Q5. Find the mean and the median of 38, 70, 48, 40, 42, 55, 63, 46, 54, 44 (NCERT Exercise 13.1, Q2 data).
The sum is 500, so the mean is 50. Sorted, the 5th and 6th values are 46 and 48, so the median is 47.
Q6. Find the mean of: xi = 5, 10, 15, 20, 25 with frequencies 7, 4, 6, 3, 5 (NCERT Exercise 13.1, Q5 data).
N = 25 and Σfixi = 35 + 40 + 90 + 60 + 125 = 350, so x̄ = 14.
Q7. Find the median of: xi = 5, 7, 9, 10, 12, 15 with frequencies 8, 6, 2, 2, 2, 6 (NCERT Exercise 13.1, Q7 data).
N = 26, and the cumulative frequencies are 8, 14, 16, 18, 20, 26. The 13th and 14th observations are both 7, so the median is 7.
Q8. Find the median of the marks: 0–10, 10–20, 20–30, 30–40, 40–50, 50–60 with 6, 8, 14, 16, 4, 2 girls (NCERT Exercise 13.1, Q11 data).
N = 50, N/2 = 25; the cumulative frequencies are 6, 14, 28, …, so the median class is 20–30 and M = 20 + (25 − 14)/14 × 10 = 20 + 55/7 ≈ 27.86.
Q9. The mean of 10 observations is 15. If each observation is multiplied by 3 and then 4 is subtracted, find the new mean.
3 × 15 − 4 = 41.
Q10. Find the mean of the first n natural numbers.
(1 + 2 + … + n)/n = (n(n + 1)/2)/n = (n + 1)/2.
Q11. In a class of 30 students the mean mark is 52. The 18 boys have mean 50. Find the girls’ mean mark.
The total is 30 × 52 = 1560 and the boys’ total is 900, so the 12 girls have 660 and a mean of 55.
Q12. Give a data set of five numbers with mean 10 and median 4.
For example 1, 2, 4, 15, 28: the middle value is 4 and the sum is 50. A few large values pull the mean far above the median.
Q13. Find the mode of: xi = 5, 7, 9, 10, 12, 15 with frequencies 8, 6, 2, 2, 2, 6 (NCERT Exercise 13.1, Q7 data).
The largest frequency, 8, is at 5, so the mode is 5.