Statistics — Mathematics textbook solutions

13.2 Mean of Grouped Data

12 q

Solved Examples

Worked · 3

13.1 Eg.1 $x_i$
13.1 Eg.2 $The table below gives the percentage distribution of female teachers in the primary schools of rural areas of various states and union territories (U.T.) of India. Find the mean percentage of female teachers by all the three methods discussed in this section. Percentage of female teachers 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75 - 85 Number of States/U.T. 61174421 *Source : Seventh All India School Education Survey conducted by NCERT*$
13.1 Eg.3 $The distribution below shows the number of wickets taken by bowlers in one-day cricket matches. Find the mean number of wickets by choosing a suitable method. What does the mean signify? Number of wickets 20 - 60 60 - 100 100 - 150 150 - 250 250 - 350 350 - 450 Number of bowlers 75161223$

Marks obtained $x_i$	10	20	36	40	50	56	60	70	72	80	88	92	95
Number of students $f_i$	1	1	3	4	3	2	4	4	1	1	2	3	1

Percentage of female teachers	15 - 25	25 - 35	35 - 45	45 - 55	55 - 65	65 - 75	75 - 85
Number of States/U.T.	6	11	7	4	4	2	1

Number of wickets	20 - 60	60 - 100	100 - 150	150 - 250	250 - 350	350 - 450
Number of bowlers	7	5	16	12	2	3

Exercise 13.1

Practice · 9

Ex 13.1 Q1 $A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house. Number of plants 0 - 2 2 - 4 4 - 6 6 - 8 8 - 10 10 - 12 12 - 14 Number of houses 1215623 Which method did you use for finding the mean, and why?$
Ex 13.1 Q2 $Consider the following distribution of daily wages of 50 workers of a factory. Daily wages (in ₹) 500 - 520 520 - 540 540 - 560 560 - 580 580 - 600 Number of workers 12148610 Find the mean daily wages of the workers of the factory by using an appropriate method.$
Ex 13.1 Q3 $f$
Ex 13.1 Q4 $Thirty women were examined in a hospital by a doctor and the number of heartbeats per minute were recorded and summarised as follows. Find the mean heartbeats per minute for these women, choosing a suitable method. Number of heartbeats per minute 65 - 68 68 - 71 71 - 74 74 - 77 77 - 80 80 - 83 83 - 86 Number of women 2438742$
Ex 13.1 Q5 $In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes. Number of mangoes 50 - 52 53 - 55 56 - 58 59 - 61 62 - 64 Number of boxes 1511013511525 Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?$
Ex 13.1 Q6 $The table below shows the daily expenditure on food of 25 households in a locality. Daily expenditure (in ₹) 100 - 150 150 - 200 200 - 250 250 - 300 300 - 350 Number of households 451222 Find the mean daily expenditure on food by a suitable method.$
Ex 13.1 Q7 $To find out the concentration of SO₂ in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below. Find the mean concentration of SO₂ in the air. Concentration of SO₂ (in ppm) Frequency 0.00 - 0.04 4 0.04 - 0.08 9 0.08 - 0.12 9 0.12 - 0.16 2 0.16 - 0.20 4 0.20 - 0.24 2$
Ex 13.1 Q8 $A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent. Number of days 0 - 6 6 - 10 10 - 14 14 - 20 20 - 28 28 - 38 38 - 40 Number of students 111074431$
Ex 13.1 Q9 $The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate. Literacy rate (in %) 45 - 55 55 - 65 65 - 75 75 - 85 85 - 95 Number of cities 3101183$

Number of plants	0 - 2	2 - 4	4 - 6	6 - 8	8 - 10	10 - 12	12 - 14
Number of houses	1	2	1	5	6	2	3

Daily wages (in ₹)	500 - 520	520 - 540	540 - 560	560 - 580	580 - 600
Number of workers	12	14	8	6	10

Daily pocket allowance (in ₹)	11 - 13	13 - 15	15 - 17	17 - 19	19 - 21	21 - 23	23 - 25
Number of children	7	6	9	13	$f$	5	4

Number of heartbeats per minute	65 - 68	68 - 71	71 - 74	74 - 77	77 - 80	80 - 83	83 - 86
Number of women	2	4	3	8	7	4	2

Number of mangoes	50 - 52	53 - 55	56 - 58	59 - 61	62 - 64
Number of boxes	15	110	135	115	25

Daily expenditure (in ₹)	100 - 150	150 - 200	200 - 250	250 - 300	300 - 350
Number of households	4	5	12	2	2

Concentration of SO₂ (in ppm)	Frequency
0.00 - 0.04	4
0.04 - 0.08	9
0.08 - 0.12	9
0.12 - 0.16	2
0.16 - 0.20	4
0.20 - 0.24	2

Number of days	0 - 6	6 - 10	10 - 14	14 - 20	20 - 28	28 - 38	38 - 40
Number of students	11	10	7	4	4	3	1

Literacy rate (in %)	45 - 55	55 - 65	65 - 75	75 - 85	85 - 95
Number of cities	3	10	11	8	3

13.3 Mode of Grouped Data

9 q

Solved Examples

Worked · 3

13.2 Eg.4 $The wickets taken by a bowler in 10 cricket matches are as follows: 2 6 4 5 0 2 1 3 2 3 Find the mode of the data.$
13.2 Eg.5 $A survey conducted on 20 households in a locality by a group of students resulted in the following frequency table for the number of family members in a household. Find the mode of this data. Family size 1 - 3 3 - 5 5 - 7 7 - 9 9 - 11 Number of families 78221$
The marks distribution of 30 students in a mathematics examination is given in Table 13.3 of Example 1, reproduced here:
Class interval 10 - 25 25 - 40 40 - 55 55 - 70 70 - 85 85 - 100
Number of students 2 3 7 6 6 6
The mean of this distribution was found in Example 1 to be 62.
13.2 Eg.6 $Find the mode of this data. Also compare and interpret the mode and the mean.$

Family size	1 - 3	3 - 5	5 - 7	7 - 9	9 - 11
Number of families	7	8	2	2	1

Class interval	10 - 25	25 - 40	40 - 55	55 - 70	70 - 85	85 - 100
Number of students	2	3	7	6	6	6

Exercise 13.2

Practice · 6

Ex 13.2 Q1 $The following table shows the ages of the patients admitted in a hospital during a year: Age (in years) 5 - 15 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 Number of patients 6112123145 Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.$
Ex 13.2 Q2 $The following data gives the information on the observed lifetimes (in hours) of 225 electrical components. Determine the modal lifetimes of the components. Lifetimes (in hours) 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100 100 - 120 Frequency 103552613829$
Ex 13.2 Q3 $The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure : Expenditure (in ₹) Number of families 1000 - 1500 24 1500 - 2000 40 2000 - 2500 33 2500 - 3000 28 3000 - 3500 30 3500 - 4000 22 4000 - 4500 16 4500 - 5000 7$
Ex 13.2 Q4 $The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures. Number of students per teacher Number of states / U.T. 15 - 20 3 20 - 25 8 25 - 30 9 30 - 35 10 35 - 40 3 40 - 45 0 45 - 50 0 50 - 55 2$
Ex 13.2 Q5 $The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches. Find the mode of the data. Runs scored Number of batsmen 3000 - 4000 4 4000 - 5000 18 5000 - 6000 9 6000 - 7000 7 7000 - 8000 6 8000 - 9000 3 9000 - 10000 1 10000 - 11000 1$
Ex 13.2 Q6 $A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data : Number of cars 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 Frequency 71413122011158$

Age (in years)	5 - 15	15 - 25	25 - 35	35 - 45	45 - 55	55 - 65
Number of patients	6	11	21	23	14	5

Lifetimes (in hours)	0 - 20	20 - 40	40 - 60	60 - 80	80 - 100	100 - 120
Frequency	10	35	52	61	38	29

Expenditure (in ₹)	Number of families
1000 - 1500	24
1500 - 2000	40
2000 - 2500	33
2500 - 3000	28
3000 - 3500	30
3500 - 4000	22
4000 - 4500	16
4500 - 5000	7

Number of students per teacher	Number of states / U.T.
15 - 20	3
20 - 25	8
25 - 30	9
30 - 35	10
35 - 40	3
40 - 45	0
45 - 50	0
50 - 55	2

Runs scored	Number of batsmen
3000 - 4000	4
4000 - 5000	18
5000 - 6000	9
6000 - 7000	7
7000 - 8000	6
8000 - 9000	3
9000 - 10000	1
10000 - 11000	1

Number of cars	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 - 70	70 - 80
Frequency	7	14	13	12	20	11	15	8

13.4 Median of Grouped Data

9 q

Solved Examples

Worked · 2

13.3 Eg.7 $A survey regarding the heights (in cm) of 51 girls of Class X of a school was conducted and the following data was obtained. Find the median height. Height (in cm) Number of girls Less than 140 4 Less than 145 11 Less than 150 29 Less than 155 40 Less than 160 46 Less than 165 51$
13.3 Eg.8 $x$

Height (in cm)	Number of girls
Less than 140	4
Less than 145	11
Less than 150	29
Less than 155	40
Less than 160	46
Less than 165	51

Class intervals	Frequency
0 - 100	2
100 - 200	5
200 - 300	$x$
300 - 400	12
400 - 500	17
500 - 600	20
600 - 700	$y$
700 - 800	9
800 - 900	7
900 - 1000	4

Exercise 13.3

Practice · 7

Ex 13.3 Q1 $The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them. Monthly consumption (in units) Number of consumers 65 - 85 4 85 - 105 5 105 - 125 13 125 - 145 20 145 - 165 14 165 - 185 8 185 - 205 4$
Ex 13.3 Q2 $x$
Ex 13.3 Q3 $A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 year. Age (in years) Number of policy holders Below 20 2 Below 25 6 Below 30 24 Below 35 45 Below 40 78 Below 45 89 Below 50 92 Below 55 98 Below 60 100$
Ex 13.3 Q4 $The lengths of 40 leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table : Length (in mm) Number of leaves 118 - 126 3 127 - 135 5 136 - 144 9 145 - 153 12 154 - 162 5 163 - 171 4 172 - 180 2 Find the median length of the leaves. (Hint : The data needs to be converted to continuous classes for finding the median, since the formula assumes continuous classes. The classes then change to 117.5 - 126.5, 126.5 - 135.5, . . ., 171.5 - 180.5.)$
Ex 13.3 Q5 $The following table gives the distribution of the life time of 400 neon lamps. Find the median life time of a lamp. Life time (in hours) Number of lamps 1500 - 2000 14 2000 - 2500 56 2500 - 3000 60 3000 - 3500 86 3500 - 4000 74 4000 - 4500 62 4500 - 5000 48$
Ex 13.3 Q6 $100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows: Number of letters 1 - 4 4 - 7 7 - 10 10 - 13 13 - 16 16 - 19 Number of surnames 630401644 Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.$
Ex 13.3 Q7 $The distribution below gives the weights of 30 students of a class. Find the median weight of the students. Weight (in kg) 40 - 45 45 - 50 50 - 55 55 - 60 60 - 65 65 - 70 70 - 75 Number of students 2386632$

Monthly consumption (in units)	Number of consumers
65 - 85	4
85 - 105	5
105 - 125	13
125 - 145	20
145 - 165	14
165 - 185	8
185 - 205	4

Age (in years)	Number of policy holders
Below 20	2
Below 25	6
Below 30	24
Below 35	45
Below 40	78
Below 45	89
Below 50	92
Below 55	98
Below 60	100

Length (in mm)	Number of leaves
118 - 126	3
127 - 135	5
136 - 144	9
145 - 153	12
154 - 162	5
163 - 171	4
172 - 180	2