An Article Review
Kurtosis in Descriptive Statistics
Kurtosis is defined as the measure of “peakedness” of a distribution. It is defined as the fourth population moment about the mean and calculated as per the formula given below.
β2 = – (Decarlo, 1997),
where E is expectation operator, µ is the mean, µ4 is the fourth moment about the mean, and is the standard deviation. The normal distribution has a kurtosis of 3 and β – 3 (denoted by γ2) is often calculated so as to refer to a normal distribution with kurtosis of zero.
The kurtosis is now considered an important tool in the study of descriptive statistics. Descriptive statistics fall into 3 general categories: location statistics (e.g. mean, median, mode, quantiles), dispersion statistics (e.g. variance, standard deviation, range, interquartile range), and shape statistics (e.g. skewness, kurtosis) (Larson, 2006). In a Gaussian distribution, a positive kurtosis denotes a sharper peak and fatter tails, whereas, a negative kurtosis can be seen in curves with broader shoulders and thinner tails. It is often argued that in very large samples with very small departures from normality may lead to significant skewness and kurtosis coefficients and rejection of the normality assumption (Ullman, 2006). Some statistical societies like SAS Institute are also of the opinion that the kurtosis is in actuality, a measure of heavy and larger tails, rather than that of “peakedness”.
In the write-up, we are to study an article and the descriptive statistical measures used in it and assess the effectiveness of the tool. The article selected for such a study is the “Study of the Kurtosis in Directional Sea States for Freak Wave Forecasting”, written by Nubohito Mori, Miguel Onorato and P. Janssen for the Journal of Physical Oceanography. The article examines the dependence of the kurtosis on the directional energy distribution of the initial
conditions, based on Monte Carlo simulations of the nonlinear Schrodinger equation in two horizontal dimensions.
The article begins with the need of such a study and relates the kurtosis with freak wave generation. The article begins with defining a correlation between Benjamin–Feir index (BFI) and Kurtosis. It defines the relationship as
The reason for choice of Kurtosis over Skewness is also presented. The skewness is a correction of the second-order and is hardly mildly affected by free wave dynamics (Onorato, 2005). The relation between kurtosis and the distribution of height of waves function is a correction which is nonlinear to the Rayleigh distribution (Mori, Onorato & Janssen, 2011). In other words it is when the kurtosis value approaches to 3, the same as that in a Gaussian curve.
The author argues that the framework designed for the relation between freak wave frequency and kurtosis was followed up with wave experiments of a large scale. This was an expected result as the kurtosis is the fourth order moment of probability density. It is here that the importance of the kurtosis as a better indicator of measure of the tails in the distribution curve than the peaks are brought to light. The author then concludes that for a viable model for freak wave prediction, it is very important to accurately measure the kurtosis. The paper proposes a modification in the kurtosis estimation formula and in later sections provides a validation for it.
The kurtosis is used as exactly as it is predicted to be. The kurtosis of the surface elevation is mainly determined by four wave– wave interactions, whereas bound waves only give a small contribution. Directional dispersion suppresses the kurtosis enhancement by four wave–wave interactions as indicated by several researchers. The hypothesis that the kurtosis is indicator of large tails is strengthened.
DeCarlo, L.T. (1997). On the Meaning and Use of Kurtosis. Psychological Methods, 2(3), 292-307.
Larson, M.G. (2006). Descriptive Statistics and Graphical Displays. Circulation: Journal of American Heart Association, 114, 76-81.
Mori, N., Onorato, M. & Janssen, P.A.E.M. (2011). On the Estimation of the Kurtosis in Directional Sea States for Freak Wave Forecasting. JOURNAL OF PHYSICAL OCEANOGRAPHY, 41, 1484-1497.
Onorato, M., A. (2005). Freak waves in random oceanic sea states. Phys. Rev. Lett., 86, 5831–5834.
Ullman, J.B. (2006). Structural Equation Modeling: Reviewing the Basics and Moving Forward. JOURNAL OF PERSONALITY ASSESSMENT, 87(1), 35–50.