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Jacek Wallusch _________________________________ Statistics for International Business

Explore the concepts of skewness and kurtosis in statistical distributions, including their graphical representations and interpretations. Learn how to calculate, interpret, and apply skewness and kurtosis measures in data analysis.

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Jacek Wallusch _________________________________ Statistics for International Business

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  1. Jacek Wallusch_________________________________Statistics for International Business Lecture 3: Skewness and Kurtosis

  2. 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

  3. 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)

  4. Skewness____________________________________________________________________________________________Skewness____________________________________________________________________________________________ positive skewness Graphical presentation: Statistics: 3 http://azzalini.stat.unipd.it/SN/plot-SN1.html

  5. Skewness____________________________________________________________________________________________Skewness____________________________________________________________________________________________ negative skewness Graphical presentation: Statistics: 3 http://azzalini.stat.unipd.it/SN/plot-SN1.html

  6. 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

  7. 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

  8. 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

  9. Interpretation____________________________________________________________________________________________Interpretation____________________________________________________________________________________________ skewness Histogram and skewness sk(Wlkp) = 0.423, sk(Maz) = –0.294 Statistics: 3 unemployment rate in voivodships: interpret the results

  10. 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:

  11. 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)

  12. 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

  13. Interpretation____________________________________________________________________________________________Interpretation____________________________________________________________________________________________ kurtosis Histogram and kurtosis Whatto lookat: Arethereanyclusters of volatility? kurtosis = 20.238 Statistics: 3 Huge value, thus: leptokurtic

  14. Interpretation____________________________________________________________________________________________Interpretation____________________________________________________________________________________________ kurtosis Histogram and kurtosis WhernhamHoggand thediscountpolicyagain: Isthediscountpolicyconsistent? kurtosis= 1.406 Statistics: 3

  15. 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

  16. 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

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