Mathematics
The daily wages of 80 workers in a project are given below:
| Wages (in ₹) | Number of workers |
|---|---|
| 400 - 450 | 2 |
| 450 - 500 | 6 |
| 500 - 550 | 12 |
| 550 - 600 | 18 |
| 600 - 650 | 24 |
| 650 - 700 | 13 |
| 700 - 750 | 5 |
Use a graph paper to draw an ogive for the above distribution. (Use a scale of 2 cm = ₹ 50 on x-axis and 2 cm = 10 workers on y-axis). Use your ogive to estimate:
(i) the median wage of the workers.
(ii) the lower quartile wage of the workers.
(iii) the number of workers who earn more than ₹ 625 daily.
Measures of Central Tendency
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Answer
Cumulative frequency distribution table :
| Wages (in ₹) | Number of workers | Cumulative frequency |
|---|---|---|
| 400 - 450 | 2 | 2 |
| 450 - 500 | 6 | 8 (2 + 6) |
| 500 - 550 | 12 | 20 (8 + 12) |
| 550 - 600 | 18 | 38 (20 + 18) |
| 600 - 650 | 24 | 62 (38 + 24) |
| 650 - 700 | 13 | 75 (62 + 13) |
| 700 - 750 | 5 | 80 (75 + 5) |
Here, n = 80, which is even.
Steps of construction:
Take 2 cm along x-axis = ₹ 50
Take 2 cm along y-axis = 10 workers
Since, scale on x-axis starts at 400, a kink is shown near the origin on x-axis to indicate that the graph is drawn to scale beginning at 400.
Plot the points (450, 2), (500, 8), (550, 20), (600, 38), (650, 62), (700, 75), (750, 80) representing upper class limits and the respective cumulative frequencies. Also plot the point representing lower limit of the first class i.e, 400 - 450.
Joint the points by a free hand curve.
(i) Here, n = 80
To find the median :
Let A be the point on y-axis representing frequency = = 40.
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 points M represents 600.5.
Hence, the median is ₹ 600.5.
(ii) To find lower quartile:
Let B be the point on y-axis representing frequency = = 20.
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 550.
Hence, lower quartile wage = ₹ 550.
(iii) Let T be the point on x-axis representing wage = ₹ 625.
Through T, draw a vertical line to meet the ogive at S. Through S, draw a horizontal line to meet the y-axis at C. The ordinate of the point C. The ordinate of point C represents 51.
Workers who earn less than ₹ 625 = 51.
So, workers earning more than ₹ 625 = Total workers - workers who earn less than ₹ 625 = 80 - 51 = 29.
Hence, there are 29 workers earning more than ₹ 625 daily.
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