Measures of Dispersion — Mathematics textbook solutions

8.1.1 Range

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Solved Examples

Worked · 2

8.1.1 SolvedEx.1 $Following data gives weights of 10 students (in kgs) in a certain school. Find the range of the data. 70, 62, 38, 55, 43, 73, 36, 58, 65, 47$
8.1.1 SolvedEx.2 $Calculate range for the following data. Salary (00's Rs.) 30-50 50-70 70-90 90-110 110-130 130-150 No. of Employees 71530241811$

Salary (00's Rs.)	30-50	50-70	70-90	90-110	110-130	130-150
No. of Employees	7	15	30	24	18	11

Exercise 8.1

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Ex 8.1 Q1 $Find range of the following data: 19, 27, 15, 21, 33, 45, 7, 12, 20, 26$
Ex 8.1 Q2 $Find range of the following data: 575, 609, 335, 280, 729, 544, 852, 427, 967, 250$
Ex 8.1 Q3 $The following data gives number of typing mistakes done by Radha during a week. Find the range of the data. Day Monday Tuesday Wednesday Thursday Friday Saturday No. of mistakes 152021121710$
Ex 8.1 Q4 $Following results were obtained by rolling a die 25 times. Find the range of the data. Score 123456 Frequency 462733$
Ex 8.1 Q5 $Find range for the following data. Classes 62-64 64-66 66-68 68-70 70-72 Frequency 53453$

Day	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
No. of mistakes	15	20	21	12	17	10

Score	1	2	3	4	5	6
Frequency	4	6	2	7	3	3

Classes	62-64	64-66	66-68	68-70	70-72
Frequency	5	3	4	5	3

8.2.3 Change of Origin and Scale

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Solved Examples

Worked · 6

8.2.3 SolvedEx.1 $Compute variance and standard deviation of the following data observations. 9, 12, 15, 18, 21, 24, 27$
8.2.3 SolvedEx.2 $Given below are the marks out of 25 of 5 students in mathematics test. Calculate the variance and standard deviation of these observations. Marks : 10, 13, 17, 20, 23$
8.2.3 SolvedEx.3 $(X)$
8.2.3 SolvedEx.4 $x$
8.2.3 SolvedEx.5 $f$
8.2.3 SolvedEx.6 $Find the standard deviation of the following frequency distribution which gives distribution of heights of 500 plants in centimeters. Height of plants (in cm) 20-25 25-30 30-35 35-40 40-45 45-50 No. of plants 14512590404555$

Score $(X)$	1	2	3	4	5	6
Frequency $(f)$	2	6	2	5	10	5

$x$	15	20	25	30	35	40	45
$f$	13	12	15	18	17	10	15

C.I.	45-55	55-65	65-75	75-85	85-95	95-105	105-115	115-125
$f$	7	20	27	23	13	6	3	1

Height of plants (in cm)	20-25	25-30	30-35	35-40	40-45	45-50
No. of plants	145	125	90	40	45	55

Exercise 8.2

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Ex 8.2 Q1 $Find variance and S.D. for the following set of numbers. 7, 11, 2, 4, 9, 6, 3, 7, 11, 2, 5, 8, 3, 6, 8, 8, 2, 6$
Ex 8.2 Q2 $Find variance and S.D. for the following set of numbers. 65, 77, 81, 98, 100, 80, 129$
Ex 8.2 Q3 $X$
Ex 8.2 Q4 $X$
Ex 8.2 Q5 $Following data gives age of 100 students in a college. Calculate variance and S.D. Age (In years) 161718192021 No. of Students 20711173015$
Ex 8.2 Q6 $Find mean, variance and S.D. of the following data. Classes 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 Freq. 714613915111015$
Ex 8.2 Q7 $Find the variance and S.D. of the following frequency distribution which gives the distribution of 200 plants according to their height. Height (in cm) 14-18 19-23 24-28 29-33 34-38 39-43 44-48 No. of plants 518447036225$
Ex 8.2 Q8 $The mean of 5 observations is 4.8 and the variance is 6.56. If three of the five observations are 1, 3 and 8, find the other two observations.$

$X$	2	4	6	8	10	12	14	16	18	20
$F$	8	10	10	7	6	4	3	4	2	6

$X$	31	32	33	34	35	36	37
Frequency	15	12	10	8	9	10	6

Age (In years)	16	17	18	19	20	21
No. of Students	20	7	11	17	30	15

Classes	10-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90	90-100
Freq.	7	14	6	13	9	15	11	10	15

Height (in cm)	14-18	19-23	24-28	29-33	34-38	39-43	44-48
No. of plants	5	18	44	70	36	22	5

8.3 Standard Deviation for Combined Data

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Solved Examples

Worked · 2

8.3 SolvedEx.1 $The means of two samples of sizes 10 and 20 are 24 and 45 respectively and the standard deviations are 6 and 11. Obtain the standard deviation of the sample of size 30 obtained by combining the two samples.$
8.3 SolvedEx.2 $The first group has 100 items with mean 45 and variance 49. If the combined group has 250 items with mean 51 and variance 130, find the mean and standard deviation of second group.$

8.3.1 Coefficient of Variation

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Solved Examples

Worked · 2

8.3.1 SolvedEx.1 $The arithmetic mean of runs scored by 3 batsmen Varad, Viraj and Akhilesh in the same series are 50, 52 and 21 respectively. The standard deviation of their runs are 11, 16 and 5 respectively. Who is the most consistent of the three? If one of the three is to be selected, who will be selected?$
8.3.1 SolvedEx.2 $The following values are calculated in respect of prices of shares of companies X and Y. State the share of which company is more stable in value. Share of Y Share of Y Mean 50105 Variance 74$

	Share of Y	Share of Y
Mean	50	105
Variance	7	4

Exercise 8.3

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Ex 8.3 Q1 $The means of two samples of sizes 60 and 120 respectively are 35.4 and 30.9 and the standard deviations 4 and 5. Obtain the standard deviation of the sample of size 180 obtained by combining the two sample.$
Ex 8.3 Q2 $For a certain data, following information is available. X Y Mean 1317 S. D. 32 Size 2030 Obtain the combined standard deviation.$
Ex 8.3 Q3 $Calculate coefficient of variation of marks secured by a student in the exam, where the marks are: 85, 91, 96, 88, 98, 82$
Ex 8.3 Q4 $Find the coefficient of variation of a sample which has mean equal to 25 and standard deviation of 5.$
Ex 8.3 Q5 $A group of 65 students of class XI have their average height is 150.4 cm with coefficient of variance 2.5%. What is the standard deviation of their height?$
Two workers on the same job show the following results:
Worker P Worker Q
Mean time for completing the job (hours) 33 21
Standard Deviation (hours) 9 7
Ex 8.3 Q6(i) $Regarding the time required to complete the job, which worker is more consistent?$
Ex 8.3 Q6(ii) $Which worker seems to be faster in completing the job?$
A company has two departments with 42 and 60 employees respectively. Their average weekly wages are Rs. 750 and Rs. 400. The standard deviations are 8 and 10 respectively.
Ex 8.3 Q7(i) $Which department has a larger bill?$
Ex 8.3 Q7(ii) $Which department has larger variability in wages?$
Ex 8.3 Q8 $The following table gives weights of the students of two classes. Calculate the coefficient of variation of the two distributions. Which series is more variable? Weight (in kg) Class A Class B 30-40 2213 40-50 1610 50-60 1217$
Ex 8.3 Q9 $Compute coefficient of variation for team A and team B. No. of goals 01234 No. of matches played by team A 19651614 No. of matches played by team B 1614101416 Which team is more consistent?$
Ex 8.3 Q10 $Given below is the information about marks obtained in Mathematics and Statistics by 100 students in a class. Which subject shows the highest variability in marks? Mathematics Statistics Mean 2025 S.D. 23$

	X	Y
Mean	13	17
S. D.	3	2
Size	20	30

	Worker P	Worker Q
Mean time for completing the job (hours)	33	21
Standard Deviation (hours)	9	7

Weight (in kg)	Class A	Class B
30-40	22	13
40-50	16	10
50-60	12	17

No. of goals	0	1	2	3	4
No. of matches played by team A	19	6	5	16	14
No. of matches played by team B	16	14	10	14	16

	Mathematics	Statistics
Mean	20	25
S.D.	2	3

Miscellaneous Exercise 8

32 q

(I) Select the correct option

Practice · 10

Misc I Q1 $If there are 10 values each equal to 10, then S.D. of these values is : ______.$
1. A. $100$
2. B. $20$
3. C. $0$
4. D. $10$
Misc I Q2 $Number of patients who visited cardiologists are 13, 17, 11, 15 in four days then standard deviation (approximately) is$
1. A. $5 patients$
2. B. $4 patients$
3. C. $10 patients$
4. D. $15 patients$
Misc I Q3 $-4$
1. A. $-48$
2. B. $40$
3. C. $-40$
4. D. $48$
Misc I Q4 $The standard deviation of a distribution divided by the mean of the distribution and expressing in percentage is called:$
1. A. $Coefficient of Standard deviation$
2. B. $Coefficient of skewness$
3. C. $Coefficient of quartile deviation$
4. D. $Coefficient of variation$
Misc I Q5 $\sqrt{2}$
1. A. $5$
2. B. $4$
3. C. $7$
4. D. $6$
Misc I Q6 $The positive square root of the mean of the squares of the deviations of observations from their mean is called:$
1. A. $Variance$
2. B. $Range$
3. C. $S.D.$
4. D. $C.V.$
Misc I Q7 $The variance of 19, 21, 23, 25 and 27 is 8. The variance of 14, 16, 18, 20 & 22 is:$
1. A. $Greater than 8$
2. B. $8$
3. C. $Less than 8$
4. D. $8 - 5 = 3$
Misc I Q8 $For any two numbers SD is always$
1. A. $Twice the range$
2. B. $Half of the range$
3. C. $Square of the range$
4. D. $None of these$
Misc I Q9 $Given the heights (in cm) of two groups of students: Group A : 131 cm, 150 cm, 147 cm, 138 cm, 144 cm Group B : 139 cm, 148 cm, 132 cm, 151 cm, 140 cm Which of the following is / are the true? I) The ranges of the heights of the two groups of students are the same. II) The means of the heights of the two groups of students are the same.$
1. A. $I only$
2. B. $II only$
3. C. $Both I and II$
4. D. $None$
Misc I Q10 $Standard deviation of data is 12 and mean is 72 then coefficient of variation is$
1. A. $13.67%$
2. B. $16.67%$
3. C. $14.67%$
4. D. $15.67%$

(II) Answer the following

Practice · 22

Misc II Q1 $76, 57, 80, 103, 61, 63, 89, 96, 105, 72 Find the range for the following data.$
Misc II Q2 $116, 124, 164, 150, 149, 114, 195, 128, 138, 203, 144$
Misc II Q3 $Given below the frequency distribution of weekly wages of 400 workers. Find the range. Weekly wages (in '00 Rs.) 10152025303540 No. of workers 456310255743625$
Misc II Q4 $Find the range of the following data Classes 115-125 125-135 135-145 145-155 155-165 165-175 Frequency 146135$
Misc II Q5 $Find variance and S.D. for the following set of numbers. 25, 21, 23, 29, 27, 22, 28, 23, 21, 25$
Misc II Q6 $Find variance and S.D. for the following set of numbers. 125, 130, 150, 165, 190, 195, 210, 230, 245, 260$
Misc II Q7 $Following data gives no. of goals scored by a team in 100 matches. Compute the standard deviation. No. of Goals Scored 012345 No. of matches 5202515205$
Misc II Q8 $Compute the variance and S.D. for the following data: X 623064476346352860 F 583457837$
Misc II Q9 $Calculate S.D. from following data. Age (In yrs) 20-29 30-39 40-49 50-59 60-69 70-79 80-89 Freq. 651005587423813$
Misc II Q10 $Given below is the frequency distribution of marks obtained by 100 students. Compute arithmetic mean and S.D. Marks 40-49 50-59 60-69 70-79 80-89 90-99 No. of students 4122528265$
Misc II Q11 $The arithmetic mean and standard deviation of a series of 20 items were calculated by a student as 20 cms and 5 cms respectively. But while calculating them, an item 13 was misread as 30. Find the corrected mean and standard deviation.$
Misc II Q12 $The mean and S.D. of a group of 50 observation are 40 and 5 respectively. If two more observations 60 and 72 are added to the set, find the mean and S.D. of 52 items.$
Misc II Q13 $The mean height of 200 students is 65 inches. The mean heights of boys and girls are 70 inches and 62 inches respectively and the standard deviations are 8 and 10 respectively. Find the number of boys and the combined S.D.$
Misc II Q14 $\sum_{i=1}^{n} x_i = 30$
Misc II Q15 $Calculate coefficient of variation of the following data. 23, 27, 25, 28, 21, 14, 16, 12, 18, 16$
Misc II Q16 $Following data relates to the distribution of weights of 100 boys and 80 girls in a school. Boys Girls Mean 6047 Variance 169 Which of the two is more variable?$
Misc II Q17 $The mean and standard deviations of two bands of watches are given below : Brand-I Brand-II Mean 36 months 48 months S.D. 8 months 10 months Calculate coefficient of variation of the two brands and interpret the results.$
Misc II Q18 $Calculate coefficient of variation for the data given below Size (cm) 5-8 8-11 11-14 14-17 17-20 20-23 23-26 No of items 3141316192411$
Misc II Q19 $Calculate coefficient of variation for the data given below Income (Rs.) 3000-4000 4000-5000 5000-6000 6000-7000 7000-8000 8000-9000 9000-10000 No. of families 2413152812810$
Misc II Q20 $Compute coefficient of variations for the following data to show whether the variation is greater in the yield or in the area of the field. Year Area (in acres) Yield (in lakhs) 2011-12 15662 2012-13 13570 2013-14 12868 2014-15 11776 2015-16 14165 2016-17 15469 2017-18 14271$
Misc II Q21 $There are two companies U and V which manufacture cars. A sample of 40 cars each from these companies are taken and the average running life (in years) is recorded. Life (in years) No of Cars — Company U No of Cars — Company V 0-5 514 5-10 188 10-15 1718 Which company shows greater consistency?$
Misc II Q22 $The means and S.D. of weights and heights of 100 students of a school are as follows. Weights Heights Mean 56.5 kg 61 inches S.D. 8.76 kg 12.18 inches Which shows more variability, weights or heights?$

Year	Area (in acres)	Yield (in lakhs)
2011-12	156	62
2012-13	135	70
2013-14	128	68
2014-15	117	76
2015-16	141	65
2016-17	154	69
2017-18	142	71

	Boys	Girls
Mean	60	47
Variance	16	9

	Brand-I	Brand-II
Mean	36 months	48 months
S.D.	8 months	10 months

Life (in years)	No of Cars — Company U	No of Cars — Company V
0-5	5	14
5-10	18	8
10-15	17	18

	Weights	Heights
Mean	56.5 kg	61 inches
S.D.	8.76 kg	12.18 inches