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Unformatted text preview: More on quartiles Stemandleaf plots Distribution shapes MATH1005 Statistics Lecture 2 M. Stewart School of Mathematics and Statistics University of Sydney More on quartiles Stemandleaf plots Distribution shapes Outline More on quartiles Stemandleaf plots Distribution shapes More on quartiles Stemandleaf 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 Stemandleaf 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 4th and ( n 4 + 1 )th smallest; • if n not a multiple of 4, the mth 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 Stemandleaf 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 4th 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 Stemandleaf 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 ith smallest xvalue . • 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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 Harshad number, Decimal, Distribution shapes

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