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ECON 2B03 (45)
Lecture 5

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McMaster University
Jeff Racine

Lecture 5 Dispersion Measures Overall Range Equals the difference between the largest and smallest observation in a data set (in R we could use the range() function) Range(x) [1] 64 114 Interfractile Ranges Measure difference between 2 values in the ordered array (called ‘fractiles’) Quartiles: divide the array into 4 quarters Interquartile range: difference between 3 and 1 quartiles (contains middle 50% of data) In R we use the IQR() and quantile() functions IQR(X) [1] 20.25 Quantile(X, seq(0,1, by = 0.25)) ## compute quartiles 0% 25% 50% 75% 100% 64.0 91.75 96.00 112.00 114.00 Shape Measure: Skewness A frequency distribution’s degree of distortion from horizontal symmetry Pearson’s (first) coefficient of skewness is (we want to know is it +/-?) Skewness = (mean – mode) / Standard deviation Skewness = 0 for symmetric distributions For right-skewed distributions the mean is bigger than the median which is bigger than the mode (for left-skewed just the opposite holds) Shape Measure: Kurtosis A frequency curve’s degree of peakedness Coefficient of kurtosis n ∑ (X −X ) Kurtosis = i=1 i In order to compute certain higher-order “moments” (e.g. skewness and kurtosis) of a quantitative variable we need to install an optional R package Library(moment) -> ?skewness In RStudio we first install the moments package via the Install Packages icon in the packages tab (lower right pane by default) The functions kurtosis() and skewness() can then be accessed after we load the moments package via library(moment) The Five-Number Summary The five-number summary of a set of observations consists of the smallest observation, the first quartile, the median, the third
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