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Jacek Wallusch _________________________________ Statistics for International Business. Lecture 3 : Skewness and Kurtosis. Shape of distribution ____________________________________________________________________________________________. graphical presentation. Skewness:.
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Jacek Wallusch_________________________________Statistics for International Business Lecture 3: Skewness and Kurtosis
Shape of distribution____________________________________________________________________________________________ graphical presentation Skewness: measures the skewness of a distribution; positive or negative skewness Kurtosis: measures thepeackednessof a distribution; leptokurtic (positive excess kurtosis, i.e. fatter tails), mesokurtic, platykurtic (negative excess kurtosis, i.e. thinner tails), Statistics: 3 fat tails – to be found in e.g. recent financial econometrics and chaotic dynamics
Shape of distribution____________________________________________________________________________________________ graphical presentation Symetrical distribution: mean = median = mode = 3, thus skewness = 0; kurtosis is small (0.003) Statistics: 3 MODE – a value that occurs most frequently (in the upper figure mode = 3)
Skewness____________________________________________________________________________________________Skewness____________________________________________________________________________________________ positive skewness Graphical presentation: Statistics: 3 http://azzalini.stat.unipd.it/SN/plot-SN1.html
Skewness____________________________________________________________________________________________Skewness____________________________________________________________________________________________ negative skewness Graphical presentation: Statistics: 3 http://azzalini.stat.unipd.it/SN/plot-SN1.html
Skewness___________________________________________________________________________________Skewness___________________________________________________________________________________ a bit of history Relationship between location measures: mean – mode = 3(mean – median) Coefficient of skewness: independent of measurment units Combining both: We will be using it Statistics: 3 Karl Pearson (1857-1938) xM – mode, a value that occurs most frequently in the sample or population
Skewness____________________________________________________________________________________________Skewness____________________________________________________________________________________________ formulas Skweness: sum of deviation from mean value devided by the cubed standard deviation Excel formula: adjusted Fisher-Pearson standardised moment coefficient Statistics: 3 compare both formulas
Interpretation____________________________________________________________________________________________Interpretation____________________________________________________________________________________________ skewness Histogram and skewness Whatto lookat: Whereistheaverage? Whereisthe ‘majority’ of observations? average = 1 267 690 USD median = 660 000 USD skewness = 1.907 Statistics: 3 relatively large value, thus: positively skewed
Interpretation____________________________________________________________________________________________Interpretation____________________________________________________________________________________________ skewness Histogram and skewness sk(Wlkp) = 0.423, sk(Maz) = –0.294 Statistics: 3 unemployment rate in voivodships: interpret the results
Interpretation____________________________________________________________________________________________Interpretation____________________________________________________________________________________________ skewness WernhamHogg’sDiscount Policy [1] no strict rules regarding the discount policy [2] guidelines – volume offered vs. discount [1] calculate the skewness [2] evaluate the discount policy in Swindon and Slough Statistics: 3 Alternative way of calculating skewnes:
Kurtosis____________________________________________________________________________________________Kurtosis____________________________________________________________________________________________ formulas Kurtosis: sum of deviation from mean value divided by the standard deviation to the 4th power Excel formula: Statistics: 3 population excess kurtosisin comparison to the normal distribution (bell-shaped distribution)
Kurtosis____________________________________________________________________________________________Kurtosis____________________________________________________________________________________________ interpretation Positive and large: leptokurtic distribution (high-frequency financial data, abnormal rate or returs, long time-series covering periods of crisises and expansions) Negative and large: platykurtic distribution (large variability) Statistics: 3 mesokurtikzero-excess kurtosis
Interpretation____________________________________________________________________________________________Interpretation____________________________________________________________________________________________ kurtosis Histogram and kurtosis Whatto lookat: Arethereanyclusters of volatility? kurtosis = 20.238 Statistics: 3 Huge value, thus: leptokurtic
Interpretation____________________________________________________________________________________________Interpretation____________________________________________________________________________________________ kurtosis Histogram and kurtosis WhernhamHoggand thediscountpolicyagain: Isthediscountpolicyconsistent? kurtosis= 1.406 Statistics: 3
Repetition____________________________________________________________________________________________Repetition____________________________________________________________________________________________ one week to 1st. mid-term Arithmetic mean; Geometric mean; when to use them? how to interpret them? Weighted average; how to calculate the weights? how to interpret? Variance; Standard deviation; how to interpret? how to detect outliers? Statistics: 3
Repetition____________________________________________________________________________________________Repetition____________________________________________________________________________________________ one week to 1st. mid-term Skewness; Kurtosis; how to interpret? Histogram; Ogive; relation to measures of location, dispersion, skewness and kurtosis Statistics: 3