1. Lecture 3: Skewness and Kurtosis
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 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

3. 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)

4. Skewness ____________________________________________________________________________________________
positive skewness
Graphical presentation:
Statistics: 3

5. Skewness ____________________________________________________________________________________________
negative skewness
Graphical presentation:
Statistics: 3

6. 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 ( )
xM – mode, a value that occurs most frequently in the sample or population

7. 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

8. 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 = USD
median = USD
skewness = 1.907
Statistics: 3
relatively large value, thus: positively skewed

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

10. 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:

11. 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)

12. 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

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

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

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