Question

The following table gives frequency distribution of marks of 100 students in an examination.

Marks	15 –20	20 – 25	25 – 30	30 –35	35 –  40	40 – 45	45 – 50
No. of students	9	12	23	31	10	8	7

Determine D₆, Q_1, and P₈₅ graphically.

Answer 1

To draw an ogive curve, we construct the less than cumulative frequency table as given below:

Marks	No. of students (f)	Less than cumulative Frequency (c.f.)
15 – 20	9	9
20 – 25	12	21
25 – 30	23	44
30 – 35	31	75
35 – 40	10	85
40 – 45	8	93
45 – 50	7	100
Total	100

The points to be plotted for less than ogive are (20, 9), (25, 21), (30 , 44), (35, 75), (40, 85), (45, 93), (50, 100).

Here, N = 100
For D₆, `(6"N")/(10) = (6(100))/(10)` = 60

For Q₁, `"N"/4=100/4` = 25

For P₈₅, `(85"N")/100=(85xx100)/100` = 85

∴ We take the points having Y coordinates 60, 25, and 85 on Y-axis. From these points, we draw lines parallel to X-axis. From the points where these lines intersect the curve, we draw perpendiculars on X-axis.
X coordinates of these points give the values of D₆, Q₁, and P₈₅.

∴ D₆ = 32.5, Q₁ = 26, P₈₅ = 40