The daily wages of 30 employees in an establishment are distributed as follows: Daily wages (in ₹ ) 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 No. of employees 1 8 10 5 4 2 Estimate the modal daily wages for this distribution by a graphical method.

Steps of construction: ( i ) Take the scale along x - axis as 2 cm = 10 units , and along y - axis as 2 cm = 2 units . ( i i ) Construct the rectangles according to the data given in the question. The required histrogram is as shown in the figure above. ( i i i ) In the highest rectangle, draw two straight lines AB and CD from the corners of the rectangles on either side of the highest rectangle to the opposite corners of the highest rectangle. ( i v ) Let M be the point of intersection of the lines AB and CD . ( v ) Through the point M , draw a vertical line that meets the x - axis at point L . The abscissa of the point L represents the value 23 . ∴ The required modal wages = ₹ 23

Draw a cumulative frequency curve for the following data: Marks obtained 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 No. of students 8 10 22 40 20 Hence, determine the median.

The cumulative frequency distribution for the given data is as follows: Marks obtained No. of students Cumulative frequency 0 - 10 8 8 10 - 20 10 18 20 - 30 22 40 30 - 40 40 80 40 - 50 20 100 To draw the cumulative frequency curve: Plot the points ( 10 , 8 ) , ( 20 , 18 ) , ( 30 , 40 ) , ( 40 , 80 ) , ( 50 , 100 ) and ( 0 , 0 ) . The cumulative frequency curve for the given data is as follows: Here, n (total number of students) = 100 , which is even. To find the Median: ( i ) Let A be the point on y - axis representing frequency = n 2 = 50 ( i i ) Through A , we draw a horizontal line to meet the cumulative frequency curve at P . ( i i i ) Through P , draw a vertical line to meet the x - axis at Q . The abscissa of the point Q represents 32 . 5 marks . ∴ The required median marks = 32 . 5 .

Draw a cumulative frequency curve for the following data: Marks obtained 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 No. of students 8 10 22 40 20 Hence, determine the pass marks, if 85 % of the students pass.

The cumulative frequency distribution for the given data is as follows: Marks obtained No. of students Cumulative frequency 0 - 10 8 8 10 - 20 10 18 20 - 30 22 40 30 - 40 40 80 40 - 50 20 100 To draw the cumulative frequency curve: Plot the points ( 10 , 8 ) , ( 20 , 18 ) , ( 30 , 40 ) , ( 40 , 80 ) , ( 50 , 100 ) and ( 0 , 0 ) . The cumulative frequency curve for the given data is as follows: Here, n (total number of students) = 100 , which is even. To find the pass marks: ( i ) Let D be the point on y - axis representing frequency = 15 % of 100 = 15 ( i i ) Through D , we draw a horizontal line to meet the cumulative frequency curve at E . ( i i i ) Through E , draw a vertical line to meet the x - axis at F . The abscissa of the point F represents 18 marks . ∴ The required pass marks = 18 marks .

Draw a cumulative frequency curve for the following data: Marks obtained 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 No. of students 8 10 22 40 20 Hence, determine the marks which 45 % of the students exceed.

The cumulative frequency distribution for the given data is as follows: Marks obtained No. of students Cumulative frequency 0 - 10 8 8 10 - 20 10 18 20 - 30 22 40 30 - 40 40 80 40 - 50 20 100 To draw the cumulative frequency curve: Plot the points ( 10 , 8 ) , ( 20 , 18 ) , ( 30 , 40 ) , ( 40 , 80 ) , ( 50 , 100 ) and ( 0 , 0 ) . The cumulative frequency curve for the given data is as follows: Here, n (total number of students) = 100 . ∵ 45 % of total students exceed a certain value of marks, it implies that ( 100 - 45 ) % = 55 % of students are either below that value, or equals to that value of marks. To find the required marks: ( i ) Let B be the point on y - axis representing frequency = 55 % of 100 = 55 ( i i ) Through B , we draw a horizontal line to meet the cumulative frequency curve at M . ( i i i ) Through M , draw a vertical line to meet the x - axis at N . The abscissa of the point N represents 34 marks . ∴ The required marks = 34 marks .