Q5. Find the quartile deviation of 38, 70, 48, 40, 42, 55, 63, 46, 54, 44 (NCERT Exercise 13.1, Q2 data).
Sorted: 38, 40, 42, 44, 46 | 48, 54, 55, 63, 70. Q1 = 42 and Q3 = 55, so QD = 6.5.
Q6. Find the mean of the sizes of the deviations from the mean for 2, 4, 6, 8, 10.
The mean is 6; the sizes 4, 2, 0, 2, 4 add up to 12, so the mean deviation is 12/5 = 2.4.
Q7. For 2, 4, 6, 8, 10, find the mean of the squared deviations from the mean, and its square root.
The squares 16, 4, 0, 4, 16 add up to 40, so the mean square is 8 and its square root is 2√2 ≈ 2.83.
Q8. Two classes took the same test. Class P has mean 60 and quartile deviation 4; class Q has mean 70 and quartile deviation 12. Which class is more consistent?
Class P: its middle half is spread over a much narrower band. The higher mean of Q says nothing about consistency.
Q9. Each of the values 12, 15, 18, 21 is increased by 10. How do the mean and the mean deviation about the mean change?
The mean rises from 16.5 to 26.5, but every deviation from the mean stays the same (−4.5, −1.5, 1.5 and 4.5), so the mean deviation stays 12/4 = 3.
Q10. Show that for any data set, Σ(xi − x̄) = 0.
Σ(xi − x̄) = Σxi − nx̄, and Σxi = nx̄ by the definition of the mean.
Q11. Find the quartile deviation of 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17 (NCERT Exercise 13.1, Q3 data).
Sorted: 10, 11, 11, 12, 13, 13 | 14, 16, 16, 17, 17, 18. Q1 is the median of the lower half, (11 + 12)/2 = 11.5, and Q3 = (16 + 17)/2 = 16.5, so QD = 2.5.
Q12. Find the quartile deviation of the seven values 3, 5, 6, 8, 9, 11, 14.
Leave the middle value 8 out of both halves. The lower half 3, 5, 6 has median 5 and the upper half 9, 11, 14 has median 11, so QD = (11 − 5)/2 = 3.