Mathematics
Marks obtained by 200 students in an examination are given below :
| Marks | No. of students |
|---|---|
| 0 - 10 | 5 |
| 10 - 20 | 11 |
| 20 - 30 | 10 |
| 30 - 40 | 20 |
| 40 - 50 | 28 |
| 50 - 60 | 37 |
| 60 - 70 | 40 |
| 70 - 80 | 29 |
| 80 - 90 | 14 |
| 90 - 100 | 6 |
Draw an ogive for the given distribution taking 2 cm = 10 marks on one axis and 2 cm = 20 students on the other axis. Using the graph, determine :
(i) The median marks
(ii) The number of students who failed if minimum marks required to pass is 40.
(iii) If scoring 85 and more marks is considered as grade one, find the number of students who secured grade one in the examination.
Measures of Central Tendency
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Answer
- The cumulative frequency table for the given continuous distribution is :
| Marks | No. of students | Cumulative frequency |
|---|---|---|
| 0 - 10 | 5 | 5 |
| 10 - 20 | 11 | 16 |
| 20 - 30 | 10 | 26 |
| 30 - 40 | 20 | 46 |
| 40 - 50 | 28 | 74 |
| 50 - 60 | 37 | 111 |
| 60 - 70 | 40 | 151 |
| 70 - 80 | 29 | 180 |
| 80 - 90 | 14 | 194 |
| 90 - 100 | 6 | 200 |
Take 1 cm along x-axis = 10 scores
Take 1 cm along y-axis = 20 (students)
Plot the points (10, 5), (20, 16), (30, 26), (40, 46), (50, 74), (60, 111), (70, 151), (80, 180), (90, 194) and (100, 200) representing upper class limits and the respective cumulative frequencies.
Also plot the point representing lower limit of the first class i.e. 0 - 10.Join these points by a freehand drawing.
The required ogive is shown in figure above.
(i) Here, n (no. of students) = 200.
To find the median :
Let A be the point on y-axis representing frequency = = 100.
Through A draw a horizontal line to meet the ogive at P. Through P, draw a vertical line to meet the x-axis at M. The abscissa of the point M represents 57.
Hence, the required median marks = 57.
(ii) Let N be the point on x-axis representing marks = 40.
Through N, draw a vertical line to meet the ogive at Q. Through Q, draw a horizontal line to meet the y-axis at B. The ordinate of the point B represents 46.
Students who scored less than 40 = 46.
Hence, 46 students failed in the examination.
(iii) Let O be the point on x-axis representing marks = 85.
Through O, draw a vertical line to meet the ogive at R. Through R, draw a horizontal line to meet the y-axis at C. The ordinate of the point C represents 187.
Students who scored less than 85 = 187.
So, students scoring more than 85 = Total students - students scoring less than 85 = 200 - 187 = 13.
Hence, 13 students secured grade one in examination.
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