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

Wages (in ₹)	No. of workers
130-140	48
140-150	34
150-160	26
160-170	32
170-180	20

Draw a cumulative frequency curve and use it to estimate :

(i) the median wage

(ii) semi-inter quartile range

(iii) percentage of workers who earn between ₹ 145 and ₹ 165.

Cumulative frequency distribution table :

Since, n = 160, which is even.

Median = $\dfrac{n}{2}$ th term = $\dfrac{160}{2}$ = 80th term.

Steps of construction :

1. Take 1 cm = ₹ 20 on x-axis.

2. Take 2 cm = 10 workers on y-axis.

3. Plot the points (130, 0), (140, 48), (150, 82), (160, 108), (170, 140) and (180, 160).

4. Join the points by a free hand curve.

5. Mark point A = 80 on y-axis, from point A draw a horizontal line parallel to x-axis touching the graph at point B, from point B draw a vertical line parallel to y-axis touching x-axis at point C (equal to 149).
   By formula,
   Lower quartile = $\dfrac{n}{4}$ th term = $\dfrac{160}{4}$ = 40th term.
   Upper quartile = $\dfrac{3n}{4}$ th term = $\dfrac{3 \times 160}{4}$ = 120th term.

6. Mark point D = 40 on y-axis, from point D draw a horizontal line parallel to x-axis touching the graph at point E, from point E draw a vertical line parallel to y-axis touching x-axis at point F (equal to 138).

7. Mark point G = 120 on y-axis, from point G draw a horizontal line parallel to x-axis touching the graph at point H, from point H draw a vertical line parallel to y-axis touching x-axis at point I (equal to 163).
   Inter quartile = Upper quartile - Lower quartile = 163 - 138 = 25.
   Semi-inter quartile range = $\dfrac{\text{Inter quartile range}}{2} = \dfrac{25}{2}$ = 12.5

8. Mark point J = 145 on x-axis, from point J draw a vertical line parallel to y-axis touching the graph at point K, from point K draw a horizontal line parallel to x-axis touching y-axis at point L (equal to 66). L represents no. of workers earning less than or equal to ₹ 145.

9. Mark point M = 165 on x-axis, from point M draw a vertical line parallel to y-axis touching the graph at point N, from point N draw a horizontal line parallel to x-axis touching y-axis at point O (equal to 124). O represents no. of workers earning less than or equal to ₹ 165.

No. of workers who earn between ₹ 145 and ₹ 165 = 124 - 66 = 58.

Percentage of workers who earn between ₹ 145 and ₹ 165

$= \dfrac{\text{No. of workers earning between ₹ 145 and ₹ 165}}{\text{Total no. of workers}} \times 100 = \dfrac{58}{160} \times 100$ = 36.25%

(i) Hence, median wage = ₹ 149.

(ii) Hence, semi-inter quartile range = 12.5

(iii) Hence, percentage of workers who earn between ₹ 145 and ₹ 165 = 36.25%.