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Business Mathematics and Statistics · Class 12 Commerce

Ch 8Measures of Central Tendency — Class 12 Business Mathematics and Statistics, concept-first.

An average (also called a measure of central tendency) is a single value that represents the general level of a whole mass of data — it summarises dozens or hundreds of individual figures into one number that speaks for the entire distribution.

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Median

Imagine you and four friends are comparing pocket money. The amounts are: ₹20, ₹50, ₹30, ₹100, ₹40. If you line these up from smallest to largest — ₹20, ₹30, ₹40, ₹50, ₹100 — the middle value is ₹40.

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CA Foundation

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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 Requisites of a Good Average

An average (also called a measure of central tendency) is a single value that represents the general level of a whole mass of data — it summarises dozens or hundreds of individual figures into one num…

2

Arithmetic Mean — Individual and Discrete Series

The Arithmetic Mean (AM), usually written , is the sum of all observations divided by the number of observations.

3

Arithmetic Mean — Continuous Series

In a continuous (grouped) series, observations are grouped into class intervals (e.g., 0–10, 10–20, …), each with a stated frequency .

4

Combined Mean and Weighted Mean

Combined mean. When two (or more) groups of data are merged into one, the mean of the combined group is not simply the average of the two individual means — it must be weighted by how many observation…

5

Properties of the Arithmetic Mean

The arithmetic mean has several algebraic properties that are frequently tested and that also explain why it behaves the way it does.

6

Median — Individual, Discrete and Continuous Series

The median is the value that divides an array of data into two exactly equal halves — half the observations lie below it and half lie above it, once the data is arranged in ascending (or descending) o…

7

Quartiles and the Graphic Location of the Median (the Ogive)

Quartiles divide an ordered dataset into four equal parts instead of the two the median divides it into.

8

Mode — Individual, Discrete and Continuous Series

The mode is the value that occurs most frequently in a set of data — the single most "typical" or "fashionable" value, in the literal sense of the French word from which it is derived.

9

The Empirical Relation Between Mean, Median and Mode

For a moderately (asymmetrically) skewed distribution — one that is not perfectly symmetric but is not wildly lopsided either — Karl Pearson observed that the three measures of central tendency mainta…

10

Geometric Mean and Harmonic Mean

Not every situation calls for the arithmetic mean, even when a single "average" figure is genuinely needed — the geometric mean and harmonic mean exist precisely because certain kinds of business data…

11

Merits, Limitations and Choice of an Appropriate Average

Every average studied in this chapter trades off some of the six requisites of §1 against the others. The table below brings the comparison together.

Exercises

+Show 5 questions5 questions
  1. Q6The weekly sales (in ₹'000) of a salesperson over 7 weeks were: 12, 18, 9, 25, 15, 20, 10. Find the median weekly sales.Free
  2. Q9Explain how the median of a continuous series can be located graphically using an ogive, and verify the graphic method against the algebraic…Free
  3. Q11For a moderately skewed distribution, the mean is 145 and the mode is 130. Estimate the median using the empirical relation, and verify your…Preview
  4. Q14Which of the following measures of central tendency is most affected by extreme (outlier) values in a dataset? (a) Arithmetic Mean (b) Media…Preview
  5. Q15In the mode formula for a continuous series, $Mode = L + \dfrac{f_1-f_0}{2f_1-f_0-f_2}\times h$, what does $f_1$ represent? (a) The frequenc…Preview

Sample & Board Papers

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

+Show 56 questions56 questions
  1. Q1Rectify the underlined portions of the following sentences: (i) Median is a <u>mathematical</u> average.Preview
  2. Q2Fill in the blanks: (viii) The median of $198, 205, 179, 146, 210, 186$ and $190$ is __________.Preview
  3. Q3(b) Median of $72, 78, 81, 88, 92, 83$ and $100$ is (a) 88 (b) 78 (c) 81 (d) 83Preview
  4. Q4(h) The values that divide a given data into four equal parts are called (a) Deciles (b) Quartiles (c) Percentiles (d) Mathematical averagesPreview
  5. Q5(i) Arithmetic mean of Rs. 10, Rs. 25 and Rs. 37 is (a) Rs. 24 (b) Rs. 36 (c) Rs. 24 (d) Rs. 36Preview
  6. Q6(b) From the following particulars of Firm $A$ and Firm $B$, find out the firm that pays larger amount as monthly wages: | Particulars | Fir…Preview
  7. Q7(e) Explain weighted arithmetic mean.Preview
  8. Q8(c) If the sum and arithmetic mean of several observations are 133 and 19 respectively, then find the number of observations.Preview
  9. Q9(i) Find the mode from the following values: $10, 12, 14, 15, 17, 15, 17, 18, 20, 22, 17, 15, 14, 12, 15$Preview
  10. Q10(k) Find the Harmonic Mean of $2, 4, 5, 10$.Preview
  11. Q11From the following data calculate the arithmetic mean: | Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | | --- | ---…Preview
  12. Q12(i) Show the relationship between arithmetic mean, geometric mean and harmonic mean with example. (ii) Explain the meaning of dispersion.Preview
  13. Q13From the following the one which is not a measure of central value, is : (a) Quartile (b) Median (c) Mode (d) Quartile deviationPreview
  14. Q14The sum of the deviations of any given set of observations from their arithmetic mean is equal to : (a) $-1$ (b) $0$ (c) $1$ (d) $2$Preview
  15. Q15The geometric mean of 2 and 8 is equal to : (a) $\pm 4$ (b) $\pm 3$ (c) $\pm 2$ (d) $\pm 1$Preview
  16. Q16The average of the reciprocals of 5, 7, 8 is equal to : (a) $\frac{131}{280}$ (b) $\frac{131}{840}$ (c) $\frac{131}{140}$ (d) $\frac{131}{70…Preview
  17. Q17From the following the one which is not a positional measure, is : (a) Median (b) Quartile (c) Range (d) Standard deviationPreview
  18. Q18The median of $2, 4, 6, 8, 10, 12, 14, 16, 18, 20$ and $22$ is : (a) Fifth value (b) Sixth value (c) Eighth value (d) Tenth valuePreview
  19. Q19The value that occurs most often is called : (a) Geometric mean (b) Harmonic mean (c) Median (d) ModePreview
  20. Q20The number of parts, a set of observation is divided by quartiles, is : (a) $2$ (b) $4$ (c) $10$ (d) $100$Preview
  21. Q21The modal value of 6, 7, 8, 9, 8, 7, 8, 9, 8 is : (a) $6$ (b) $7$ (c) $8$ (d) $9$Preview
  22. Q22The median of 98.13, 98.01, 98.1, 98.42 and 98.08 is : (a) $98.08$ (b) $98.13$ (c) $98.01$ (d) $98.1$Preview
  23. Q23The harmonic mean of $\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{5}$ is : (a) $0.2$ (b) $0.3$ (c) $0.4$ (d) $0.5$Preview
  24. Q24The geometric mean of $8, 125$ and $343$ is : (a) $140$ (b) $120$ (c) $100$ (d) $70$Preview
  25. Q25The percentile that equals median, is : (a) 50th percentile (b) 49th percentile (c) 48th percentile (d) 47th percentilePreview
  26. Q26Name any two measures of central tendency.Preview
  27. Q27State any two merits of arithmetic mean.Preview
  28. Q28State the two empirical relation between mean, median and mode in case of a symmetrical distribution and in case of a moderately asymmetrica…Preview
  29. Q29From the following data, state the modal class with reason : | Marks | Number of students | | --- | --- | | Below 10 | 4 | | Below 20 | 13 |…Preview
  30. Q30From the following data, calculate the median, second quartile and the fifth decile : | Value (x) | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |…Preview
  31. Q31Calculate the arithmetic mean from the following data : | Age (in years) | Persons vaccinated | | --- | --- | | 14 – 15 | 35 | | 15 – 16 | 4…Preview
  32. Q32Explain the merits and demerits of mode.Preview
  33. Q33The median of 5, 15, 12, 7, 14 and 18 is equal to : (a) 12 (b) 14 (c) 13 (d) 15Preview
  34. Q34The arithmetic mean and geometric mean of two numbers are respectively 5 and 4, then the numbers are : (a) 4, 4 (b) 6, 4 (c) 7, 3 (d) 2, 8Preview
  35. Q35Decile divides a distribution into : (a) Two (b) Four (c) Eight (d) TenPreview
  36. Q36Fill in the blanks : Median is a ______ average.Preview
  37. Q37Express each of the following in one word / term : A series with two modal valuesPreview
  38. Q38Rectify the underlined portions of the following sentences : Second quartile and _mean_ of a series are equal.Preview
  39. Q39Write any two features of an ideal measure of central tendency.Preview
  40. Q40Find the arithmetic mean of natural numbers from 1 to 20.Preview
  41. Q41The arithmetic mean and harmonic mean of two numbers are 32 and 28 respectively. Find out the geometric mean of the two numbers.Preview
  42. Q42Write any two limitations of arithmetic mean.Preview
  43. Q43Find out the median of the following distribution : | Marks | 20 | 25 | 30 | 35 | 40 | 45 | 50 | | --- | --- | --- | --- | --- | --- | --- |…Preview
  44. Q44From the following, the one which is a positional average, is : (a) Arithmetic mean (b) Geometric mean (c) Harmonic mean (d) ModePreview
  45. Q45From the following, the one which is not equal to median, is : (a) Second quartile (b) Sixth decile (c) Fiftieth percentile (d) Fifth decilePreview
  46. Q46For two numbers the relation between arithmetic mean (A), geometric mean (G) and harmonic mean (H), is : (a) $A^2 = G \times H$ (b) $G^2 = A…Preview
  47. Q47Express each of the following in one word / term : A measure of central value which has greatest frequency density.Preview
  48. Q48Answer each of the following question in one sentence each : What do you mean by median of a distribution ?Preview
  49. Q49Rectify the underlined portions of the following sentences : Sum of the values divided by the number of the values is equal to _harmonic_ me…Preview
  50. Q50Fill in the blanks : The median of 25, 27, 30, 35, 38, 32 and 40 is ________.Preview
  51. Q51Name any two mathematical averages and any two positional averages.Preview
  52. Q52Calculate the geometric mean of 27, 64 and 343.Preview
  53. Q53Write any two demerits of mode.Preview
  54. Q54Using the empirical relationship among mean, median and mode in a moderately asymmetrical distribution, calculate the mode when mean and med…Preview
  55. Q55State any three merits of arithmetic mean.Preview
  56. Q56Calculate the median from the following distribution : | Marks | Number of students | | --- | --- | | 0 – 5 | 5 | | 5 – 10 | 7 | | 10 – 15 |…Preview

More questions

+Show 10 questions10 questions
  1. Example 1The daily wages (in ₹) of 6 workers in a small unit are: 180, 200, 220, 240, 260, 280. Find the arithmetic mean of their daily wages.Free
  2. Example 2The marks obtained by 30 students in a class test are shown below. Calculate the arithmetic mean by (i) the direct method and (ii) the step-…Free
  3. Example 3Calculate the arithmetic mean of the following continuous series by (i) the direct method and (ii) the step-deviation method. | Class interv…Free
  4. Example 4The average marks of 40 students of Section A is 25, and the average marks of 60 students of Section B is 35. Find the combined average mark…Preview
  5. Example 5A student obtained 80, 70, 90 and 60 marks in four subjects carrying credit weights of 3, 4, 2 and 1 respectively. Find (i) the weighted mea…Preview
  6. Example 7Find the median from the following discrete series (the same data as Worked Example 2): | Marks (x) | 10 | 20 | 30 | 40 | 50 | |---|---|---|…Preview
  7. Example 8For the continuous series below (the same data as Worked Example 3), find (i) the median, (ii) the lower quartile Q₁ and (iii) the upper qua…Preview
  8. Example 10Calculate the mode of the following continuous series (the same data as Worked Example 3), and verify your answer approximately using the em…Preview
  9. Example 12Find the geometric mean of the numbers 2, 4 and 8.Preview
  10. Example 13A person travels a certain distance at a speed of 40 km/h and returns over the same distance at a speed of 60 km/h. Find the average speed f…Preview