Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to have heavy tails, or outliers. Data sets with low kurtosis tend to have light tails, or lack of outliers. A uniform distribution would be the extreme case.
A negative kurtosis means that your distribution is flatter than a normal curve with the same mean and standard deviation. The easiest way to visualise this is to plot a histogram with a fitted normal curve.
In statistics, kurtosis is used to describe the shape of a probability distribution. Specifically, it tells us the degree to which data values cluster in the tails or the peak of a distribution. The kurtosis for a distribution can be negative, equal to zero, or positive.
Kurtosis is a statistical measure that defines how heavily the tails of a distribution differ from the tails of a normal distribution. In other words, kurtosis identifies whether the tails of a given distribution contain extreme values. In finance, kurtosis is used as a measure of financial risk.
m2 is the variance, the square of the standard deviation. The kurtosis can also be computed as a4 = the average value of z4, where z is the familiar z-score, z = (x−x¯)/σ.
Skewness characterizes the degree of asymmetry of a distribution around its mean. Positive skewness indicates a distribution with an asymmetric tail extending toward more positive values.
In order to obtain a coefficient of kurtosis that is independent of the units of measurement, the fourth-order moment is divided by the standard deviation of the population σ raised to the fourth power. The coefficient of kurtosis then becomes equal to: eta_2=frac{mu_4}{sigma^4}:.
What Does Platykurtic Mean? The term "platykurtic" refers to a statistical distribution in which the excess kurtosis value is negative. For this reason, a platykurtic distribution will have thinner tails than a normal distribution, resulting in fewer extreme positive or negative events.
Calculation. The formula given in most textbooks is Skew = 3 * (Mean – Median) / Standard Deviation. This is known as an alternative Pearson Mode Skewness. You could calculate skew by hand.
Leptokurtic distributions are distributions with kurtosis larger than that of a normal distribution. A leptokurtic distribution means that the investor can experience broader fluctuations (e.g. three or more standard deviations from the mean) resulting in greater potential for extremely low or high returns.
The excess kurtosis should be zero for a perfectly normal distribution. Distributions with positive excess kurtosis are called leptokurtic distribution meaning high peak, and distributions with negative excess kurtosis are called platykurtic distribution meaning flat-topped curve.
(2010) and Bryne (2010) argued that data is considered to be normal if Skewness is between -2 to +2 and Kurtosis is between -7 to +7. Multi-normality data tests are performed using leveling asymmetry tests (skewness < 3), (Kurtosis between -2 and 2) and Mardia criterion (< 3).
Kurtosis is only useful when used in conjunction with standard deviation. It is possible that an investment might have a high kurtosis (bad), but the overall standard deviation is low (good). Conversely, one might see an investment with a low kurtosis (good), but the overall standard deviation is high (bad).
In general, kurtosis tells you nothing about the "peak" of a distribution, and also tells you nothing about its "shoulders." It measures outliers (tails) only. For an outlier-prone (heavy tailed) distribution, this percentage is typically higher, like 2.0%.
As a general rule of thumb: If skewness is less than -1 or greater than 1, the distribution is highly skewed. If skewness is between -1 and -0.5 or between 0.5 and 1, the distribution is moderately skewed. If skewness is between -0.5 and 0.5, the distribution is approximately symmetric.
The normal curve is called Mesokurtic curve. If the curve of a distribution is peaked than a normal or mesokurtic curve then it is referred to as a Leptokurtic curve. If a curve is less peaked than a normal curve, it is called as a Platykurtic curve. That's why kurtosis of normal distribution equal to three.
