Class 9 Maths Chapter 12 Statistics – Exercise 12.3 NCERT Solutions

Introduction

Exercise 12.3 focuses on mode of grouped data. Students learn how to calculate the mode using the mode formula for frequency distributions. This exercise is essential for understanding the most frequent value in a dataset and applying it to real‑life problems.

Key Formula

For grouped data, mode is calculated using:

$$\text{Mode}=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\cdot h$$

Where:

• $l$ = lower limit of modal class

• $h$ = class size

• $f_1$ = frequency of modal class

• $f_0$ = frequency of class before modal class

• $f_2$ = frequency of class after modal class

Common Mistakes

• Not identifying the correct modal class (highest frequency).

• Forgetting to use class size $h$.

• Arithmetic errors in numerator and denominator.

• Confusing mode with mean or median.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Find mode of data: Class intervals 0–10, 10–20, 20–30, 30–40; frequencies 5, 7, 10, 8. Modal class = 20–30, $l=20,h=10,f_1=10,f_0=7,f_2=8$.

$$\text{Mode}=20+\left(\frac{10-7}{2\cdot 10-7-8}\right)\cdot 10=20+\frac{3}{5}\cdot 10=26$$

Q2. Find mode of data: Class intervals 0–20, 20–40, 40–60, 60–80; frequencies 10, 15, 20, 5. Modal class = 40–60, $l=40,h=20,f_1=20,f_0=15,f_2=5$.

$$\text{Mode}=40+\left(\frac{20-15}{40-15-5}\right)\cdot 20=40+\frac{5}{20}\cdot 20=45$$

Q3. Find mode of data: Class intervals 0–10, 10–20, 20–30; frequencies 3, 4, 5. Modal class = 20–30, $l=20,h=10,f_1=5,f_0=4,f_2=0$.

$$\text{Mode}=20+\left(\frac{5-4}{10-4-0}\right)\cdot 10=20+\frac{1}{6}\cdot 10=21.67$$

Q4. Find mode of data: Class intervals 10–20, 20–30, 30–40, 40–50; frequencies 6, 8, 10, 6. Modal class = 30–40, $l=30,h=10,f_1=10,f_0=8,f_2=6$.

$$\text{Mode}=30+\left(\frac{10-8}{20-8-6}\right)\cdot 10=30+\frac{2}{6}\cdot 10=33.33$$

Q5. Find mode of data: Class intervals 0–10, 10–20, 20–30, 30–40; frequencies 2, 6, 8, 4. Modal class = 20–30, $l=20,h=10,f_1=8,f_0=6,f_2=4$.

$$\text{Mode}=20+\left(\frac{8-6}{16-6-4}\right)\cdot 10=20+\frac{2}{6}\cdot 10=23.33$$

Q6. Find mode of data: Class intervals 0–20, 20–40, 40–60; frequencies 5, 10, 15. Modal class = 40–60, $l=40,h=20,f_1=15,f_0=10,f_2=0$.

$$\text{Mode}=40+\left(\frac{15-10}{30-10-0}\right)\cdot 20=40+\frac{5}{20}\cdot 20=45$$

Q7. Find mode of data: Class intervals 10–30, 30–50, 50–70; frequencies 7, 9, 4. Modal class = 30–50, $l=30,h=20,f_1=9,f_0=7,f_2=4$.

$$\text{Mode}=30+\left(\frac{9-7}{18-7-4}\right)\cdot 20=30+\frac{2}{7}\cdot 20=35.71$$

Q8. Find mode of data: Class intervals 0–25, 25–50, 50–75, 75–100; frequencies 5, 10, 15, 20. Modal class = 75–100, $l=75,h=25,f_1=20,f_0=15,f_2=0$.

$$\text{Mode}=75+\left(\frac{20-15}{40-15-0}\right)\cdot 25=75+\frac{5}{25}\cdot 25=80$$

Q9. Find mode of data: Class intervals 0–10, 10–20, 20–30, 30–40, 40–50; frequencies 4, 6, 8, 10, 12. Modal class = 40–50, $l=40,h=10,f_1=12,f_0=10,f_2=0$.

$$\text{Mode}=40+\left(\frac{12-10}{24-10-0}\right)\cdot 10=40+\frac{2}{14}\cdot 10=41.43$$

Q10. Find mode of data: Class intervals 0–20, 20–40, 40–60, 60–80; frequencies 8, 12, 20, 10. Modal class = 40–60, $l=40,h=20,f_1=20,f_0=12,f_2=10$.

$$\text{Mode}=40+\left(\frac{20-12}{40-12-10}\right)\cdot 20=40+\frac{8}{18}\cdot 20=48.89$$

FAQs (10)

FAQ1. What is mode? Most frequent value in dataset.

FAQ2. What is modal class? Class interval with highest frequency.

FAQ3. Why use mode formula? To estimate mode for grouped data.

FAQ4. What is class size? Difference between upper and lower class limits.

FAQ5. Why subtract frequencies in formula? To adjust for distribution around modal class.

FAQ6. Can mode be unique? Yes, if one class has highest frequency.

FAQ7. Can data have more than one mode? Yes, in bimodal or multimodal distributions.

FAQ8. Why is Exercise 12.3 important? It teaches calculation of mode for grouped data.

FAQ9. What is unit of mode? Same as unit of data values.

FAQ10. What is practical use of mode? Used in identifying most popular choice (e.g., product demand).

Conclusion

Exercise 12.3 covers mode of grouped data with solved examples and FAQs. Mastering these problems helps students analyze frequency distributions and apply statistics in real‑life contexts.

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