Statistics — Algebra textbook solutions

Mean of grouped data - Direct method

3 q

Direct method

Worked · 1

Direct method Ex.1 $The following table shows the frequency distribution of the time required for each worker to complete a work. From the table find the mean time required to complete the job for a worker. Time (Hrs.) for each to complete the work 15-19 20-24 25-29 30-34 35-39 No. of workers 10151285$

Time (Hrs.) for each to complete the work	15-19	20-24	25-29	30-34	35-39
No. of workers	10	15	12	8	5

Solved Examples

Worked · 2

Direct method SolvedEx.1 $The percentage of marks of 50 students in a test is given in the following table. Find the mean of the percentage. Percentage of marks 0-20 20-40 40-60 60-80 80-100 No. of students 3715205$
Direct method SolvedEx.2 $^\circ$

Percentage of marks	0-20	20-40	40-60	60-80	80-100
No. of students	3	7	15	20	5

Temperature ( $^\circ$ C)	24-28	28-32	32-36	36-40	40-44
No. of towns	4	5	7	8	6

Mean of grouped data - Assumed mean method

2 q

Assumed mean method

Worked · 1

Assumed mean Ex.1 $The daily sale of 100 vegetable vendors is given in the following table. Find the mean of the sale by assumed mean method. Daily sale (Rupees) 1000-1500 1500-2000 2000-2500 2500-3000 No. of vendors 15203530$

Daily sale (Rupees)	1000-1500	1500-2000	2000-2500	2500-3000
No. of vendors	15	20	35	30

Solved Examples

Worked · 1

Assumed mean SolvedEx.1 $The following table shows the frequency table of daily wages of 50 workers in a trading company. Find the mean wages of a worker, by assumed mean method. Daily Wages (Rs) 200-240 240-280 280-320 320-360 360-400 Frequency 51015128$

Daily Wages (Rs)	200-240	240-280	280-320	320-360	360-400
Frequency	5	10	15	12	8

Mean of grouped data - Step deviation method

2 q

Step deviation method

Worked · 1

Step deviation Ex.1 $The amount invested in health insurance by 100 families is given in the following frequency table. Find the mean of investments using step deviation method. Amount invested (₹) 800-1200 1200-1600 1600-2000 2000-2400 2400-2800 2800-3200 No. of families 3152025307$

Amount invested (₹)	800-1200	1200-1600	1600-2000	2000-2400	2400-2800	2800-3200
No. of families	3	15	20	25	30	7

Solved Example

Worked · 1

Step deviation SolvedEx.1 $The following table shows the funds collected by 50 students for flood affected people. Find the mean of the funds. Fund (Rupees) 0-500 500-1000 1000-1500 1500-2000 2000-2500 2500-3000 No. of students 24241811$

Fund (Rupees)	0-500	500-1000	1000-1500	1500-2000	2000-2500	2500-3000
No. of students	2	4	24	18	1	1

Practice set 6.1

6 q

Ex 6.1 Q.1 $The following table shows the number of students and the time they utilized daily for their studies. Find the mean time spent by students for their studies by direct method. Time (hrs.) 0-2 2-4 4-6 6-8 8-10 No. of students 71812103$
Ex 6.1 Q.2 $In the following table, the toll paid by drivers and the number of vehicles is shown. Find the mean of the toll by 'assumed mean' method. Toll (Rupees) 300-400 400-500 500-600 600-700 700-800 No. of vehicles 801101207040$
Ex 6.1 Q.3 $A milk centre sold milk to 50 customers. The table below gives the number of customers and the milk they purchased. Find the mean of the milk sold by direct method. Milk Sold (Litre) 1-2 2-3 3-4 4-5 5-6 No. of Customers 17131073$
Ex 6.1 Q.4 $A frequency distribution table for the production of oranges of some farm owners is given below. Find the mean production of oranges by 'assumed mean' method. Production (Thousand rupees) 25-30 30-35 35-40 40-45 45-50 No. of farm owners 2025151010$
Ex 6.1 Q.5 $A frequency distribution of funds collected by 120 workers in a company for the drought affected people are given in the following table. Find the mean of the funds by 'step deviation' method. Fund (Rupees) 0-500 500-1000 1000-1500 1500-2000 2000-2500 No. of workers 3528321510$
Ex 6.1 Q.6 $The following table gives the information of frequency distribution of weekly wages of 150 workers of a company. Find the mean of the weekly wages by 'step deviation' method. Weekly wages (Rupees) 1000-2000 2000-3000 3000-4000 4000-5000 No. of workers. 25455030$

Time (hrs.)	0-2	2-4	4-6	6-8	8-10
No. of students	7	18	12	10	3

Toll (Rupees)	300-400	400-500	500-600	600-700	700-800
No. of vehicles	80	110	120	70	40

Milk Sold (Litre)	1-2	2-3	3-4	4-5	5-6
No. of Customers	17	13	10	7	3

Production (Thousand rupees)	25-30	30-35	35-40	40-45	45-50
No. of farm owners	20	25	15	10	10

Fund (Rupees)	0-500	500-1000	1000-1500	1500-2000	2000-2500
No. of workers	35	28	32	15	10

Weekly wages (Rupees)	1000-2000	2000-3000	3000-4000	4000-5000
No. of workers.	25	45	50	30

Median of grouped data

1 q

Solved Example

Worked · 1

Median SolvedEx.1 $The following table shows frequency distribution of marks of 100 students of 10th class which they obtained in a practice examination. Find the median of the marks. Marks in exam 0-20 20-40 40-60 60-80 80-100 No. of students 42030406$

Marks in exam	0-20	20-40	40-60	60-80	80-100
No. of students	4	20	30	40	6

Median from a cumulative frequency table

2 q

Solved Examples

Worked · 2

Median cf SolvedEx.1 $Observe the following frequency distribution table. It shows the distances travelled by 60 public transport buses in a day. Find the median of the distance travelled. Daily distance travelled (in Km) 200-209 210-219 220-229 230-239 240-249 No. of buses 41426106$
Median cf SolvedEx.2 $The following table shows the ages of persons who visited a museum on a certain day. Find the median age of the persons visiting the museum. Age (Years) No. of persons Less than 10 3 Less than 20 10 Less than 30 22 Less than 40 40 Less than 50 54 Less than 60 71$

Daily distance travelled (in Km)	200-209	210-219	220-229	230-239	240-249
No. of buses	4	14	26	10	6

Age (Years)	No. of persons
Less than 10	3
Less than 20	10
Less than 30	22
Less than 40	40
Less than 50	54
Less than 60	71

Practice set 6.2

4 q

Ex 6.2 Q.1 $The following table shows classification of number of workers and the number of hours they work in a software company. Find the median of the number of hours they work. Daily No. of hours 8-10 10-12 12-14 14-16 Number of workers 15050030050$
Ex 6.2 Q.2 $The frequency distribution table shows the number of mango trees in a grove and their yield of mangoes. Find the median of data. No. of Mangoes 50-100 100-150 150-200 200-250 250-300 No. of trees 3330908017$
Ex 6.2 Q.3 $The following table shows the classification of number of vehicles and their speeds on Mumbai-Pune express way. Find the median of the data. Average Speed of Vehicles (Km/hr) 60-64 64-69 70-74 75-79 79-84 84-89 No. of vehicles 10345585106$
Ex 6.2 Q.4 $The production of electric bulbs in different factories is shown in the following table. Find the median of the productions. No. of bulbs produced (Thousands) 30-40 40-50 50-60 60-70 70-80 80-90 90-100 No. of factories 12352015878$

Daily No. of hours	8-10	10-12	12-14	14-16
Number of workers	150	500	300	50

No. of Mangoes	50-100	100-150	150-200	200-250	250-300
No. of trees	33	30	90	80	17

Average Speed of Vehicles (Km/hr)	60-64	64-69	70-74	75-79	79-84	84-89
No. of vehicles	10	34	55	85	10	6

No. of bulbs produced (Thousands)	30-40	40-50	50-60	60-70	70-80	80-90	90-100
No. of factories	12	35	20	15	8	7	8

Mode of grouped data

2 q

Solved Examples

Worked · 2

Mode SolvedEx.1 $The classification of children according to their ages, playing on a ground is shown in the following table. Find the mode of ages of the children. Age-group of children (Yrs) 6-8 8-10 10-12 12-14 14-16 No. of children 4358704227$
Mode SolvedEx.2 $The following frequency distribution table shows the classification of the number of vehicles and the volume of petrol filled in them. Find the mode of the volume. Petrol filled (Litre) 1-3 4-6 7-9 10-12 13-15 No. of vehicle 3340271812$

Age-group of children (Yrs)	6-8	8-10	10-12	12-14	14-16
No. of children	43	58	70	42	27

Petrol filled (Litre)	1-3	4-6	7-9	10-12	13-15
No. of vehicle	33	40	27	18	12

Practice set 6.3

4 q

Ex 6.3 Q.1 $The following table shows the information regarding the milk collected from farmers on a milk collection centre and the content of fat in the milk, measured by a lactometer. Find the mode of fat content. Content of fat (%) 2-3 3-4 4-5 5-6 6-7 Milk collected (Litre) 3070806020$
Ex 6.3 Q.2 $Electricity used by some families is shown in the following table. Find the mode for use of electricity. Use of electricity (Unit) 0-20 20-40 40-60 60-80 80-100 100-120 No. of families 1350701008017$
Ex 6.3 Q.3 $Grouped frequency distribution of supply of milk to hotels and the number of hotels is given in the following table. Find the mode of the supply of milk. Milk (Litre) 1-3 3-5 5-7 7-9 9-11 11-13 No. of hotels 7515203518$
Ex 6.3 Q.4 $The following frequency distribution table gives the ages of 200 patients treated in a hospital in a week. Find the mode of ages of the patients. Age (years) Less than 5 5-9 10-14 15-19 20-24 25-29 No. of patients 383250362420$

Content of fat (%)	2-3	3-4	4-5	5-6	6-7
Milk collected (Litre)	30	70	80	60	20

Use of electricity (Unit)	0-20	20-40	40-60	60-80	80-100	100-120
No. of families	13	50	70	100	80	17

Milk (Litre)	1-3	3-5	5-7	7-9	9-11	11-13
No. of hotels	7	5	15	20	35	18

Age (years)	Less than 5	5-9	10-14	15-19	20-24	25-29
No. of patients	38	32	50	36	24	20

Histogram

1 q

Histogram Ex.1 $The table below shows the net asset value (NAV) per unit of mutual funds of some companies. Draw a histogram representing the information. NAV (₹) 8-9 10-11 12-13 14-15 16-17 No. of mutual funds 2040302515$

NAV (₹)	8-9	10-11	12-13	14-15	16-17
No. of mutual funds	20	40	30	25	15

Practice set 6.4

4 q

Ex 6.4 Q.1 $Draw a histogram of the following data. Height of student (cm) 135-140 140-145 145-150 150-155 No. of students 412168$
Ex 6.4 Q.2 $The table below shows the yield of jowar per acre. Show the data by histogram. Yield per acre (quintal) 2-3 4-5 6-7 8-9 10-11 No. of farmers 3050554020$
Ex 6.4 Q.3 $In the following table, the investment made by 210 families is shown. Present it in the form of a histogram. Investment (Thousand Rupees) 10-15 15-20 20-25 25-30 30-35 No. of families 3050605515$
Ex 6.4 Q.4 $Time alloted for the preparation of an examination by some students is shown in the table. Draw a histogram to show the information. Time (minutes) 60-80 80-100 100-120 120-140 140-160 No. of students 1420242216$

Height of student (cm)	135-140	140-145	145-150	150-155
No. of students	4	12	16	8

Yield per acre (quintal)	2-3	4-5	6-7	8-9	10-11
No. of farmers	30	50	55	40	20

Investment (Thousand Rupees)	10-15	15-20	20-25	25-30	30-35
No. of families	30	50	60	55	15

Time (minutes)	60-80	80-100	100-120	120-140	140-160
No. of students	14	20	24	22	16

Frequency polygon

2 q

Solved Examples

Worked · 2

Frequency polygon SolvedEx.1 $x = 5$
Frequency polygon SolvedEx.2 $The following table shows the weights of children and the number of children. Draw a frequency polygon showing the information. Weight of children (kg) 18-19 19-20 20-21 21-22 22-23 23-24 No. of children 4131519176$

Weight of children (kg)	18-19	19-20	20-21	21-22	22-23	23-24
No. of children	4	13	15	19	17	6

Practice set 6.5

7 q

Observe the adjacent frequency polygon (fig 6.6) and write the answers of the following questions. The polygon is drawn with class marks (marks obtained) on the X-axis and number of students on the Y-axis; it starts on the X-axis at $x = 25$ , passes through the points (35, 20), (45, 40), (55, 50), (65, 60), (75, 35), (85, 15) and returns to the X-axis at $x = 95$ .
Ex 6.5 Q.1(1) $Which class has the maximum number of students?$
Ex 6.5 Q.1(2) $Write the classes having zero frequency.$
Ex 6.5 Q.1(3) $What is the class-mark of the class, having frequency of 50 students?$
Ex 6.5 Q.1(4) $Write the lower and upper class limits of the class whose class mark is 85.$
Ex 6.5 Q.1(5) $How many students are in the class 80-90?$
Ex 6.5 Q.2 $Show the following data by a frequency polygon. Electricity bill (₹) 0-200 200-400 400-600 600-800 800-1000 Families 240300450350160$
Ex 6.5 Q.3 $The following table shows the classification of percentages of marks of students and the number of students. Draw a frequency polygon from the table. Result (Percentage) 30-40 40-50 50-60 60-70 70-80 80-90 90-100 No. of students 733456547185$

Electricity bill (₹)	0-200	200-400	400-600	600-800	800-1000
Families	240	300	450	350	160

Result (Percentage)	30-40	40-50	50-60	60-70	70-80	80-90	90-100
No. of students	7	33	45	65	47	18	5

Pie diagram

3 q

To read a Pie diagram

Worked · 1

Pie diagram Ex.1 $138^\circ$

Solved Examples

Worked · 2

Pie diagram SolvedEx.1 $In a bicycle shop, number of bicycles purchased and choice of their colours was as follows. Find the measures of sectors of a circle to show the information by a pie diagram. Colour White Black Blue Grey Red Number of bicycles 109674$
Pie diagram SolvedEx.2 $The following table shows the daily supply of electricity to different places in a town. Show the information by a pie diagram. Places Factories Houses Roads Shops Offices Others Supply of electricity (Thousand units) 24147564$

Colour	White	Black	Blue	Grey	Red
Number of bicycles	10	9	6	7	4

Places	Factories	Houses	Roads	Shops	Offices	Others
Supply of electricity (Thousand units)	24	14	7	5	6	4

Practice set 6.6

11 q

Ex 6.6 Q.1 $The age group and number of persons, who donated blood in a blood donation camp is given below. Draw a pie diagram from it. Age group (Yrs) 20-25 25-30 30-35 35-40 No. of persons 80603525$
Ex 6.6 Q.2 $The marks obtained by a student in different subjects are shown. Draw a pie diagram showing the information. Subject English Marathi Science Mathematics Social science Hindi Marks 507080906050$
Ex 6.6 Q.3 $In a tree plantation programme, the number of trees planted by students of different classes is given in the following table. Draw a pie diagram showing the information. Standard 5th 6th 7th 8th 9th 10th No. of trees 405075507075$
Ex 6.6 Q.4 $The following table shows the percentages of demands for different fruits registered with a fruit vendor. Show the information by a pie diagram. Fruits Mango Sweet lime Apples Cheeku Oranges Percentages of demand 3015252010$
The pie diagram in figure 6.13 shows the proportions of different workers in a town. Answer the following questions with its help. The central angles of the sectors are: Production $90^\circ$ , Construction $72^\circ$ , Hotels $72^\circ$ , Food $48^\circ$ , Administration (Admin) $36^\circ$ , Agriculture $24^\circ$ , Electricity $18^\circ$ .
Ex 6.6 Q.5(1) $If the total workers is 10,000; how many of them are in the field of construction?$
Ex 6.6 Q.5(2) $How many workers are working in the administration?$
Ex 6.6 Q.5(3) $What is the percentage of workers in production?$
The annual investments of a family are shown in the adjacent pie diagram (fig 6.14). Answer the following questions based on it. The central angles of the sectors are: Immovable Property $120^\circ$ , Deposit in bank $90^\circ$ , Shares $60^\circ$ , Mutual fund $60^\circ$ , Post $30^\circ$ .
Ex 6.6 Q.6(1) $If the investment in shares is ₹ 2000/-, find the total investment.$
Ex 6.6 Q.6(2) $How much amount is deposited in bank?$
Ex 6.6 Q.6(3) $How much more money is invested in immovable property than in mutual fund?$
Ex 6.6 Q.6(4) $How much amount is invested in post?$

Age group (Yrs)	20-25	25-30	30-35	35-40
No. of persons	80	60	35	25

Subject	English	Marathi	Science	Mathematics	Social science	Hindi
Marks	50	70	80	90	60	50

Standard	5th	6th	7th	8th	9th	10th
No. of trees	40	50	75	50	70	75

Fruits	Mango	Sweet lime	Apples	Cheeku	Oranges
Percentages of demand	30	15	25	20	10

Miscellaneous Problems - 6

26 q

Find the correct answer from the alternatives given.
PS6 Q.1(1) $The persons of O- blood group are 40%. The classification of persons based on blood groups is to be shown by a pie diagram. What should be the measures of angle for the persons of O- blood group?$
1. A. $114^\circ$
2. B. $140^\circ$
3. C. $104^\circ$
4. D. $144^\circ$
PS6 Q.1(2) $75^\circ$
1. A. $2,16,000$
2. B. $3,60,000$
3. C. $4,50,000$
4. D. $7,50,000$
PS6 Q.1(3) $Cumulative frequencies in a grouped frequency table are useful to find . . .$
1. A. $Mean$
2. B. $Median$
3. C. $Mode$
4. D. $All of these$
PS6 Q.1(4) $\overline{X} = A + \dfrac{\Sigma f_i u_i}{\Sigma f_i} \times h$
1. A. $\dfrac{x_i + A}{g}$
2. B. $(x_i - A)$
3. C. $\dfrac{x_i - A}{g}$
4. D. $\dfrac{A - x_i}{g}$
PS6 Q.1(5) $Distance Covered per litre (km) 12-14 14-16 16-18 18-20 No. of cars 1112207 The median of the distances covered per litre shown in the above data is in the group . . . . . .$
1. A. $12-14$
2. B. $14-16$
3. C. $16-18$
4. D. $18-20$
PS6 Q.1(6) $No. of trees planted by each student 1-3 4-6 7-9 10-12 No. of students 7864 The above data is to be shown by a frequency polygon. The coordinates of the points to show number of students in the class 4-6 are . . . .$
1. A. $(4, 8)$
2. B. $(3, 5)$
3. C. $(5, 8)$
4. D. $(8, 4)$
PS6 Q.2 $The following table shows the income of farmers in a grape season. Find the mean of their income. Income (Thousand Rupees) 20-30 30-40 40-50 50-60 60-70 70-80 Farmers 101115161814$
PS6 Q.3 $The loans sanctioned by a bank for construction of farm ponds are shown in the following table. Find the mean of the loans. Loan (Thousand Rupees) 40-50 50-60 60-70 70-80 80-90 No. of farm ponds 132024367$
PS6 Q.4 $The weekly wages of 120 workers in a factory are shown in the following frequency distribution table. Find the mean of the weekly wages by 'assumed mean' method. Weekly wages (Rupees) 0-2000 2000-4000 4000-6000 6000-8000 No. of workers 15355020$
PS6 Q.5 $The following frequency distribution table shows the amount of aid given to 50 flood affected families. Find the mean of the amount of aid. Amount of aid (Thosand rupees) 50-60 60-70 70-80 80-90 90-100 No. of families 7132064$
PS6 Q.6 $The distances covered by 250 public transport buses in a day is shown in the following frequency distribution table. Find the median of the distances. Distance (km) 200-210 210-220 220-230 230-240 240-250 No. of buses 4060805020$
PS6 Q.7 $The prices of different articles and demand for them is shown in the following frequency distribution table. Find the median of the prices. Price (Rupees) 20 less than 20-40 40-60 60-80 80-100 No. of articles 140100806020$
PS6 Q.8 $The following frequency table shows the demand for a sweet and the number of customers. Find the mode of demand of sweet. Weight of sweet (gram) 0-250 250-500 500-750 750-1000 1000-1250 No. of customers 1060252015$
PS6 Q.9 $Draw a histogram for the following frequency distribution. Use of electricity (Unit) 50-70 70-90 90-110 110-130 130-150 150-170 No. of families 150400460540600350$
PS6 Q.10 $In a handloom factory different workers take different periods of time to weave a saree. The number of workers and their required periods are given below. Present the information by a frequency polygon. No. of days 8-10 10-12 12-14 14-16 16-18 18-20 No. of workers 51630403514$
PS6 Q.11 $The time required for students to do a science experiment and the number of students is shown in the following grouped frequency distribution table. Show the information by a histogram and also by a frequency polygon. Time required for experiment (minutes) 20-22 22-24 24-26 26-28 28-30 30-32 No. of students 81622181412$
PS6 Q.12 $Draw a frequency polygon for the following grouped frequency distribution table. Age of the donor (Yrs.) 20-24 25-29 30-34 35-39 40-44 45-49 No. of blood doners 384635241512$
PS6 Q.13 $The following table shows the average rainfall in 150 towns. Show the information by a frequency polygon. Average rainfall (cm) 0-20 20-40 40-60 60-80 80-100 No. of towns 1412364840$
Observe the adjacent pie diagram (fig 6.15). It shows the percentages of number of vehicles passing a signal in a town between 8 am and 10 am. The sectors are labelled: Cars 30%, Tempos 12%, Buses 8%, Auto-rikshaws 10%, Two wheelers 40%.
PS6 Q.14(1) $Find the central angle for each type of vehicle.$
PS6 Q.14(2) $If the number of two-wheelers is 1200, find the number of all vehicles.$
PS6 Q.15 $The following table shows causes of noise pollution. Show it by a pie diagram. Construction Traffic Aircraft take offs Industry Trains 10% 50% 9% 20% 11%$
A survey of students was made to know which game they like. The data obtained in the survey is presented in the adjacent pie diagram (fig 6.16), whose sectors have central angles Cricket $81^\circ$ , Football $63^\circ$ , Hockey $54^\circ$ , Other $72^\circ$ , Kho-kho $45^\circ$ and Kabaddi $45^\circ$ . If the total number of students are 1000,
PS6 Q.16(1) $How many students like cricket?$
PS6 Q.16(2) $How many students like football?$
PS6 Q.16(3) $How many students prefer other games?$
PS6 Q.17 $Medical check up of 180 women was conducted in a health centre in a village. 50 of them were short of haemoglobin, 10 suffered from cataract and 25 had respiratory disorders. The remaining women were healthy. Show the information by a pie diagram.$
PS6 Q.18 $On an environment day, students in a school planted 120 trees under plantation project. The information regarding the project is shown in the following table. Show it by a pie diagram. Tree name Karanj Behada Arjun Bakul Kadunimb No. of trees 2028242226$