Draw an ogive to represent the following frequency distribution: Class-interval Number of students 0 - 4 2 5 - 9 6 10 - 14 10 15 - 19 5 20 - 24 3

The given frequency distribution is not continuous. So, we first make it continuous and prepare the cumulative frequency distribution as under: Class-interval Number of students Upper class limits Cumulative frequency 0 - 4 . 5 2 Less than 4 . 5 2 4 . 5 - 9 . 5 6 Less than 9 . 5 8 9 . 5 - 14 . 5 10 Less than 14 . 5 18 14 . 5 - 19 . 5 5 Less than 19 . 5 23 19 . 5 - 24 . 5 3 Less than 24 . 5 26 Now, we mark the upper class limits or upper boundaries along x -axis and cumulative frequencies along y -axis. Thus, we plot the points ( 4 . 5 , 2 ) , ( 9 . 5 , 8 ) , ( 14 . 5 , 18 ) , ( 19 . 5 , 23 ) and 24 . 5 , 26 .

The monthly profits ( in ₹ ), of 100 shops are distributed as follows: Profits per shop No. of shops 0 - 50 12 50 - 100 18 100 - 150 27 150 - 200 20 200 - 250 17 250 - 300 6 Draw the frequency polygon for it.

We have, Profit per shop Mid-value No. of shops ( - 50 ) - 0 - 25 0 0 - 50 25 12 50 - 100 75 18 100 - 150 125 27 150 - 200 175 20 200 - 250 225 17 250 - 300 275 6 300 - 350 325 0 Now, we mark the mid-value along x -axis and number of shops along y -axis. Thus, we plot the points ( - 25 , 0 ) , ( 25 , 12 ) , ( 75 , 18 ) , ( 125 , 27 ) , ( 175 , 20 ) , ( 225 , 17 ) , ( 275 , 6 ) & ( 325 , 0 ) .

The following distribution given the daily income of 50 workers of a factory: Daily income (in ₹ ) Number of workers: 100 - 120 12 120 - 140 14 140 - 160 8 160 - 180 6 180 - 200 10 Convert the above distribution to a less than type cumulative frequency distribution and draw its ogive.

We first prepare the cumulative frequency table by less than method as given below: Daily income (in ₹ ) Cumulative frequency Less than - 120 12 Less than - 140 26 Less than - 160 34 Less than - 180 40 Less than - 200 50 Now, we mark the upper-class limits along x -axis and cumulative frequencies along y -axis. Thus, we plot the points ( 120 , 12 ) , ( 140 , 26 ) , ( 160 , 34 ) , ( 180 , 40 ) & ( 200 , 50 ) .

The following table gives production yield per hectare of wheat of 100 farms of a village: Production yield in kg per hectare: 50 - 55 55 - 60 60 - 65 65 - 70 70 - 75 75 - 80 Number of farms: 2 8 12 24 38 16 Draw 'less than' ogive and 'more than' ogive.

Less than method: We first prepare the cumulative frequency table by less than method as given below: Production yield (in kg per hectare) Number of farms Production yield less than Cumulative frequency 50 - 55 2 55 2 55 - 60 8 60 10 60 - 65 12 65 22 65 - 70 24 70 46 70 - 75 38 75 84 75 - 80 16 80 100 Now, mark the upper class limits on x -axis and cumulative frequencies along y -axis. We plot the points ( 55 , 2 ) , ( 60 , 10 ) , ( 65 , 22 ) , ( 70 , 46 ) , ( 75 , 84 ) and ( 80 , 100 ) . More than method: We prepare the cumulative frequency table by more than method as given below: Production yield (in kg per hectare) Number of farms Production yield more than Cumulative frequency 50 - 55 2 50 100 55 - 60 8 55 98 60 - 65 12 60 90 65 - 70 24 65 78 70 - 75 38 70 54 75 - 80 16 75 16 Now, mark the lower class limits on x -axis and cumulative frequencies along y -axis. We plot the points ( 50 , 100 ) , ( 55 , 98 ) , ( 60 , 90 ) , ( 65 , 78 ) , ( 70 , 54 ) and ( 75 , 16 ) .

During the medical check-up of 35 students of a class, their weights were recorded as follows: Daily income (in Rs) Cumulative frequency Less than 38 0 Less than 40 3 Less than 42 5 Less than 44 9 Less than 46 14 Less than 48 28 Less than 50 32 Less than 52 35 Draw a less than type ogive for the given data. Hence, obtain the median weight from the graph and verify the result by using the formula.

Less than method: Now, mark the upper class limits on x -axis and cumulative frequencies along y -axis. We plot the points ( 38 , 0 ) , ( 40 , 3 ) , ( 42 , 5 ) , ( 44 , 9 ) , ( 46 , 14 ) , ( 48 , 28 ) , ( 50 , 32 ) & ( 52 , 35 ) . Weight (in kg) Number of students Weight less than Cumulative frequency 36 - 38 0 38 0 38 - 40 3 40 3 40 - 42 2 42 5 42 - 44 4 44 9 44 - 46 5 46 14 46 - 48 14 48 28 48 - 50 4 50 32 50 - 52 3 52 35 Now, mark the point A whose ordinate is N 2 = 35 2 = 17 . 5 . Its x -coordinate is 46 . 5 . So median of the given data is 46 . 5 . Verification: We have, Weight (in kg) Number of students Cumulative frequency 36 - 38 0 0 38 - 40 3 3 40 - 42 2 5 42 - 44 4 9 44 - 46 5 14 46 - 48 14 28 48 - 50 4 32 50 - 52 3 35 Now, N = 35 ⇒ N 2 = 35 2 = 17 . 5 The cumulative frequency just greater than 17 . 5 is 28 and the corresponding class is 46 - 48 . Thus, 46 - 48 is the median class such that. l = 46 , f = 14 , cf = 14 & h = 2 . ∴ Median = l + N 2 - cf f × h = 46 + 17 . 5 - 14 14 × 2 = 46 + 3 . 5 14 × 2 = 46 + 7 14 = 46 + 0 . 5 = 46 . 5 ⇒ Median = 46 . 5 kg Hence, verified.

The annual rainfall record of a city for 66 days if given in the following table: Rainfall (in cm): 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 Number of days: 22 10 8 15 5 6 Calculate the median rainfall using ogives of more than type and less than type.

Less than method: Upper limits Less than cumulative frequency Less than 10 22 Less than 20 32 Less than 30 40 Less than 40 55 Less than 50 60 Less than 60 66 Considering upper limit of class on x -axis and less than cumulative frequency on y -axis to draw the ogive. We plot the points, ( 10 , 22 ) , ( 20 , 32 ) , ( 30 , 40 ) , ( 40 , 55 ) , ( 50 , 60 ) & ( 60 , 66 ) to get less than ogive. More than method: Lower limits More than cumulative frequency More than 0 66 More than 10 44 More than 20 34 More than 30 26 More than 40 11 More than 50 6 Considering lower limit on x -axis and more than cumulative frequency on y -axis to draw the ogive. We plot the points ( 0 , 66 ) , ( 10 , 44 ) , ( 20 , 24 ) , ( 30 , 26 ) , ( 40 , 11 ) & ( 50 , 6 ) to get more than ogive. We find that the two types of cumulative frequency curves intersect point P . From point P perpendicular PM is drawn on x -axis. The amount of rainfall corresponding to M is 21 . 25 cm . Hence, the median is 21 . 25 cm .

The following table gives the height of trees: Height No. of trees Less than 7 26 Less than 14 57 Less than 21 92 Less than 28 134 Less than 35 216 Less than 42 287 Less than 49 341 Less than 56 360 Draw 'less than' ogive and 'more than' ogive.

Less than method : Height Cumulative frequency Less than 7 26 Less than 14 57 Less than 21 92 Less than 28 134 Less than 35 216 Less than 42 287 Less than 49 341 Less than 56 360 Now, we mark the upper class limits along x -axis and less than cumulative frequency along y -axis. Thus, we plot ( 7 , 26 ) , ( 14 , 57 ) , ( 21 , 92 ) , ( 28 , 134 ) , ( 35 , 216 ) , ( 42 , 287 ) , ( 49 , 341 ) & ( 56 , 360 ) . More than method : We prepare the cumulative frequency table by more than method as given below : Height Frequency Height Cumulative frequency 0 - 7 26 More than 0 360 7 - 14 31 More than 7 334 14 - 21 35 More than 14 303 21 - 28 42 More than 21 268 28 - 35 82 More than 28 226 35 - 42 71 More than 35 144 42 - 49 54 More than 42 73 49 - 56 19 More than 49 19 Now, we mark the lower class limits on x -axis and more than cumulative frequency along y -axis. Thus, we plot the ( 0 , 360 ) , ( 7 , 334 ) , ( 14 , 303 ) , ( 21 , 268 ) , ( 28 , 226 ) , ( 35 , 144 ) , ( 42 , 73 ) & ( 49 , 19 ) .

The annual profits earned by 30 shops of a shopping complex in a locality give rise to the following distribution: Profit (in lakhs in ₹) Number of shops More than or equal to 5 30 More than or equal to 10 28 More than or equal to 15 16 More than or equal to 20 14 More than or equal to 25 10 More than or equal to 30 7 More than or equal to 35 3 Draw both ogives for the above data and hence obtain the median.

More than method: Profit (in lakhs in ₹) Number of shops More than equal to 5 30 More than equal to 10 28 More than equal to 15 16 More than equal to 20 14 More than equal to 25 10 More than equal to 30 7 More than equal to 35 3 Now, we mark the lower class limits on x -axis and the more than cumulative frequencies along y -axis. Thus, we plot the points ( 5 , 30 ) , ( 10 , 28 ) , ( 15 , 16 ) , ( 20 , 14 ) , ( 25 , 10 ) , ( 30 , 7 ) & ( 35 , 3 ) . Less than method: Profit (in lakhs in ₹) Number of shops (frequency) Profit less than Cumulative frequency 0 - 10 2 10 2 10 - 15 12 15 14 15 - 20 2 20 16 20 - 25 4 25 20 25 - 30 3 30 23 30 - 35 4 35 27 35 - 40 3 40 30 Now, we mark the upper class limits on x -axis and the less than cumulative frequencies along y -axis. Thus, we plot the points ( 10 , 2 ) , ( 15 , 14 ) , ( 20 , 16 ) , ( 25 , 20 ) , ( 30 , 23 ) , ( 35 , 27 ) & ( 40 , 30 ) . We find that the two types of cumulative frequency curves intersect at point P . From point P , we draw perpendicular PM to x -axis. The value of profit corresponding to M is ₹ 17 . 5 lakh . Hence, the median is ₹ 17 . 5 lakh .

R. D. Sharma Solutions for Chapter: Statistics, Exercise 6: EXERCISE 15.6

Author:R. D. Sharma

R. D. Sharma Mathematics Solutions for Exercise - R. D. Sharma Solutions for Chapter: Statistics, Exercise 6: EXERCISE 15.6

Attempt the free practice questions on Chapter 15: Statistics, Exercise 6: EXERCISE 15.6 with hints and solutions to strengthen your understanding. MATHEMATICS CLASS X solutions are prepared by Experienced Embibe Experts.

Questions from R. D. Sharma Solutions for Chapter: Statistics, Exercise 6: EXERCISE 15.6 with Hints & Solutions

HARD

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 15 - Statistics>EXERCISE 15.6>Q 3

Draw an ogive to represent the following frequency distribution:

Class-interval	Number of students
$0 - 4$	$2$
$5 - 9$	$6$
$10 - 14$	$10$
$15 - 19$	$5$
$20 - 24$	$3$

HARD

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 15 - Statistics>EXERCISE 15.6>Q 4

The monthly profits ( in $₹$ ), of $100$ shops are distributed as follows:

Profits per shop	No. of shops
$0 - 50$	$12$
$50 - 100$	$18$
$100 - 150$	$27$
$150 - 200$	$20$
$200 - 250$	$17$
$250 - 300$	$6$

Draw the frequency polygon for it.

HARD

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 15 - Statistics>EXERCISE 15.6>Q 5

The following distribution given the daily income of $50$ workers of a factory:

Daily income (in $₹$ )	Number of workers:
$100 - 120$	$12$
$120 - 140$	$14$
$140 - 160$	$8$
$160 - 180$	$6$
$180 - 200$	$10$

Convert the above distribution to a less than type cumulative frequency distribution and draw its ogive.

HARD

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 15 - Statistics>EXERCISE 15.6>Q 6

The following table gives production yield per hectare of wheat of $100$ farms of a village:

Production yield in kg per hectare:	$50 - 55$	$55 - 60$	$60 - 65$	$65 - 70$	$70 - 75$	$75 - 80$
Number of farms:	$2$	$8$	$12$	$24$	$38$	$16$

Draw 'less than' ogive and 'more than' ogive.

HARD

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 15 - Statistics>EXERCISE 15.6>Q 7

During the medical check-up of $35$ students of a class, their weights were recorded as follows:

Daily income (in Rs)	Cumulative frequency
Less than $38$	$0$
Less than $40$	$3$
Less than $42$	$5$
Less than $44$	$9$
Less than $46$	$14$
Less than $48$	$28$
Less than $50$	$32$
Less than $52$	$35$

Draw a less than type ogive for the given data. Hence, obtain the median weight from the graph and verify the result by using the formula.

HARD

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 15 - Statistics>EXERCISE 15.6>Q 8

The annual rainfall record of a city for $66$ days if given in the following table:

Rainfall (in cm):	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$
Number of days:	$22$	$10$	$8$	$15$	$5$	$6$

Calculate the median rainfall using ogives of more than type and less than type.

HARD

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 15 - Statistics>EXERCISE 15.6>Q 9

The following table gives the height of trees:

Height	No. of trees
Less than $7$	$26$
Less than $14$	$57$
Less than $21$	$92$
Less than $28$	$134$
Less than $35$	$216$
Less than $42$	$287$
Less than $49$	$341$
Less than $56$	$360$

Draw 'less than' ogive and 'more than' ogive.

HARD

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 15 - Statistics>EXERCISE 15.6>Q 10

The annual profits earned by $30$ shops of a shopping complex in a locality give rise to the following distribution:

Profit (in lakhs in ₹)	Number of shops
More than or equal to $5$	$30$
More than or equal to $10$	$28$
More than or equal to $15$	$16$
More than or equal to $20$	$14$
More than or equal to $25$	$10$
More than or equal to $30$	$7$
More than or equal to $35$	$3$

Draw both ogives for the above data and hence obtain the median.

R. D. Sharma Solutions for Chapter: Statistics, Exercise 6: EXERCISE 15.6

R. D. Sharma Mathematics Solutions for Exercise - R. D. Sharma Solutions for Chapter: Statistics, Exercise 6: EXERCISE 15.6

Questions from R. D. Sharma Solutions for Chapter: Statistics, Exercise 6: EXERCISE 15.6 with Hints & Solutions

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