Question

The daily wages of 160 workers in a building project are given below:

Wages (in ₹)	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60	60 – 70	70 – 80
No. of workers	12	20	30	38	24	16	12	8

Using a graph paper, draw an Ogive for the above distribution. Use your Ogive to estimate:

1. the median wage of the workers.
2. the upper quartile wage of the workers.
3. the lower quartile wages of the workers.
4. the percentage of workers who earn more than ₹ 45 a day.

Answer 1

1. Construct the cumulative frequency table

To draw an Ogive, we first calculate the cumulative frequency (CF) by adding the number of workers in each successive wage bracket.

Wages (₹)	No. of Workers (f)	Cumulative Frequency (CF)	Points to Plot (x, y)
0 – 10	12	12	(10, 12)
10 – 20	20	32	(20, 32)
20 – 30	30	62	(30, 62)
30 – 40	38	100	(40, 100)
40 – 50	24	124	(50, 124)
50 – 60	16	140	(60, 140)
60 – 70	12	152	(70, 152)
70 – 80	8	160	(80, 160)

2. Estimate values from the Ogive

Using the total number of workers (N = 160), we locate specific points on the Y-axis and drop perpendiculars to the X-axis to find the corresponding wage estimates.

i. Median wage: The median is the `(N/2)^(th)` term, which is the 80^th observation. From the Ogive, at CF = 80, the corresponding wage is approximately ₹ 34.74.

ii. Upper quartile wage (Q₃): The upper quartile is the `((3N)/4)^(th)` term, which is the 120^th observation. From the Ogive, at CF = 120, the corresponding wage is approximately ₹ 48.33.

iii. Lower quartile wage (Q₁): The lower quartile is the `(N/4)^(th)` term, which is the 40^th observation. From the Ogive, at CF = 40, the corresponding wage is approximately ₹ 22.67.

iv. Percentage of workers earning more than ₹ 45:

Locate ₹ 45 on the X-axis and find its corresponding CF on the Y-axis, which is 112 workers.

Workers earning more than ₹ 45 = Total workers – Workers earning up to ₹ 45 = 160 – 112 = 48 workers.

Percentage = `48/160 xx 100` = 30%.

The estimated median wage is ₹ 34.74, the upper quartile wage is ₹ 48.33, the lower quartile wage is ₹ 22.67 and 30% of workers earn more than ₹ 45 a day.