Mathematics
The following distribution represents the height of 160 students of a school.
| Height (in cm) | No. of students |
|---|---|
| 140 - 145 | 12 |
| 145 - 150 | 20 |
| 150 - 155 | 30 |
| 155 - 160 | 38 |
| 160 - 165 | 24 |
| 165 - 170 | 16 |
| 170 - 175 | 12 |
| 175 - 180 | 8 |
Draw an ogive for the given distribution taking 2 cm = 5 cm of height on one axis and 2 cm = 20 students on the other axis. Using the graph, determine :
(i) The median height.
(ii) The inter quartile range.
(iii) The number of students whose height is above 172 cm.
Measures of Central Tendency
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Answer
- The cumulative frequency table for the given continuous distribution is :
| Height (in cm) | No. of students | Cumulative frequency |
|---|---|---|
| 140 - 145 | 12 | 12 |
| 145 - 150 | 20 | 32 |
| 150 - 155 | 30 | 62 |
| 155 - 160 | 38 | 100 |
| 160 - 165 | 24 | 124 |
| 165 - 170 | 16 | 140 |
| 170 - 175 | 12 | 152 |
| 175 - 180 | 8 | 160 |
Take 2 cm along x-axis = 5 cm (height)
Take 1 cm along y-axis = 20 (students)
Since, scale on x-axis starts at 140, a kink is shown near the origin on x-axis to indicate that the graph is drawn to scale beginning at 140.
Plot the points (145, 12), (150, 32), (155, 62), (160, 100), (165, 124), (170, 140), (175, 152) and (180, 160) representing upper class limits and the respective cumulative frequencies. Also plot the point representing lower limit of the first class i.e. 140 - 145.
Join these points by a freehand drawing.
The required ogive is shown in figure above.
(i) Here, n (no. of students) = 160.
To find the median :
Let A be the point on y-axis representing frequency = = 80.
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 height = 157.5 cm.
Hence, the median height = 157.5 cm.
(ii) To find lower quartile :
Let B be the point on y-axis representing frequency = = 40
Through B, draw a horizontal line to meet the ogive at Q. Through Q, draw a vertical line to meet the x-axis at N. The abscissa of the point N represents 151.5.
To find upper quartile :
Let C be the point on y-axis representing frequency = = 120
Through C, draw a horizontal line to meet the ogive at R. Through R, draw a vertical line to meet the x-axis at O. The abscissa of the point O represents 164.
Inter quartile range = Upper quartile - Lower quartile = 164 - 151.5 = 12.5.
Hence, the inter quartile range = 12.5 cm.
(iii) Let T be the point on x-axis representing height = 172 cm.
Through T, draw a vertical line to meet the ogive at S. Through S, draw a horizontal line to meet the y-axis at D. The ordinate of the point D represents 144.
No. of students shorter than 172 cm = 144.
So, no. of students taller than 172 cm = Total students - No. of students shorter than 172 cm = 160 - 144 = 16.
Hence, there are 16 students taller than 172 cm.
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