Question

The marks obtained by 200 students in an examination are given below : 

Marks	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90	90-100
No.of students	5	10	11	20	27	38	40	29	14	6

Using a graph paper, draw an Ogive for the above distribution. Use your Ogive to estimate:
(i) the median;
(ii) the lower quartile;
(iii) the number of students who obtained more than 80% marks in the examination and
(iv) the number of students who did not pass, if the pass percentage was 35.
Use the scale as 2 cm = 10 marks on one axis and 2 cm = 20 students on the other axis.

Answer 1

We construct cumulative frequency table of the given distribution : 

Marks	No. of students (f)	Cumulative Frequency
0-10	5	5
10-20	10	15
20-30	11	26
30-40	20	46
40-50	27	73
50-60	38	111
60-70	40	151
70-80	29	180
80-90	14	194
90-100	6	200

Take a graph paper and draw both the axes. 

On the x - axis , take a scale of 1cm= 20 to represent the marks. 

On the y - axis , take a scale of 1 cm =50 to represent the no. of students. 

Now, plot the points (10 ,5) ,(20 ,15) ,(30 ,26) ,( 40 ,46) ,(50 ,73) ,(60 ,111), (70 ,151) ,(80 ,180) ,(90 ,194) ,(100 ,200) . 

Join them by a smooth curve to get the ogive. 

(i) No. of terms = 200 

∴ Median = `(100 + 101)/2` = 100.5^th term

Through mark of 100.5 on y-axis draw a line parallel to x-axis which meets the curve at A. From A, draw a perpendicular to x-axis which meets it at B. 

The value of B is the median which is 58.5. 

(ii) Lower Quartile (Q₁) = `n/4 = 200/4` = 50^th term

Through mark of 50 on y-axis draw a line parallel to x-axis which meets the curve at P. From P, draw a perpendicular to x-axis which meets it at Q. 

The value of Q is the lower quartile which is 41 . 

(iii) From marks = 80 draw a line parallel to y-axis and meet the curve at R. From R, Draw a perpendicular on y-axis which meets it at S. The difference of the value obtained when subtracted from 200 gives the number of students who scored more than 80%. 

⇒ 200 - 180 = 20 

20 students scored more than 80%

Answer 2

Less than	C.F.	Points
10	5	(10, 5)
20	15	(20, 15)
30	26	(30, 26)
40	46	(40, 46)
50	73	(50, 73)
60	111	(60, 111)
70	151	(70, 151)
80	180	(80, 180)
90	194	(90, 194)
100	200	(100, 200)

(i) Median = `"N"/(2) = (200)/(2)`
= 100 → On seeing the corresponding value.
Median = 57
(ii) Q₁ = `"N"/(4) = (200)/(4)`
= 50 → On seeing the corresponding value.
= 38
(iii) 200 - 180 = 20 students
(iv) 38 students did not pass.