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Unformatted text preview: Math 211 Introduction to Statistics Chapter 4 Measures of Dispersion Dispersion: The degree to which numerical raw data tend to spread about an average value is called the Dispersion, or Variation of the data. The most common measures of dispersion is the range, mean deviation, semiinterquartile range, and standard deviation. The Range: The difference between the largest and smallest numbers in the set. The Mean Deviation: (Average deviation) The mean deviation of a set of numbers 1 2 , ,... N X X X is denoted by MD and defined as 1 1 N i N i i i X X MD X X N = = = = where X is the arithmetic mean, i X X is the absolute value of the deviations of i X from X . If 1 2 , ,... k X X X occur with frequencies 1 2 , ,... k f f f respectively, the mean deviation can be written as 1 k i i i f X X MD N = = where 1 k i i N X = = . This form is useful for grouped data, where i X s represent class marks and j f s are the corresponding class frequencies. The SemiInterquartile Range: (Quartile Deviation) 3 1 2 Q Q Q = The 1090 Percentile Range: 90 10 P P Semi 1090 Percentile Range: 90 10 2 P P The Standard Deviation: The standard deviation of a set of N numbers 1 2 , ,... N X X X is denoted by S and defined as 2 1 ( ) 1 N i i X X S N = = . 1 Sonu Zorlu Lecture Notes If 1 2 , ,... k X X X occur with frequencies 1 2 , ,... k f f f respectively, the standard deviation can be written as 2 1 ( ) 1 N i i i f X X S N = = where 1 k i i N X = = . This form is useful for grouped data, where i X s represent class marks and j f s are the corresponding class frequencies. The Variance: The variance of a set of N numbers 1 2 , ,... N X X X is denoted by 2 S and defined as 2 2 1 ( ) 1 N i i X X S N = = . Properties of the Standard Deviation (1) The standard deviation can be defined as 2 1 ( ) 1 N i i X a S N = = ~ a X . S is minimum when a X = . (2) For moderately skewed distributions, the percentages below may hold approximately....
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 Spring '11
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