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Lecture 5

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McMaster University

Economics

ECON 2B03

Jeff Racine

Fall

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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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