Leptokurtic distribution

Kurtosis K4 characterizes the proportion of the tails in relation to the center. When compared with the normal distribution, platykurtic distributions have more values in the tails and leptokurtic distribution have less. 

There is a fourth parameter to describe the characteristics of a distribution, which is the kurtosis. This is the peakedness of the data set. If the data set has a flat peak in the distribution curve it is called platykurtic. If the peak is very sharp it is called leptokurtic. Distributions with peaks in-between are called mesokurtic. 

Some literature has defined kurtosis in terms of pdfs that are relatively flat versus relatively peaked at the mode. A tendency in more recent literature is to emphasize the idea of tail weight. Leptokurtic distributions have relatively heavier pdf tails, while platykurtic distributions have relatively lighter tails (Balanda and MacGillivray 1998). 

Leptokurtic distributions are more outlier-prone. When fitting distributions to data, it may sometimes be difficult to decide whether one should assume a leptokurtic distribution (say, a Student t distribution with relatively few degrees of freedom) or assume the presence of a few outliers. 

All the cations in the octahedral position have a normal-type distribution. Octahedral A1 has a very distinct mode. Twenty-eight percent of the values fall in one class range (0.40—0.50) out of a total range of thirteen classes. The Fe2+ is even more leptokurtic having 63% of the values in the 0.15—0.25 range. Octahedral Mg also has a leptokurtic distribution but no narrow mode occurs. Eighty-nine percent of the values are in the range of 0.25-0.50. 

To avoid problems related to infinite (or very high) kurtosis values, obtained when along a principal axis all the atoms are projected in the centre (or near the centre, i.e. leptokurtic distribution), the inverse of the kurtosis is used. 

FIGURE 4.3 Graphical representation of kurtosis, K, and skewness, S, in comparison to the Gaussian (standard) distribution (upper left). The right-hand side shows leptokurtic (peaked) or platykurtic (flatted) distribution as well as positive skewed distribution (fronting) and negative skewed distribution (tailing). 

If the kurtosis is substantially different from zero, then the distribution is either flatter (platykurtic, K <0) or more peaked (leptokurtic, K > 0) than the Gaussian distribution that should have a kurtosis of zero (Figure 4.3). 

Leptokurtic Refers to a particle size distribution that is excessively peaked (cf. Platykur-tic). 

The kurtosis measure reveals the peakedness or flatness of a distribution. A kurtosis value greater than that of a normal distribution means that the distribution is leptokurtic, or simply more peaked than... 

Many authors will define kurtosis as the normalized fourth central moment minus 3. This is done primarily to reference the kurtosis to the normal distribution which has a kurtosis of 3. This form is also known as and, in the authors opinion, more properly known as the excess kurtosis. The excess kurtosis is primarily used to relate the flatness of a distribution to that of the normal distribution. A distribution that is sharper or more bunched in the middle than the normal distribution has a negative excess kurtosis and is referred to as a leptocurtical or leptokurtic distribution. A distribution that is flatter or more spread out from the normal distribution has a positive excess kurtosis and is referred to as a platicurtical or platykurtic distribution. A distribution that has the same... 

Kurtosis is a measure of the weight of the tails of a distribution or die peakedness of a distribution. A normal distribution is defined as having zero kurtosis (mesokurtic). When q(x) values are closer to the mean, the distribution is narrower or sharper than the normal distribution and the kurtosis (g2) is positive (leptokurtic). When q(x) values tend towards the extremes, the distribution is broader than the normal distribution and the kurtosis is negative (platykurtic). These three situations are illustrated in Figure 1.14. The second approximate equality in Eq. 1.13 is held when N is large. 

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