Q5. Find the mode of: xi = 5, 10, 15, 20, 25 with frequencies 7, 4, 6, 3, 5 (NCERT Exercise 13.1, Q5 data).
The largest frequency, 7, is at 5, so the mode is 5.
Q6. Find the mode of: xi = 10, 30, 50, 70, 90 with frequencies 4, 24, 28, 16, 8 (NCERT Exercise 13.1, Q6 data).
The largest frequency, 28, is at 50, so the mode is 50.
Q7. Find the modal income: ₹0–100, 100–200, …, 700–800 per day for 4, 8, 9, 10, 7, 5, 4, 3 persons (NCERT Exercise 13.1, Q9 data).
The modal class is 300–400: 300 + (10 − 9)/(20 − 9 − 7) × 100 = 300 + 25 = ₹325.
Q8. Ages 16–20, 21–25, …, 51–55 years have frequencies 5, 6, 12, 14, 26, 12, 16, 9 (NCERT Exercise 13.1, Q12 data). Find the modal age.
Make the classes continuous: 15.5–20.5, 20.5–25.5, …. The modal class is 35.5–40.5, with f0 = 14 and f2 = 12: 35.5 + (26 − 14)/(52 − 14 − 12) × 5 = 35.5 + 30/13 ≈ 37.81 years.
Q9. Classes 0–30, 30–60, …, 180–210 have frequencies 2, 3, 5, 10, 3, 5, 2 (NCERT Exercise 13.2, Q7 data). Find the mode.
The modal class is 90–120: 90 + (10 − 5)/(20 − 5 − 3) × 30 = 90 + 12.5 = 102.5.
Q10. Classes 0–10, 10–20, 20–30, 30–40, 40–50 have frequencies 5, 8, 15, 16, 6 (NCERT Exercise 13.2, Q8 data). Find the mode.
The modal class is 30–40: 30 + (16 − 15)/(32 − 15 − 6) × 10 = 30 + 10/11 ≈ 30.91.
Q11. Heights 70–75, 75–80, …, 110–115 cm of 3, 4, 7, 7, 15, 9, 6, 6, 3 children (NCERT Exercise 13.2, Q9 data). Find the modal height.
The modal class is 90–95: 90 + (15 − 7)/(30 − 7 − 9) × 5 = 90 + 20/7 ≈ 92.86 cm.
Q12. Classes 0–10, 10–20, 20–30, 30–40 have frequencies 5, x, 15, 9, and the mode is 26. Find x.
The mode 26 lies in 20–30, so that is the modal class: 20 + (15 − x)/(30 − x − 9) × 10 = 26, that is (15 − x)/(21 − x) = 0.6. Then 15 − x = 12.6 − 0.6x, so 0.4x = 2.4 and x = 6. Check: 6 < 15, so 20–30 is indeed the modal class.