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.

Question

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.

KnowledgeBoat · Accepted Answer

Cumulative frequency distribution table : Since, n = 160, which is even. Median = th term = = 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 = th term = = 40th term. Upper quartile = th term = = 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 = = 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 = 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%.

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