Lecture 3: Skewness and Kurtosis 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 the peackedness of a distribution; leptokurtic (positive excess kurtosis, i.e. fatter tails), mesokurtic, platykurtic (negative excess kurtosis, i.e. thinner tails), Probability distribution and uncertainty and risk – this topic will be reconsidered soon Statistics: 3 fat tails – to be found in e.g. recent financial econometrics and chaotic dynamics

thus skewness = 0; kurtosis is small (0.003) Shape of distribution ____________________________________________________________________________________________ graphical presentation Symetrical distribution: Probability distribution and uncertainty and risk – this topic will be reconsidered soon 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 ____________________________________________________________________________________________ positive skewness Graphical presentation: Statistics: 3 http://azzalini.stat.unipd.it/SN/plot-SN1.html

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

Skewness ___________________________________________________________________________________ a bit of history Relationship between location measures: mean – mode = 3(mean – median) Coefficient of skewness: independent of measurment units Combining both: Probability distribution and uncertainty and risk – this topic will be reconsidered soon We will be using it Statistics: 3 Karl Pearson (1857-1938) xM – mode, a value that occurs most frequently in the sample or population

adjusted Fisher-Pearson standardised moment coefficient Skewness ____________________________________________________________________________________________ formulas Skweness: sum of deviation from mean value devided by the cubed standard deviation Excel formula: Probability distribution and uncertainty and risk – this topic will be reconsidered soon adjusted Fisher-Pearson standardised moment coefficient Statistics: 3 compare both formulas

Where is the ‘majority’ of observations? Interpretation ____________________________________________________________________________________________ skewness Histogram and skewness What to look at: Where is the average? Where is the ‘majority’ of observations? average = 1 267 690 USD median = 660 000 USD skewness = 1.907 Statistics: 3 relatively large value, thus: positively skewed

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

Interpretation ____________________________________________________________________________________________ skewness Wernham Hogg’s Discount Policy [1] no strict rules regarding the discount policy [2] guidelines – volume offered vs. discount Swindon Slough Avg. 0.500 Median 0.550 0.495 Std. Dev. 0.100 [1] calculate the skewness [2] evaluate the discount policy in Swindon and Slough Statistics: 3 Alternative way of calculating skewnes:

Kurtosis ____________________________________________________________________________________________ formulas Kurtosis: sum of deviation from mean value divided by the standard deviation to the 4th power Excel formula: Statistics: 3 population excess kurtosis in comparison to the normal distribution (bell-shaped distribution)

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 mesokurtik zero-excess kurtosis

Are there any clusters of volatility? Interpretation ____________________________________________________________________________________________ kurtosis Histogram and kurtosis What to look at: Are there any clusters of volatility? kurtosis = 20.238 Statistics: 3 Huge value, thus: leptokurtic

Interpretation ____________________________________________________________________________________________ kurtosis Histogram and kurtosis Whernham Hogg and the discount policy again: Is the discount policy consistent? kurtosis = 1.406 Statistics: 3

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 ____________________________________________________________________________________________ one week to 1st. mid-term Skewness; Kurtosis; how to interpret? Histogram; Ogive; relation to measures of location, dispersion, skewness and kurtosis Statistics: 3