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Box and Whisker Diagrams Objectives: B Grade Construct and interpret a Box & Whisker diagram (also known as a Box Plot) Compare two set of data using box plots Prior knowledge Draw a cumulative frequency diagram Find the upper and lower quartiles from a cumulative frequency diagram Box and Whisker Diagrams Averages use one number to describe a set of data, this can be misleading, depending upon which average you use. Box and whisker diagrams use the • median, • upper and lower quartiles and • range to draw a picture that can be used to interpret data Box plots can be drawn in two ways, from • cumulative frequency diagrams, or • the raw data Box and Whisker Diagrams Cumulative frequency Box plots from cumulative frequency diagrams xand UQ from 80 We have already seen how to find the median, LQ x x a cumulative frequency diagram 70 x Extend the 60 the working out 50 x lines below the 40 x axis x 30 Median UQ x 20 LQ Minimum value 10 0 Maximum value x x x 0 10 20 30 40 50 60 Weight (g) 70 80 90 100 Draw the box plot like this Box and Whisker Diagrams Looking at the box plot on its own: Median Minimum value UQ LQ whisker Maximum value whisker Box and Whisker Diagrams Box plots from raw data First them in order of size Fred put keeps a record of his journey times to school each morning These are his time to the nearest minute for 25 days: 29, 16, 21, 16, 18, 25, 18, 21, 18 19, 18, 19, 30, 19, 21, 19, 21, 20, 12, 20, 26, 21, 19, 12, 12 18, 21 21, 21, 20, 21, 19, 21, 30, 21, 29, 25, 16, 26, 21, 27, 18, 29, 18, 29, 27, 30, 18, 30 20 25 Minimum value Maximum value Median of pieces of data, n Lowernumber Quartile Median positionposition n + 1 = n25+ +1 13+1 = 6.25 Lower Quartile 1 ==25 Upper Quartile 2 42 4 Upper Quartile position 3(n + 1) = 3(25 +1) = 18.75 4 4 Box and Whisker Diagrams Minimum value Maximum value Median Lower Quartile Upper Quartile 12 30 21 18 25 Draw a sensible scale: 12 14 16 18 20 22 Time (minutes) 24 26 28 30 Box and Whisker Diagrams Now do this: Draw a box plot for this data: 18, 8, 19, 15, 27, 13, 10, 4, 8, 31, 26, 11, 29, 28, 23 Solution: Arrange the data in order of size: 4, 8, 8, 10, 11, 13, 15, 18, 19, 23, 26, 27, 28, 29, 31 Median position n + 1 = 15 +1 = 8 Median = 18 2 2 LQ position n + 1 = 15 +1 = 4 4 4 LQ = 10 UQ position 3(n + 1) = 3(15 +1) = 12 2 2 UQ = 27 0 4 8 10 14 16 20 24 28 32 Box and Whisker Diagrams Interpreting Box and Whisker Diagrams What can you say about the ages of the shoppers and what kind of shop they are? Dressnice has a much lower median age, so the shoppers are younger 75% of the Dressnice shoppers are younger than 75% of the Clotheswell shoppers Clotheswell has broader appeal because the IQR is bigger Dressnice is a shop for younger people, perhaps more fashionable