000/23 = 4000 + 608.695

= 4608.695

Hence, the mode of the given data is 4608.7 runs

Question 6. A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarized it in the table given below. Find the mode of the data:

Number of 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:

According to the question

Modal class = 40 – 50, l = 40,

Class width (h) = 10, and the frequencies are

f_m = 20, f₁ = 12 and f₂ = 11

Now, we find the mode using the given formula

Mode = $l+ \left[\frac{(f_m-f_1)}{(2f_m-f_1-f_2)}\right]×h$

On substituting the values in the formula, we get

Mode = $40+\frac{(20-12)}{(40-12-11)}×10$

Mode = 40 + 80/17 = 40 + 4.7 = 44.7

Hence, the mode of the given data is 44.7 cars

Suggest improvement

Class 10 NCERT Solutions - Chapter 14 Statistics - Exercise 14.1

Class 10 NCERT Solutions- Chapter 14 Statistics - Exercise 14.3

Share your thoughts in the comments

Class Interval	Frequency (f_i)	Mid-point (x_i)	f_ix_i
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
	Sum f_i = 80		Sum f_ix_i = 2830

Class Interval	f_i	x_i	d_i = x_i – a	u_i = d_i/h	f_iu_i
1000-1500	24	1250	-1500	-3	-72
1500-2000	40	1750	-1000	-2	-80
2000-2500	33	2250	-500	-1	-33
2500-3000	28	2750	0	0	0
3000-3500	30	3250	500	1	30
3500-4000	22	3750	1000	2	44
4000-4500	16	4250	1500	3	48
4500-5000	7	4750	2000	4	28
	f_i = 200				f_iu_i = -35

Chapter 1: Real Numbers

Chapter 2: Polynomials

Chapter 3: Pair of Linear Equations in Two Variables

Chapter 4: Quadratic Equations

Chapter 5: Arithmetic Progressions

Chapter 6: Triangles

Chapter 7: Coordinate Geometry

Chapter 8: Introduction to Trigonometry

Chapter 9: Some Applications of Trigonometry

Chapter 10: Circles

Chapter 11: Constructions

Chapter 12: Areas Related to Circles

Chapter 13: Surface Areas and Volumes

Chapter 14: Statistics

Chapter 15: Probability

Chapter 1: Real Numbers

Chapter 2: Polynomials

Chapter 3: Pair of Linear Equations in Two Variables

Chapter 4: Quadratic Equations

Chapter 5: Arithmetic Progressions

Chapter 6: Triangles

Chapter 7: Coordinate Geometry

Chapter 8: Introduction to Trigonometry

Chapter 9: Some Applications of Trigonometry

Chapter 10: Circles

Chapter 11: Constructions

Chapter 12: Areas Related to Circles

Chapter 13: Surface Areas and Volumes

Chapter 14: Statistics

Chapter 15: Probability

Class 10 NCERT Solutions- Chapter 14 Statistics – Exercise 14.2

Question 1. The following table shows the ages of the patients admitted in a hospital during a year:

Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Question 2. The following data gives the information on the observed lifetimes (in hours) of 225 electrical components:

Determine the modal lifetimes of the components.

Question 3. The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure:

Question 4. The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures

Question 5. The given distribution shows the number of runs scored by some top batsmen of the world in one- day international cricket matches.

Find the mode of the data.

Question 6. A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarized it in the table given below. Find the mode of the data:

Please Login to comment...

Similar Reads

What kind of Experience do you want to share?