--- title: "Dispersion measures; construct box plot" subject: "Probability and Statistics" chapter: 1 key: "semester_5/probability_and_statistics:absolute-relative-dispersion-box-plot" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/absolute-relative-dispersion-box-plot site: BCT Engineering --- # Dispersion measures; construct box plot Probability and Statistics · Chapter 1: Descriptive Statistics and Basic Probability - Asked in: 2072 Kartik, 1+5 marks - Video: [How To Make Box and Whisker Plots](https://www.youtube.com/watch?v=mhaGAaL6Abw) — The Organic Chemistry Tutor - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/absolute-relative-dispersion-box-plot - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1 ## Question What is absolute and relative Measure of Dispersion? Construct a Box plot from the following data of marks of students as: | Marks | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | |---|---|---|---|---|---| | No. of students | 2 | 6 | 22 | 13 | 7 | ## Short answer Absolute dispersion (range, SD) has data units; relative dispersion (CV) is a dimensionless ratio for comparison. The box plot uses min 10, Q1 32.05, median 37.73, Q3 45.77, max 60; IQR = 13.72. ## Answer #### Absolute and relative measures of dispersion - **Absolute measure:** expresses the spread in the same unit as the data. Examples: range, quartile deviation, mean deviation, standard deviation. - **Relative measure:** a unit-free ratio or percentage used to compare the spread of different data sets. Examples: coefficient of range, coefficient of quartile deviation, coefficient of variation. #### Box plot from grouped data N = 50. Cumulative frequencies: 2, 8, 30, 43, 50. Class width = 10. | Marks | f | cf | |---|---|---| | 10-20 | 2 | 2 | | 20-30 | 6 | 8 | | 30-40 | 22 | 30 | | 40-50 | 13 | 43 | | 50-60 | 7 | 50 | **Q₁:** position N/4 = 12.5, class 30-40 (cf before = 8, f = 22) ```text Q1 = 30 + (12.5 − 8)/22 × 10 = 32.05 ``` **Median:** position 25, class 30-40 ```text Median = 30 + (25 − 8)/22 × 10 = 37.73 ``` **Q₃:** position 3N/4 = 37.5, class 40-50 (cf before = 30, f = 13) ```text Q3 = 40 + (37.5 − 30)/13 × 10 = 45.77 ``` The smallest and largest values are not known exactly, so the whiskers are drawn to the ends of the first and last classes: 10 and 60. | Minimum | Q₁ | Median | Q₃ | Maximum | |---|---|---|---|---| | 10 | 32.05 | 37.73 | 45.77 | 60 | IQR = 45.77 − 32.05 = 13.72. ```text 10 20 30 40 50 60 | | | | | | 10---------[32.05==37.73==45.77]-----------60 min Q1 median Q3 max ``` The median (37.73) is nearer to Q₁ (distance 5.68) than to Q₃ (distance 8.04), which hints at a slight right skew. However, the lower whisker (10 to 32.05, length 22.05) is longer than the upper whisker (45.77 to 60, length 14.23), which points the other way. Because the whiskers are drawn only to the class ends, they are approximate, so the marks are roughly symmetric and no clear skew can be claimed. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which statement correctly distinguishes absolute and relative measures of dispersion? A. Absolute measures are unit-free, relative ones have units B. Absolute measures have the data's unit, relative measures are unit-free ratios C. Both are always unit-free D. Relative measures include only the range Answer: B. Range and SD are in data units; coefficient of variation or coefficient of range are unit-free and used to compare data sets. (Key point: Absolute vs relative) 2. For N = 50, in which class does Q1 lie and why? A. 40-50, because it is the median class B. 20-30, because N/4 = 12.5 is below 20 C. 30-40, because position 12.5 exceeds cf 8 and is within cf 30 D. 10-20, because it is the first class Answer: C. Q1 position is N/4 = 12.5; cumulative frequency reaches 8 before 30-40 and 30 at its end, so Q1 is in 30-40. (Key point: Quartile position) 3. What are the five-number summary values used in the box plot? A. 2, 8, 30, 43, 50 B. 10, 30, 40, 50, 60 C. 10, 32.05, 37.73, 45.77, 60 D. 20, 32.05, 37.73, 45.77, 50 Answer: C. Whiskers go to the ends of the first and last classes (10 and 60); the box runs from Q1 32.05 to Q3 45.77 with median 37.73. 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