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Business Mathematics and Statistics · Class 12 Commerce

Ch 9Measures of Dispersion — Class 12 Business Mathematics and Statistics, concept-first.

An average — the mean, median or mode studied in the previous chapter — condenses an entire data set into a single representative number, but it says nothing about how the individual values are actually spread around that number.

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Range and Coefficient of Range

Range and Coefficient of Range

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Meaning, Objectives and Types of Dispersion

An average — the mean, median or mode studied in the previous chapter — condenses an entire data set into a single representative number, but it says nothing about how the individual values are actual…

2

Range and Coefficient of Range

The Range is the simplest possible measure of dispersion — the difference between the largest and the smallest value in a data set.

3

Quartiles and Quartile Deviation

Quartiles divide an array of data (arranged in ascending order) into four equal parts. (the lower quartile) is the value below which a quarter of the observations lie; is simply the median (half the o…

4

Mean Deviation and its Coefficient

Range and Quartile Deviation each use only a couple of specific data points (the extremes, or the quartiles) and ignore the rest.

5

Standard Deviation and Variance — Direct Method

Mean Deviation uses absolute values to stop positive and negative deviations from cancelling out. An alternative — mathematically far more convenient, and the one on which almost all further statistic…

6

Standard Deviation — Assumed-Mean and Step-Deviation Methods

When the actual mean is not a convenient round number, computing or directly can involve large, awkward decimals.

7

Combined Standard Deviation of Two Series

A business often needs to combine two separately-recorded groups of data into one overall picture — the combined output of two factory shifts, or the combined test scores of two class sections — witho…

8

Coefficient of Variation

Standard deviation is measured in the same unit as the original data — rupees, marks, kilograms — which makes it impossible to fairly compare the relative variability of two data sets recorded in diff…

Exercises

+Show 5 questions5 questions
  1. Q11The marks scored by 8 students in a Business Statistics unit test (out of 50) are: 32, 45, 28, 38, 41, 25, 36, 48. Find the Range and the Co…Free
  2. Q12The number of units produced per day by 11 workers in a factory is: 18, 22, 15, 25, 20, 17, 23, 19, 21, 16, 24. Find the Quartile Deviation…Free
  3. Q13The weekly income (in ₹ hundred) of 45 shopkeepers in a market is grouped as follows. Find the Standard Deviation (a) by the assumed-mean me…Preview
  4. Q14Which of the following is a relative (unit-free) measure of dispersion, suitable for comparing the variability of two series recorded in dif…Preview
  5. Q15If a constant $k$ is added to every observation in a data series, the standard deviation of the new series is: (a) increased by $k$ (b) decr…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 46 questions46 questions
  1. Q1Rectify the underlined portions of the following sentences: (ii) <u>Standard</u> deviation is half of the difference between the third and f…Preview
  2. Q2Rectify the underlined portions of the following sentences: (iii) Standard deviation is <u>one</u> if all the values of a variable are equal…Preview
  3. Q3Fill in the blanks: (vii) Semi-interquartile range is also known as __________ deviation.Preview
  4. Q4Fill in the blanks: (ix) Coefficient of __________ $= \dfrac{\text{Standard Deviation}}{\text{Mean}} \times 100$.Preview
  5. Q5From the alternatives given under each bit, write serially the correct answer along with its serial number against each bit: (a) The differe…Preview
  6. Q6(g) An absolute measure of dispersion is (a) Range (b) Coefficient of quartile deviation (c) Coefficient of mean deviation (d) Coefficient o…Preview
  7. Q7(a) From the following data find the mean deviation from its median: $10, 15, 18, 20, 30$Preview
  8. Q8(b) State any two merits of standard deviation.Preview
  9. Q9(f) If the coefficient of variation and standard deviation of 10 observations are 40% and 2 respectively, find their arithmetic mean.Preview
  10. Q10(m) Find the interquartile range and quartile deviation from the three quartiles 36, 54 and 72.Preview
  11. Q11From the following data calculate the quartile deviation: | Marks | 0-15 | 15-30 | 30-45 | 45-60 | 60-75 | 75-90 | 90-105 | | --- | --- | --…Preview
  12. Q12From the following the one which is a relative measure of dispersion is : (a) Range (b) Quartile deviation (c) Coefficient of mean deviation…Preview
  13. Q13The difference between the greatest and the smallest value of a distribution is known as : (a) Coefficient of range (b) Range (c) Mean devia…Preview
  14. Q14The difference between the largest and the smallest quartiles is equal to : (a) Quartile deviation (b) $2 \times$ Quartile deviation (c) Sec…Preview
  15. Q15The mean deviation of 10, 12, 20 and 30, is : (a) $7$ (b) $18$ (c) $14$ (d) $6$Preview
  16. Q16If the mean and standard deviation are ₹75 and ₹15 respectively, then their coefficient of variation is : (a) $20\%$ (b) $33\frac{1}{3}\%$ (…Preview
  17. Q17The standard deviation of 2 and 4 is : (a) $1$ (b) $-1$ (c) $2$ (d) $-2$Preview
  18. Q18The formula for calculating coefficient of mean deviation is : (a) $\frac{\text{Mean}}{\text{Mean deviation}}$ (b) $\frac{\text{Mean deviati…Preview
  19. Q19If the first and the third quartiles are ₹31.75 and ₹62.63 respectively, then their coefficient of quartile deviation is : (a) $0.11$ (b) $0…Preview
  20. Q20If $Q_1$ and $Q_3$ are the first and third quartiles respectively, then the formula for calculating their interquartile range is : (a) $Q_3…Preview
  21. Q21A measure of dispersion, that does not base on all the values of a distribution, is : (a) Standard deviation (b) Coefficient of variation (c…Preview
  22. Q22From the following the one which is not the value of a relative measure of dispersion, is : (a) ₹$10$ (b) $0.22$ (c) $0.33$ (d) $0.44$Preview
  23. Q23Write the formulae for calculating absolute and relative measures of quartile deviation.Preview
  24. Q24If mean deviation about median of 12.95 is 54.8, then find the coefficient of mean deviation.Preview
  25. Q25What is meant by measure of dispersion ?Preview
  26. Q26If mean and coefficient of variation of a set of observations are 5 and 40% respectively, then find out the value of the standard deviation.Preview
  27. Q27Name any three measures of dispersion.Preview
  28. Q28Calculate the standard deviation from the following data : | x | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | | --- | --- | --- | --- | --- | ---…Preview
  29. Q29If the values of a distribution are 10, 15, 5, 25 and 30, then the co-efficient of range is : (a) $\dfrac{25}{35}$ (b) 25 (c) $\dfrac{35}{25…Preview
  30. Q30If the standard deviation and mean of a distribution are 5 and 20 respectively. The co-efficient of variation of the distribution is : (a) 4…Preview
  31. Q31Standard deviation is a measure of : (a) Central tendency (b) Dispersion (c) Variation (d) AssociationPreview
  32. Q32Rectify the underlined portions of the following sentences : Standard deviation is a _relative_ measure of dispersion.Preview
  33. Q33Answer the following questions within one sentence each : What do you mean by a measure of dispersion ?Preview
  34. Q34The highest limit of a distribution is 60 and the co-efficient of range is 0.2. Find the lowest limit of the distribution.Preview
  35. Q35Write any three merits of standard deviation.Preview
  36. Q36Calculate standard deviation and co-efficient of standard deviation from the following data : | x | 5 | 15 | 25 | 35 | 45 | 55 | | --- | ---…Preview
  37. Q37From the following the one which is not a measure of dispersion, is : (a) Quartile (b) Mean deviation (c) Standard deviation (d) RangePreview
  38. Q38If mean and standard deviation of some numbers are 10 and 3 respectively, then their coefficient of variation, is : (a) 10% (b) 20% (c) 30%…Preview
  39. Q39Rectify the underlined portions of the following sentences : Standard deviation is _relative_ measure of dispersion.Preview
  40. Q40Fill in the blanks : The formula for determining semi-interquartile range is ________.Preview
  41. Q41What is range ?Preview
  42. Q42Write the two formulae for determining the relative measure of mean deviation about mean and about median.Preview
  43. Q43State the formula for calculating coefficient of variation and explain the meaning of symbols used.Preview
  44. Q44Write the other name of semi-interquartile range and state the formula for its determination.Preview
  45. Q45Calculate the quartile deviation for the following distribution : | Class interval | Frequency | | --- | --- | | 0 – 15 | 7 | | 15 – 30 | 26…Preview
  46. Q46Explain the characteristics of an ideal measure of dispersion.Preview

More questions

+Show 10 questions10 questions
  1. Example 1The daily wages (in ₹) of 7 workers in a small enterprise are: 250, 300, 275, 320, 260, 290, 310. Find the Range and the Coefficient of Rang…Free
  2. Example 2The marks obtained (out of 25) by 9 students in a class test are: 12, 18, 9, 22, 15, 20, 11, 17, 14. Find the Quartile Deviation and its Coe…Free
  3. Example 3The marks obtained by 50 students in a Business Mathematics test are grouped as follows. Find the Quartile Deviation and its Coefficient. |…Free
  4. Example 4The number of complaint calls received per day by a customer-care centre, recorded over 50 days, is given below. Find the Mean Deviation abo…Preview
  5. Example 5Using the same 50-student marks distribution as Worked Example 3, find the Median and hence the Mean Deviation about the Median, along with…Preview
  6. Example 6The daily cash sales (in ₹ thousand) of a shopkeeper over 6 days are: 20, 24, 18, 26, 22, 28. Find the Variance and Standard Deviation (a) b…Preview
  7. Example 7The number of customers served per hour at a bank counter, recorded over 30 hours, is given below. Find the Variance and Standard Deviation…Preview
  8. Example 8The profit (in ₹ lakh) earned by 40 firms in an industry survey is grouped as follows. Find the Variance and Standard Deviation (a) by the a…Preview
  9. Example 9In an examination, Section A (50 students) scored a mean of 60 marks with SD 8, while Section B (50 students) scored a mean of 55 marks with…Preview
  10. Example 10The daily sales of two branches of a retail chain over the past month show: Branch A: mean $=₹40{,}000$, SD $=₹8{,}000$; Branch B: mean $=₹6…Preview