Exercise 14.2 Page: 275
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:
To find out the modal class, let us the consider the class interval with high frequency
Here, the greatest frequency = 23, so the modal class = 35 – 45,
l = 35,
class width (h) = 10,
fm = 23,
f1 = 21 and f2 = 14
The formula to find the mode is
Mode = l+ [(fm-f1)/(2fm-f1-f2)]×h
Substitute the values in the formula, we get
Mode = 35+[(23-21)/(46-21-14)]×10
Mode = 35+(20/11) = 35+1.8
Mode = 36.8 year
So the mode of the given data = 36.8 year
Calculation of Mean:
First find the midpoint using the formula, xi = (upper limit +lower limit)/2
| Class Interval | Frequency (fi) | Mid-point (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 |
| Sum fi = 80 | Sum fixi = 2830 |
The mean formula is
Mean = x̄ = ∑fixi /∑fi
= 2830/80
= 35.37 years
Therefore, the mean of the given data = 35.37 years
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:
From the given data the modal class is 60–80.
l = 60,
The frequencies are:
fm = 61, f1 = 52, f2 = 38 and h = 20
The formula to find the mode is
Mode = l+ [(fm-f1)/(2fm-f1-f2)]×h
Substitute the values in the formula, we get
Mode =60+[(61-52)/(122-52-38)]×20
Mode = 60+((9 x 20)/32)
Mode = 60+(45/8) = 60+ 5.625
Therefore, modal lifetime of the components = 65.625 hours.
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:
Given data:
Modal class = 1500-2000,
l = 1500,
Frequencies:
fm = 40 f1 = 24, f2 = 33 and
h = 500
Mode formula:
Mode = l+ [(fm-f1)/(2fm-f1-f2)]×h
Substitute the values in the formula, we get
Mode =1500+[(40-24)/(80-24-33)]×500
Mode = 1500+((16×500)/23)
Mode = 1500+(8000/23) = 1500 + 347.83
Therefore, modal monthly expenditure of the families = Rupees 1847.83
Calculation for mean:
First find the midpoint using the formula, xi =(upper limit +lower limit)/2
Let us assume a mean, A be 2750
| Class Interval | fi | xi | di = xi – a | ui = di/h | fiui |
| 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 |
| fi = 200 | fiui = -35 |
The formula to calculate the mean,
Mean = x̄ = a +(∑fiui /∑fi)×h
Substitute the values in the given formula
= 2750+(-35/200)×500
= 2750-87.50
= 2662.50
So, the mean monthly expenditure of the families = Rupees 2662.50
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:
Given data:
Modal class = 30 – 35,
l = 30,
Class width (h) = 5,
fm = 10, f1 = 9 and f2 = 3
Mode Formula:
Mode = l+ [(fm-f1)/(2fm-f1-f2)]×h
Substitute the values in the given formula
Mode = 30+((10-9)/(20-9-3))×5
Mode = 30+(5/8) = 30+0.625
Mode = 30.625
Therefore, the mode of the given data = 30.625
Calculation of mean:
Find the midpoint using the formula, xi =(upper limit +lower limit)/2
| Class Interval | Frequency (fi) | Mid-point (xi) | fixi |
| 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 fi = 35 | Sum fixi = 1022.5 |
Mean = x̄ = ∑fixi /∑fi
= 1022.5/35
= 29.2
Therefore, mean = 29.2
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:
Given data:
Modal class = 4000 – 5000,
l = 4000,
class width (h) = 1000,
fm = 18, f1 = 4 and f2 = 9
Mode Formula:
Mode = l+ [(fm-f1)/(2fm-f1-f2)]×h
Substitute the values
Mode = 4000+((18-4)/(36-4-9))×1000
Mode = 4000+(14000/23) = 4000+608.695
Mode = 4608.695
Mode = 4608.7 (approximately)
Thus, the mode of the given data is 4608.7 runs
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:
Given Data:
Modal class = 40 – 50, l = 40,
Class width (h) = 10, fm = 20, f1 = 12 and f2 = 11
Mode = l+ [(fm-f1)/(2fm-f1-f2)]×h
Substitute the values
Mode = 40+((20-12)/(40-12-11))×10
Mode = 40 + (80/17) = 40 + 4.7 = 44.7
Thus, the mode of the given data is 44.7 cars
NCERT Solutions for Class 10 Maths Chapter 14 Statistics
Class 10 Maths Chapter 14, Statistics, is one of the most important of all the chapter present in the textbook. The weightage of this chapter in the CBSE Term II exam is around 11 to 12 marks. On average, there will be 3 questions which could be asked from this chapter and marks will be distributed in a manner of 3+4+4( it could vary as per question).
Topics covered in Chapter 14, Statistics are;
- Mean of Grouped Data
- Mode of Grouped Data
- Median of Grouped Data
- Graphical Representation of Cumulative Frequency Distribution
List of Exercises in class 10 Maths Chapter 14 :
Exercise 14.1 Solutions 9 Question ( 9 long)
Exercise 14.2 Solutions 6 Question ( 6 long)
Exercise 14.3 Solutions 7 Question ( 7 long)
Exercise 14.4 Solutions 3 Question ( 3 long)
NCERT solutions for Class 10 Maths Chapter 14- Statistics are made available for students who want to obtain good marks in this chapter. The methods and procedure to solve the questions have been explained clearly in these NCERT Solutions, such that, students find it easy to understand the fundamentals quickly.
The world is highly data-oriented, in fact, each and every field has a group of data, which represents the relevant information. Statistics is the branch of mathematics which deals with the representation of data in a meaningful way.
You will face many real-life scenarios where the fundamentals of statistics are used to represent a set of data in tabular form or in graphs or in pie charts. There are a number of methods you will learn from this chapter such as, step deviation methods, finding mode and median of grouped data, converting frequency distribution and the relation between mode, mean and median methods, etc. 10th Class NCERT solutions are the best study materials to prepare for the CBSE Term II exam.
Key Features of NCERT Solutions for Class 10 Maths Chapter 14- Statistics
- The solutions for the statistics chapter works as a reference for the students.
- It will help students to score marks against the questions asked from the statistics chapter.
- Students can prepare and do the revision for chapter 14 with this source.
- The questions of statistics are solved by subject experts.
- The content of the material is as per the term-wise CBSE Syllabus (2021-22) and guidelines.
Statistics can also be understood in a much better way by using the other solutions which are provided at RAJ EDUCATION HUB. The solutions are prepared to help students perform well in the CBSE exams.