The marks obtained by 120 students in a test are given below. Marks Number of students 0 - 10 5 10 - 20 9 20 - 30 16 30 - 40 22 40 - 50 26 50 - 60 18 60 - 70 11 70 - 80 6 80 - 90 4 90 - 100 3 Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for ogive to estimate the median.

Given table is, Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 - 100 Number of students 5 9 16 22 26 18 11 6 4 3 The cumulative frequency table for the given continuous distribution is given below Marks Number of students Cumulative frequency c f 0 - 10 5 5 10 - 20 9 14 20 - 30 16 30 30 - 40 22 52 40 - 50 26 78 50 - 60 18 96 60 - 70 11 107 70 - 80 6 113 80 - 90 4 117 90 - 100 3 120 On the graph paper, we plot the following points A ( 10 , 5 ) , B ( 20 , 14 ) , C ( 30 , 30 ) , D ( 40 , 52 ) , E ( 50 , 78 ) , F ( 60 , 96 ) , G ( 70 , 107 ) , H ( 80 , 113 ) , I ( 90 , 117 ) and V ( 100 , 120 ) . Join all these points by a free hand drawing. The required ogive is shown on the graph paper given below Here, the number of students ( n ) = 120 , which is even. Let P be the point on Y - axis representing frequency = n 2 = 120 2 = 60 . Through P , draw a horizontal line to meet the ogive at point Q . Through Q , draw a vertical line to meet the X - axis at T . The abscissa of the point T represents 43 marks. Hence, the median marks is 43 .

The marks obtained by 120 students in a Mathematics test are given below: Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 - 100 No. of students 5 9 16 22 26 18 11 6 4 3 Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for ogive to estimate the number of students who obtained more than 75 % marks in the test.

The cumulative frequency distribution for the given data is as follows: Marks No. of students Cumulative frequency 0 - 10 5 5 10 - 20 9 14 20 - 30 16 30 30 - 40 22 52 40 - 50 26 78 50 - 60 18 96 60 - 70 11 107 70 - 80 6 113 80 - 90 4 117 90 - 100 3 120 To draw the Ogive: Plot the points ( 10 , 5 ) , ( 20 , 14 ) , ( 30 , 30 ) , ( 40 , 52 ) , ( 50 , 78 ) , ( 60 , 96 ) , ( 70 , 107 ) , ( 80 , 113 ) , ( 90 , 117 ) , ( 100 , 120 ) and ( 0 , 0 ) . The Ogive plot for the given data is as follows: To find the number of students with marks > 75 : ( i ) Let D be the point on x - axis representing variate = 75 ( i i ) Through D , we draw a vertical line to meet the cumulative frequency curve at E . ( i i i ) Through E , draw a horizontal line to meet the y - axis at F . The ordinate of the point F represents 110 . ∴ The number of students with marks more than 75 = 120 - 110 = 10 .

The marks obtained by 120 students in a Mathematics test are given below: Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 - 100 No. of students 5 9 16 22 26 18 11 6 4 3 Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for ogive to estimate the number of students who did not pass in the test, if the pass percentage was 40 .

The cumulative frequency distribution for the given data is as follows: Marks No. of students Cumulative frequency 0 - 10 5 5 10 - 20 9 14 20 - 30 16 30 30 - 40 22 52 40 - 50 26 78 50 - 60 18 96 60 - 70 11 107 70 - 80 6 113 80 - 90 4 117 90 - 100 3 120 To draw the Ogive: Plot the points ( 10 , 5 ) , ( 20 , 14 ) , ( 30 , 30 ) , ( 40 , 52 ) , ( 50 , 78 ) , ( 60 , 96 ) , ( 70 , 107 ) , ( 80 , 113 ) , ( 90 , 117 ) , ( 100 , 120 ) and ( 0 , 0 ) . The Ogive plot for the given data is as follows: To find the number of students who did not pass: ( i ) Let C be the point on x - axis representing variate = 40 ( i i ) Through C , we draw a vertical line to meet the cumulative frequency curve at T . ( i i i ) Through T , draw a horizontal line to meet the y - axis at W . The ordinate of the point W represents 52 . ∴ The number of students with marks less than 40 = 52 .

The following distribution represents the height of 160 students of a school. Height (in cm ) 140 - 145 145 - 150 150 - 155 155 - 160 160 - 165 165 - 170 170 - 175 175 - 180 No. of students 12 20 30 38 24 16 12 8 Draw an ogive for the given distribution taking 2 cm = 5 cm of height on one axis and 2 cm = 20 students on the other axis. Using the graph, determine the median height.

The cumulative frequency distribution for the given data is as follows: Height (in cm ) No. of students Cumulative frequency 140 - 145 12 12 145 - 150 20 32 150 - 155 30 62 155 - 160 38 100 160 - 165 24 124 165 - 170 16 140 170 - 175 12 152 175 - 180 8 160 To draw the Ogive: Plot the points ( 145 , 12 ) , ( 150 , 32 ) , ( 155 , 62 ) , ( 160 , 100 ) , ( 165 , 124 ) , ( 170 , 140 ) , ( 175 , 152 ) , ( 180 , 160 ) and ( 140 , 0 ) after drawing a kink on the x - axis near to the origin, representing that it starts from 140 and not from 0 . The Ogive plot for the given data is as follows: Here, n (total number of students) = 160 , which is even. To find the Median: ( i ) Let A be the point on y - axis representing frequency = n 2 = 80 ( 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 157 . 5 cm . ∴ The required median height = 157 . 5 cm .

The following distribution represents the height of 160 students of a school. Height (in cm ) 140 - 145 145 - 150 150 - 155 155 - 160 160 - 165 165 - 170 170 - 175 175 - 180 No. of students 12 20 30 38 24 16 12 8 Draw an ogive for the given distribution taking 2 cm = 5 cm of height on one axis and 2 cm = 20 students on the other axis. Using the graph, determine the inter quartile range.

The cumulative frequency distribution for the given data is as follows: Height (in cm ) No. of students Cumulative frequency 140 - 145 12 12 145 - 150 20 32 150 - 155 30 62 155 - 160 38 100 160 - 165 24 124 165 - 170 16 140 170 - 175 12 152 175 - 180 8 160 To draw the Ogive: Plot the points ( 145 , 12 ) , ( 150 , 32 ) , ( 155 , 62 ) , ( 160 , 100 ) , ( 165 , 124 ) , ( 170 , 140 ) , ( 175 , 152 ) , ( 180 , 160 ) and ( 140 , 0 ) after drawing a kink on the x - axis near to the origin, representing that it starts from 140 and not from 0 . The Ogive plot for the given data is as follows: Here, n (total number of students) = 160 , which is even. To find the Upper Quartile ( Q 3 ): ( i ) Let B be the point on y - axis representing frequency = 3 n 4 = 120 ( 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 164 cm . ∴ The Upper Quartile height = 164 cm . To find the Lower Quartile ( Q 1 ): ( i ) Let C be the point on y - axis representing frequency = n 4 = 40 ( i i ) Through C , we draw a horizontal line to meet the cumulative frequency curve at S . ( i i i ) Through S , draw a vertical line to meet the x - axis at T . The abscissa of the point T represents 151 . 5 cm . ∴ The Lower Quartile height = 151 . 5 cm . Hence, the Inter Quartile Range = Q 3 - Q 1 = 164 - 151 . 5 = 12 . 5 cm

The following distribution represents the height of 160 students of a school. Height (in cm ) 140 - 145 145 - 150 150 - 155 155 - 160 160 - 165 165 - 170 170 - 175 175 - 180 No. of students 12 20 30 38 24 16 12 8 Draw an ogive for the given distribution taking 2 cm = 5 cm of height on one axis and 2 cm = 20 students on the other axis. Using the graph, determine the number of students whose height is above 172 cm .

The cumulative frequency distribution for the given data is as follows: Height (in cm ) No. of students Cumulative frequency 140 - 145 12 12 145 - 150 20 32 150 - 155 30 62 155 - 160 38 100 160 - 165 24 124 165 - 170 16 140 170 - 175 12 152 175 - 180 8 160 To draw the Ogive: Plot the points ( 145 , 12 ) , ( 150 , 32 ) , ( 155 , 62 ) , ( 160 , 100 ) , ( 165 , 124 ) , ( 170 , 140 ) , ( 175 , 152 ) , ( 180 , 160 ) and ( 140 , 0 ) after drawing a kink on the x - axis near to the origin, representing that it starts from 140 and not from 0 . The Ogive plot for the given data is as follows: To find the number of students with height > 172 cm : ( i ) Let D be the point on x - axis representing variate = 172 ( i i ) Through D , we draw a vertical line to meet the cumulative frequency curve at E . ( i i i ) Through E , draw a horizontal line to meet the y - axis at F . The ordinate of the point F represents 144 . ∴ The number of students with height more than 172 cm = 160 - 144 = 16 .

100 pupils in a school have heights as tabulated below: Height (in cm ) 121 - 130 131 - 140 141 - 150 151 - 160 161 - 170 171 - 180 No. of pupils 12 16 30 20 14 8 Draw the ogive for the above data and from it, determine the median (use graph paper).

Here, we see that the classes are not continuous. So we have to make them continuous with the help of adjustment factor. Adjustment factor = 131 - 130 2 = 1 2 = 0 . 5 So, the adjusted cumulative frequency distribution for the given data is as follows: Height (in cm ) No. of pupils Cumulative frequency 120 . 5 - 130 . 5 12 12 130 . 5 - 140 . 5 16 28 140 . 5 - 150 . 5 30 58 150 . 5 - 160 . 5 20 78 160 . 5 - 170 . 5 14 92 170 . 5 - 180 . 5 8 100 To draw the Ogive: Plot the points ( 130 . 5 , 12 ) , ( 140 . 5 , 28 ) , ( 150 . 5 , 58 ) , ( 160 . 5 , 78 ) , ( 170 . 5 , 92 ) , ( 180 . 5 , 100 ) and ( 120 . 5 , 0 ) after drawing a kink on the x - axis near to the origin, representing that it starts from 120 . 5 instead of 0 . The Ogive plot for the given data is as follows: Here, n (total number of pupils) = 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 147 . 5 cm . ∴ The required median height = 147 . 5 cm .

M L Aggarwal Solutions for Exercise 6: Exercise 21.6

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Exercise 6: Exercise 21.6

Attempt the practice questions from Exercise 6: Exercise 21.6 with hints and solutions to strengthen your understanding. Understanding ICSE Mathematics Class 10 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Exercise 6: Exercise 21.6 with Hints & Solutions

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Exercise 21.6>Q 1

Using the data given below construct the cumulative frequency table and draw the ogive. From the ogive, determine the median.

Marks	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$	$60 - 70$	$70 - 80$
Number of students	$3$	$8$	$12$	$14$	$10$	$6$	$n = 60$	$2$

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Exercise 21.6>Q 9

The marks obtained by $120$ students in a test are given below.

Marks	Number of students
$0 - 10$	$5$
$10 - 20$	$9$
$20 - 30$	$16$
$30 - 40$	$22$
$40 - 50$	$26$
$50 - 60$	$18$
$60 - 70$	$11$
$70 - 80$	$6$
$80 - 90$	$4$
$90 - 100$	$3$

Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for ogive to estimate the median.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Exercise 21.6>Q 9

The marks obtained by $120$ students in a Mathematics test are given below:

Marks	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$	$60 - 70$	$70 - 80$	$80 - 90$	$90 - 100$
No. of students	$5$	$9$	$16$	$22$	$26$	$18$	$11$	$6$	$4$	$3$

Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for ogive to estimate the number of students who obtained more than $75 %$ marks in the test.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Exercise 21.6>Q 9

The marks obtained by $120$ students in a Mathematics test are given below:

Marks	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$	$60 - 70$	$70 - 80$	$80 - 90$	$90 - 100$
No. of students	$5$	$9$	$16$	$22$	$26$	$18$	$11$	$6$	$4$	$3$

Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for ogive to estimate the number of students who did not pass in the test, if the pass percentage was $40$ .

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Exercise 21.6>Q 10

The following distribution represents the height of $160$ students of a school.

Height (in $cm$ )	$140 - 145$	$145 - 150$	$150 - 155$	$155 - 160$	$160 - 165$	$165 - 170$	$170 - 175$	$175 - 180$
No. of students	$12$	$20$	$30$	$38$	$24$	$16$	$12$	$8$

Draw an ogive for the given distribution taking $2 cm = 5 cm$ of height on one axis and $2 cm = 20$ students on the other axis. Using the graph, determine the median height.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Exercise 21.6>Q 10

The following distribution represents the height of $160$ students of a school.

Height (in $cm$ )	$140 - 145$	$145 - 150$	$150 - 155$	$155 - 160$	$160 - 165$	$165 - 170$	$170 - 175$	$175 - 180$
No. of students	$12$	$20$	$30$	$38$	$24$	$16$	$12$	$8$

Draw an ogive for the given distribution taking $2 cm = 5 cm$ of height on one axis and $2 cm = 20$ students on the other axis. Using the graph, determine the inter quartile range.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Exercise 21.6>Q 10

The following distribution represents the height of $160$ students of a school.

Height (in $cm$ )	$140 - 145$	$145 - 150$	$150 - 155$	$155 - 160$	$160 - 165$	$165 - 170$	$170 - 175$	$175 - 180$
No. of students	$12$	$20$	$30$	$38$	$24$	$16$	$12$	$8$

Draw an ogive for the given distribution taking $2 cm = 5 cm$ of height on one axis and $2 cm = 20$ students on the other axis. Using the graph, determine the number of students whose height is above $172 cm$ .

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 21 - Measures of Central Tendency>Exercise 21.6>Q 11

$100$ pupils in a school have heights as tabulated below:

Height (in $cm$ )	$121 - 130$	$131 - 140$	$141 - 150$	$151 - 160$	$161 - 170$	$171 - 180$
No. of pupils	$12$	$16$	$30$	$20$	$14$	$8$

Draw the ogive for the above data and from it, determine the median (use graph paper).

M L Aggarwal Solutions for Exercise 6: Exercise 21.6

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Exercise 6: Exercise 21.6

Questions from M L Aggarwal Solutions for Exercise 6: Exercise 21.6 with Hints & Solutions

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