NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.2 Statistics - PDF Download
Table of Contents
- What You Will Learn in Class 10 Maths Chapter 13 Exercise 13.2
- Key Terms in Class 10 Maths Chapter 13 Exercise 13.2
- Class 10 Maths Chapter 13 Exercise 13.2 at a Glance
- Mode of Grouped Data: Formula for Class 10 Maths Chapter 13 Exercise 13.2
- Worked Example for Class 10 Maths Chapter 13 Exercise 13.2
- Which Method to Use for the Mean in Class 10 Maths Chapter 13 Exercise 13.2
- Real-Life Uses of the Mode in Class 10 Maths Chapter 13 Exercise 13.2
- How to Write the Interpretation in Class 10 Maths Chapter 13 Exercise 13.2 (Questions 1, 3 and 4)
- Practice Questions for Class 10 Maths Chapter 13 Exercise 13.2
- Tips for Solving Class 10 Maths Chapter 13 Exercise 13.2
Class 10 Maths Chapter 13 Exercise 13.2 teaches you how to find the mode of grouped data: the value that occurs most often when the data is given as class intervals. The six questions use real data, such as the ages of hospital patients, the lifetimes of electrical components, the monthly expenditure of families, the number of students per teacher, the runs scored by batsmen, and the number of cars on a road. Three of them (Questions 1, 3 and 4) also ask for the mean, so that you can compare the two measures. Below, you get the answers to check your work, the method, a worked example, practice questions and the facts that the board exam asks.
What You Will Learn in Class 10 Maths Chapter 13 Exercise 13.2
- How to find the modal class, the class interval with the highest frequency.
- How to use the mode formula to estimate the mode inside the modal class.
- How to find the mean of grouped data by the direct method and by the step-deviation method (Questions 1, 3 and 4).
- How to compare the mean and the mode and write one line of interpretation.
- When the mode is a better measure than the mean.
Key Terms in Class 10 Maths Chapter 13 Exercise 13.2
| Term | Meaning |
|---|---|
| Grouped data | Data arranged in class intervals with a frequency for each class |
| Class interval | A range of values, for example 35–45 |
| Frequency ($f$) | The number of observations in a class |
| Class size ($h$) | The width of a class interval (the upper limit minus the lower limit) |
| Class mark ($x_i$) | The midpoint of a class: (lower limit + upper limit) ÷ 2 |
| Modal class | The class with the highest frequency |
| Mode | The estimated most frequent value, found inside the modal class |
Class 10 Maths Chapter 13 Exercise 13.2 at a Glance
| Question | Data | What to find | Modal class | Answer |
|---|---|---|---|---|
| 1 | Ages of 80 hospital patients | Mode and mean | 35 to 45 | Mode $= 36.8$ years, mean $\approx 35.38$ years |
| 2 | Lifetimes of 225 electrical components | Mode | 60 to 80 | Mode $= 65.625$ hours |
| 3 | Monthly expenditure of 200 families | Mode and mean | 1500 to 2000 | Mode ≈ ₹1847.83, mean = ₹2662.5 |
| 4 | Students per teacher in 35 states / UTs | Mode and mean | 30 to 35 | Mode $\approx 30.6$, mean $\approx 29.2$ |
| 5 | Runs scored by top batsmen | Mode | 4000 to 5000 | Mode $\approx 4608.7$ runs |
| 6 | Cars passing a spot in 100 periods | Mode | 40 to 50 | Mode $\approx 44.7$ cars |
A small difference in the last digit (for example 35.37 or 35.38, and 4608.69 or 4608.7) comes from rounding. Keep two or three decimals, and round only at the end.
Mode of Grouped Data: Formula for Class 10 Maths Chapter 13 Exercise 13.2
In ungrouped data, the mode is the value that occurs most often. In grouped data, we cannot see the exact value, so we first find the modal class, and then we estimate the mode inside it:
$$\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$
| Symbol | Meaning |
|---|---|
| $l$ | Lower limit of the modal class |
| $h$ | Class size (taken as equal for all classes) |
| $f_1$ | Frequency of the modal class |
| $f_0$ | Frequency of the class before the modal class |
| $f_2$ | Frequency of the class after the modal class |
How to find the mode of grouped data in 4 steps:
- Find the class with the highest frequency. This is the modal class.
- Write $l$, $h$, $f_1$, $f_0$ and $f_2$ from the table.
- Put the values in the formula, and simplify the bracket first.
- Check that the mode lies inside the modal class. If it does not, an error has crept in.
Syllabus note: the CBSE Class 10 Maths syllabus for 2026-27 (Statistics) asks for the mean, median and mode of grouped data, and it says that the bimodal situation is to be avoided. So the data in the exercise has one modal class.
Worked Example for Class 10 Maths Chapter 13 Exercise 13.2
Here is an eSaral worked example. The marks of 40 students are given below. Find the mode.
| Marks | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| Students | 5 | 8 | 15 | 7 | 5 |
The highest frequency is 15, so the modal class is 20–30. Here $l = 20$, $f_1 = 15$, $f_0 = 8$, $f_2 = 7$ and $h = 10$.
$$\text{Mode} = 20 + \frac{15 - 8}{2(15) - 8 - 7} \times 10 = 20 + \frac{7}{15} \times 10 \approx 20 + 4.67 = 24.67$$
The mode is about 24.67 marks, and it lies inside the modal class 20–30, as it should.
Which Method to Use for the Mean in Class 10 Maths Chapter 13 Exercise 13.2
Three questions of this exercise also ask for the mean. These checkpoints show which method each question uses, so that you can compare your own values:
| Question | Method | Values to check |
|---|---|---|
| 1 | Step-deviation | $a = 30$, $h = 10$, $\sum f_i = 80$, $\sum f_i u_i = 43$ |
| 3 | Direct method | $\sum f_i x_i = 5{,}32{,}500$, $\sum f_i = 200$ |
| 4 | Step-deviation | $a = 32.5$, $h = 5$, $\sum f_i = 35$, $\sum f_i u_i = -23$ |
Real-Life Uses of the Mode in Class 10 Maths Chapter 13 Exercise 13.2
- Shops and manufacturers use the mode to decide which shirt or shoe size to stock the most.
- Hospitals can find the most common age group of patients (Question 1).
- Road planners look at the most common number of vehicles in a period of time (Question 6).
- Television channels find the most popular programme.
- Schools and governments find the most common student-teacher ratio (Question 4).
How to Write the Interpretation in Class 10 Maths Chapter 13 Exercise 13.2 (Questions 1, 3 and 4)
Questions 1, 3 and 4 ask for both the mean and the mode, and CBSE expects one line that compares them:
- If the two are close (Question 1: mode 36.8 and mean about 35.4), say that the data is spread fairly evenly around the middle.
- If the mean is much higher than the mode (Question 3: mode about ₹1848 and mean ₹2662.5), say that a few high values pull the average up, while most families spend near the modal value.
- Always say it in the context of the question, for example “most patients are about 36.8 years old, and the average age is about 35.4 years”.
Practice Questions for Class 10 Maths Chapter 13 Exercise 13.2
Practice 1. The daily wages of 50 workers are: ₹100–150: 4 workers, ₹150–200: 10, ₹200–250: 18, ₹250–300: 12, ₹300–350: 6. Find the modal daily wage.
Practice 2. The number of books that 50 students borrowed in a month: 0–5: 3 students, 5–10: 7, 10–15: 12, 15–20: 20, 20–25: 8. Find the mode.
Answers: Practice 1: the modal class is 200–250, so the mode $= 200 + \dfrac{18 - 10}{36 - 10 - 12} \times 50 \approx$ ₹228.57. Practice 2: the modal class is 15–20, so the mode $= 15 + \dfrac{20 - 12}{40 - 12 - 8} \times 5 =$ 17 books.
Tips for Solving Class 10 Maths Chapter 13 Exercise 13.2
- Make a small table first (this is eSaral’s first tip for every Statistics question). Write the class, the frequency and the class mark (for the mean) in one table.
- Mark the modal class with a circle, and write $l$, $h$, $f_1$, $f_0$ and $f_2$ next to it before you substitute.
- Do not mix up $f_0$ and $f_2$. $f_0$ is the class before the modal class, and $f_2$ is the class after it.
- Remember $h$. The fraction is multiplied by the class size.
- Write the units (years, hours, ₹, runs, cars) and one line of interpretation.
- Practise the other two exercises of the chapter: Exercise 13.1 for the mean, and Exercise 13.3 for the median.
More from the chapter on eSaral: Exercise 13.1 (mean of grouped data), Exercise 13.3 (median of grouped data), all of Chapter 13 Statistics and the NCERT Solutions for Class 10 Maths page.
Frequently Asked Questions
Find answers to common questions.
How many questions are there in Class 10 Maths Chapter 13 Exercise 13.2?
There are 6 questions in Class 10 Maths Chapter 13 Exercise 13.2 (Statistics). All six ask for the mode of grouped data, and Questions 1, 3 and 4 also ask for the mean.
What is the modal class?
The modal class is the class interval that has the highest frequency. The mode of the data lies in this class.
What is the formula for the mode of grouped data in Class 10?
Mode $= l + \left(\dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$, where $l$ is the lower limit of the modal class, $h$ is the class size, $f_1$ is the frequency of the modal class, and $f_0$ and $f_2$ are the frequencies of the classes before and after it.
How do you find the mode of grouped data?
Find the modal class (the class with the highest frequency), write $l$, $h$, $f_1$, $f_0$ and $f_2$, put them in the mode formula, and simplify. The answer must lie inside the modal class. eSaral suggests that you write these five values in a small table before you substitute, because it avoids the common mix-up between $f_0$ and $f_2$.
What are the answers of NCERT Solutions Class 10 Maths Chapter 13 Exercise 13.2?
The answers are: Question 1, mode 36.8 years and mean about 35.38 years; Question 2, 65.625 hours; Question 3, mode about ₹1847.83 and mean ₹2662.5; Question 4, mode about 30.6 and mean about 29.2; Question 5, about 4608.7 runs; Question 6, about 44.7 cars. eSaral’s PDF has the full working.
Which method is used to find the mean in Class 10 Maths Chapter 13 Exercise 13.2?
Question 1 and Question 4 use the step-deviation method (with the assumed mean 30 and 32.5), and Question 3 uses the direct method. Any of the three methods gives the same mean.
Why does Class 10 Maths Chapter 13 Exercise 13.2 ask for both the mean and the mode?
So that you can compare them. The mode shows the value that occurs most, and the mean shows the average. If they are close, the data is spread evenly, and if the mean is much higher than the mode, a few large values pull the average up.
Is Exercise 13.2 Statistics in the CBSE Class 10 Maths syllabus for 2026-27?
Yes. The 2026-27 syllabus has mean, median and mode of grouped data, and it says that the bimodal situation is to be avoided. So the mode of grouped data is part of the syllabus.
When is the mode a better measure than the mean?
The mode is better when you want the most common or the most popular value, for example the most common shoe size, the most watched programme, or the colour of vehicle that most people use. The mean is better when you want the average.
Why is the mode of grouped data only an estimate?
Because in grouped data we do not know the exact values inside each class. The formula estimates the mode inside the modal class.