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Mathematics · Class 11 Science

Ch 8Measures of Dispersion — Class 11 Mathematics, concept-first.

Before this chapter, you have used measures of central tendency — the mean, median, and mode — to summarise a data set with a single 'typical' value. But an average, on its own, does not tell the whole story. As the statisticians George Simpson and Fritz Kafka put it: 'An average does not tell the full story.

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Range

When you first look at a set of numbers, the most natural question is: how far apart are they? Not the average, not the middle value — just the sheer distance from the smallest to the largest. That distance is the range.

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Chapter contents

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Introduction

Before this chapter, you have used measures of central tendency — the mean, median, and mode — to summarise a data set with a single 'typical' value.

8.1

Measures of Dispersion

The measures of dispersion commonly used to quantify how much a data set's observations scatter around their mean are: (i) Range, (ii) Variance, and (iii) Standard deviation.

+Exercise 8.1i5 questions
  1. Q1Find range of the following data: 19, 27, 15, 21, 33, 45, 7, 12, 20, 26Free
  2. Q2Find range of the following data: 575, 609, 335, 280, 729, 544, 852, 427, 967, 250Free
  3. Q3The following data gives number of typing mistakes done by Radha during a week. Find the range of the data. | Day | Monday | Tuesday | Wedne…Preview
  4. Q4Following results were obtained by rolling a die 25 times. Find the range of the data. | Score | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|--…Preview
  5. Q5Find range for the following data. | Classes | 62-64 | 64-66 | 66-68 | 68-70 | 70-72 | |---|---|---|---|---|---| | Frequency | 5 | 3 | 4 | 5…Preview
8.1.1

Range

Range is the simplest of all measures of dispersion. It looks only at the two extreme values of the data — the largest value (L) and the smallest value (S) — and is defined as For raw (ungrouped) data…

8.2

Variance and Standard Deviation

Range's biggest weakness is that it is computed from only the two extreme values of the data and completely ignores every other observation — two data sets with very different internal patterns can st…

+Exercise 8.2i8 questions
  1. Q6Find 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, 6Free
  2. Q7Find variance and S.D. for the following set of numbers. 65, 77, 81, 98, 100, 80, 129Free
  3. Q8Compute variance and standard deviation for the following data: | X | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | |---|---|---|---|---|---…Free
  4. Q9Compute the variance and S.D. | X | 31 | 32 | 33 | 34 | 35 | 36 | 37 | |---|---|---|---|---|---|---|---| | Frequency | 15 | 12 | 10 | 8 | 9…Preview
  5. Q10Following data gives age of 100 students in a college. Calculate variance and S.D. | Age (In years) | 16 | 17 | 18 | 19 | 20 | 21 | |---|---…Preview
  6. Q11Find 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 | |--…Preview
  7. Q12Find the variance and S.D. of the following frequency distribution which gives the distribution of 200 plants according to their height. | H…Preview
  8. Q13The 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 observation…Preview
8.2.1

Variance

The variance of a variable X, denoted or , is defined as the arithmetic mean of the squares of the deviations of all the observations from their own arithmetic mean: Working directly with this definit…

8.2.2

Standard Deviation

Standard deviation is defined as the positive square root of the variance, denoted (sigma): It is preferred to variance for interpretation because it carries the same unit as the original data, wherea…

8.2.3

Change of Origin and Scale

Variance and standard deviation respond in specific, predictable ways when the data is shifted or rescaled, and this property is used to make hand computation easier, especially for grouped data with…

8.3

Standard Deviation for Combined Data

When two data sets (groups) are pooled into one combined data set, the mean and standard deviation of the combined set can be computed directly from the size, mean and standard deviation of each group…

+Exercise 8.3i10 questions
  1. Q14The 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 deviati…Free
  2. Q15For a certain data, following information is available. | | X | Y | |---|---|---| | Mean | 13 | 17 | | S. D. | 3 | 2 | | Size | 20 | 30 | Ob…Free
  3. Q16Calculate coefficient of variation of marks secured by a student in the exam, where the marks are: 85, 91, 96, 88, 98, 82Free
  4. Q17Find the coefficient of variation of a sample which has mean equal to 25 and standard deviation of 5.Preview
  5. Q18A 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 o…Preview
  6. Q19Two workers on the same job show the following results: | | Worker P | Worker Q | |---|---|---| | Mean time for completing the job (hours) |…Preview
  7. Q20A company has two departments with 42 and 60 employees respectively. Their average weekly wages are Rs. 750 and Rs. 400. The standard deviat…Preview
  8. Q21The following table gives weights of the students of two classes. Calculate the coefficient of variation of the two distributions. Which ser…Preview
  9. Q22Compute coefficient of variation for team A and team B. | No. of goals | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | No. of matches play…Preview
  10. Q23Given below is the information about marks obtained in Mathematics and Statistics by 100 students in a class. Which subject shows the highes…Preview
8.3.1

Coefficient of Variation

Standard deviation is measured in the same unit as the data (kg, cm, ₹, months, etc.), so it cannot be used, by itself, to compare the variability of two data sets that are in different units, or that…

More questions

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+Show 10 questions10 questions
  1. Q24If there are 10 values each equal to 10, then S.D. of these values is: _____. A) 100 B) 20 C) 0 D) 10Free
  2. Q25Number of patients who visited cardiologists are 13, 17, 11, 15 in four days then standard deviation (approximately) is A) 5 patients B) 4 p…Free
  3. Q26If the observations of a variable X are, −4, −20, −30, −44 and −36, then the value of the range will be: A) -48 B) 40 C) –40 D) 48Free
  4. Q27The standard deviation of a distribution divided by the mean of the distribution and expressing in percentage is called: A) Coefficient of S…Preview
  5. Q28If the S.D. of first n natural numbers is $\sqrt{2}$, then the value of n must be _____. A) 5 B) 4 C) 7 D) 6Preview
  6. Q29The positive square root of the mean of the squares of the deviations of observations from their mean is called: A) Variance B) Range C) S.D…Preview
  7. Q30The variance of 19, 21, 23, 25 and 27 is 8. The variance of 14, 16, 18, 20 & 22 is: A) Greater than 8 B) 8 C) Less than 8 D) 8 − 5 = 3Preview
  8. Q31For any two numbers SD is always A) Twice the range B) Half of the range C) Square of the range D) None of thesePreview
  9. Q32Given 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…Preview
  10. Q33Standard deviation of data is 12 and mean is 72 then coefficient of variation is A) 13.67% B) 16.67% C) 14.67% D) 15.67%Preview
+Show 22 questions22 questions
  1. Q3476, 57, 80, 103, 61, 63, 89, 96, 105, 72 Find the range for the following data.Free
  2. Q35116, 124, 164, 150, 149, 114, 195, 128, 138, 203, 144 Find the range for the following data.Free
  3. Q36Given below the frequency distribution of weekly wages of 400 workers. Find the range. | Weekly wages (in ’00 Rs.) | 10 | 15 | 20 | 25 | 30…Free
  4. Q37Find the range of the following data | Classes | 115-125 | 125-135 | 135-145 | 145-155 | 155-165 | 165-175 | |---|---|---|---|---|---|---| |…Preview
  5. Q38Find variance and S.D. for the following set of numbers. 25, 21, 23, 29, 27, 22, 28, 23, 21, 25Preview
  6. Q39Find variance and S.D. for the following set of numbers. 125, 130, 150, 165, 190, 195, 210, 230, 245, 260Preview
  7. Q40Following data gives no. of goals scored by a team in 100 matches. Compute the standard deviation. | No. of Goals Scored | 0 | 1 | 2 | 3 | 4…Preview
  8. Q41Compute the variance and S.D. for the following data: | X | 62 | 30 | 64 | 47 | 63 | 46 | 35 | 28 | 60 | |---|---|---|---|---|---|---|---|--…Preview
  9. Q42Calculate S.D. from following data. | Age (In yrs) | 20-29 | 30-39 | 40-49 | 50-59 | 60-69 | 70-79 | 80-89 | |---|---|---|---|---|---|---|--…Preview
  10. Q43Given below is the frequency distribution of marks obtained by 100 students. Compute arithmetic mean and S.D. | Marks | 40-49 | 50-59 | 60-6…Preview
  11. Q44The 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…Preview
  12. Q45The mean and S.D. of a group of 50 observations are 40 and 5 respectively. If two more observations 60 and 72 are added to the set, find the…Preview
  13. Q46The 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 d…Preview
  14. Q47From the following data available for 5 pairs of observations of two variables x and y, obtain the combined S.D. for all 10 observations. Wh…Preview
  15. Q48Calculate coefficient of variation of the following data. 23, 27, 25, 28, 21, 14, 16, 12, 18, 16Preview
  16. Q49Following data relates to the distribution of weights of 100 boys and 80 girls in a school. | | Boys | Girls | |---|---|---| | Mean | 60 | 4…Preview
  17. Q50The mean and standard deviations of two brands of watches are given below: | | Brand-I | Brand-II | |---|---|---| | Mean | 36 months | 48 mo…Preview
  18. Q51Calculate coefficient of variation for the data given below | Size (cm) | 5-8 | 8-11 | 11-14 | 14-17 | 17-20 | 20-23 | 23-26 | |---|---|---|…Preview
  19. Q52Calculate coefficient of variation for the data given below | Income (Rs.) | 3000-4000 | 4000-5000 | 5000-6000 | 6000-7000 | 7000-8000 | 800…Preview
  20. Q53Compute coefficient of variations for the following data to show whether the variation is greater in the yield or in the area of the field.…Preview
  21. Q54There 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…Preview
  22. Q55The means and S.D. of weights and heights of 100 students of a school are as follows. | | Weights | Heights | |---|---|---| | Mean | 56.5 kg…Preview