NCERT Class 10 Maths – Exercise 13.2 Solutions

NCERT Class 10 Maths

Chapter 13 – Statistics | Exercise 13.2

(Rationalized Syllabus 2025-26)

💡 Formula for Mode

$$ \text{Mode} = l + \left( \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \right) \times h $$
  • $l$: Lower limit of the modal class
  • $h$: Size of the class interval
  • $f_1$: Frequency of the modal class
  • $f_0$: Frequency of the class preceding the modal class
  • $f_2$: Frequency of the class succeeding the modal class
Q1

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.

1. Finding Mode
Maximum frequency is 23. Thus, the Modal Class is 35 – 45.
  • $l = 35$
  • $f_1 = 23$
  • $f_0 = 21$
  • $f_2 = 14$
  • $h = 10$
$$ \text{Mode} = 35 + \left( \frac{23 – 21}{2(23) – 21 – 14} \right) \times 10 $$ $$ = 35 + \left( \frac{2}{46 – 35} \right) \times 10 = 35 + \frac{20}{11} = 35 + 1.8 $$ $$ = 36.8 \text{ years} $$
2. Finding Mean
Using the Step Deviation Method or Assumed Mean Method, the calculated Mean is 35.37 years.
✔️ Mode = 36.8 years
✔️ Mean = 35.37 years
Interpretation: Maximum number of patients are 36.8 years old, while the average age of a patient is 35.37 years.
Q2

The following data gives the information on the observed lifetimes (in hours) of 225 electrical components. Determine the modal lifetimes of the components.

Identify Variables
Maximum frequency is 61. Modal Class is 60 – 80.
  • $l = 60$
  • $f_1 = 61$
  • $f_0 = 52$
  • $f_2 = 38$
  • $h = 20$
Calculation
$$ \text{Mode} = 60 + \left( \frac{61 – 52}{2(61) – 52 – 38} \right) \times 20 $$ $$ = 60 + \left( \frac{9}{122 – 90} \right) \times 20 $$ $$ = 60 + \left( \frac{9}{32} \right) \times 20 = 60 + \frac{180}{32} $$ $$ = 60 + 5.625 $$
✔️ Modal Lifetime = 65.625 hours
Q3

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.

1. Finding Mode
Max frequency is 40. Modal Class is 1500 – 2000.
  • $l = 1500, h = 500$
  • $f_1 = 40, f_0 = 24, f_2 = 33$
$$ \text{Mode} = 1500 + \left( \frac{40 – 24}{2(40) – 24 – 33} \right) \times 500 $$ $$ = 1500 + \left( \frac{16}{80 – 57} \right) \times 500 $$ $$ = 1500 + \frac{16}{23} \times 500 = 1500 + 347.83 $$
2. Finding Mean
Calculating using Step Deviation method yields Mean $\approx$ ₹ 2662.5.
✔️ Modal Expenditure = ₹ 1847.83
Q4

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.

1. Finding Mode
Max frequency is 10. Modal Class is 30 – 35.
  • $l = 30, h = 5$
  • $f_1 = 10, f_0 = 9, f_2 = 3$
$$ \text{Mode} = 30 + \left( \frac{10 – 9}{2(10) – 9 – 3} \right) \times 5 $$ $$ = 30 + \left( \frac{1}{20 – 12} \right) \times 5 $$ $$ = 30 + \frac{5}{8} = 30 + 0.625 $$
2. Finding Mean
Calculated Mean = 29.2.
✔️ Mode = 30.6 (approx)
✔️ Mean = 29.2
Interpretation: Most states have a teacher-student ratio of 30.6, while the average ratio across states is 29.2.
Q5

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.

Identify Variables
Max frequency is 18. Modal Class is 4000 – 5000.
  • $l = 4000, h = 1000$
  • $f_1 = 18, f_0 = 4, f_2 = 9$
Calculation
$$ \text{Mode} = 4000 + \left( \frac{18 – 4}{2(18) – 4 – 9} \right) \times 1000 $$ $$ = 4000 + \left( \frac{14}{36 – 13} \right) \times 1000 $$ $$ = 4000 + \frac{14000}{23} = 4000 + 608.695… $$
✔️ Mode = 4608.7 runs
Q6

A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes… Find the mode of the data.

Identify Variables
Max frequency is 20. Modal Class is 40 – 50.
  • $l = 40, h = 10$
  • $f_1 = 20, f_0 = 12, f_2 = 11$
Calculation
$$ \text{Mode} = 40 + \left( \frac{20 – 12}{2(20) – 12 – 11} \right) \times 10 $$ $$ = 40 + \left( \frac{8}{40 – 23} \right) \times 10 $$ $$ = 40 + \frac{80}{17} = 40 + 4.705… $$
✔️ Mode = 44.7 cars
🎉 Exercise 13.2 Completed | Chapter 13 Statistics
learncbsehub.in