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# L3h - More on quartiles Stem-and-leaf plots Distribution...

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Unformatted text preview: More on quartiles Stem-and-leaf plots Distribution shapes MATH1005 Statistics Lecture 2 M. Stewart School of Mathematics and Statistics University of Sydney More on quartiles Stem-and-leaf plots Distribution shapes Outline More on quartiles Stem-and-leaf plots Distribution shapes More on quartiles Stem-and-leaf plots Distribution shapes Brief recap • Types of data • discrete • continuous • Numerical summaries • Frequency tables (based on a given set of intervals in the continuous case) • Median cuts data (roughly) in half • Quartiles (with median) cut data (roughly) into quarters • Graphical summaries • Discrete case: ordinate diagram • Continuous case: • histogram • boxplot incorporates notion of an outlier More on quartiles Stem-and-leaf plots Distribution shapes More on quartiles • Rough de nition: the lower quartile cuts o (roughly) the lower quarter of the data. • Formula: if we have n values then • if n is a multiple of 4, average the n 4-th and ( n 4 + 1 )-th smallest; • if n not a multiple of 4, the m-th smallest where m is obtained by rounding up n 4 . • Justi ed using the same values in same proportions: same quartiles (and median) principle. More on quartiles Stem-and-leaf plots Distribution shapes More precise de nition • a lower quartile is any value Q 1 such that • at least 1 4 of the data is ≤ Q 1 and • at least 3 4 of the data is ≥ Q 1 . • When n is not a multiple of 4, this uniquely determines Q 1 as per our formula. • When n is a multiple of 4, the n 4-th smallest, or ( n 4 + 1 )-th smallest, (or indeed any value in between) satis es the conditions: so we average those two values. More on quartiles Stem-and-leaf plots Distribution shapes Order statistic notation • For a generic dataset x 1 , x 2 , . . . , x n , we use the order statistic notation x ( 1 ) , x ( 2 ) , . . . , x ( n ) to refer to the same numbers arranged in increasing order . Thus x ( i ) can be read as the i-th smallest x-value . • Also, a standard notation for rounding up ( ceiling function) is e.g. d x e . Rounding down ( oor function) is e.g. b x c ....
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L3h - More on quartiles Stem-and-leaf plots Distribution...

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