# Kurtosis Definition, Types, and Importance

The excess kurtosis in a platykurtic distribution is negative that is characterized by a flat-tail distribution. The minor outliers in a distribution are indicated by the flat tails. The platykurtic distribution of investment returns is advantageous for investors in the financial context as this would mean a higher return on investment.

All measures of kurtosis are compared against a normal distribution curve. The aggregate weight of a allotment’s tails compared to the center of the distribution is measured by kurtosis. A bell peak is shown with most data within three standard deviations within + or – variations of the mean and can be seen when normal data is graphed using a histogram.

1. Rather, it means the distribution produces fewer and/or less extreme outliers than the normal distribution.
2. An extreme positive kurtosis indicates a distribution where more of the values are located in the tails of the distribution rather than around the mean.
3. It is characterized by huge tails on either side with large outliers.

Instead, it approximately follows a Laplace distribution (shown by the blue curve). Leptokurtosis is sometimes called positive kurtosis, since the excess kurtosis is positive. In this tutorial ‘The Complete Guide to Skewness and Kurtosis’, you saw the concept of Skewness and Kurtosis and how to find their mathematical values.

The kurtosis of a mesokurtic distribution is neither high nor low; rather, it is considered to be a baseline for the two other classifications. The BB pulled together a team to analyze the reason for the outliers at each end of the distribution. After identifying the reasons, the team made some adjustments in the process which resulted in the process data now being more normal. This allowed the BB to complete her analyses which had normality as an assumption. The graphic below illustrates the relative shape of the three types of kurtosis. Where denotes the th central moment (and in
particular,
is the variance).

## Understanding Kurtosis

It is common to compare the excess kurtosis (defined below) of a distribution to 0. This value 0 is the excess kurtosis of any univariate normal distribution. Distributions with negative excess kurtosis are said to be platykurtic, although this does not imply the distribution is “flat-topped” as is sometimes stated.

Some authors use the term kurtosis to mean what we have defined as excess kurtosis. Kurtosis is always positive, since we have assumed that $$\sigma \gt 0$$ (the random variable really is random), and therefore $$\P(X \ne \mu) \gt 0$$. A leptokurtic distribution is one that has kurtosis greater than a mesokurtic distribution. Leptokurtic distributions are sometimes identified by peaks that are thin and tall. The tails of these distributions, to both the right and the left, are thick and heavy.

Kurtosis is sometimes confused with a measure of the peakedness of a distribution. However, kurtosis is a measure that describes the shape of a distribution’s tails in relation to its overall shape. A distribution can be sharply peaked with low kurtosis, and a distribution can have a lower peak with high kurtosis. Kurtosis is a statistical measure used to describe a characteristic of a dataset. When normally distributed data is plotted on a graph, it generally takes the form of a bell. The plotted data that are furthest from the mean of the data usually form the tails on each side of the curve.

## Types of Kurtosis

Mathematically speaking, kurtosis is the standardized fourth moment of a distribution. Moments are a set of measurements that tell you about the shape of a distribution. Excess kurtosis can be at or near zero as well, so the chance of an extreme outcome is rare. The tails of this kind of distribution is similar to that of a normal distribution. Open the Brownian motion experiment and select the last zero. Note the shape of the probability density function in relation to the moment results in the last exercise.

## Scientific definitions for kurtosis

A mesokurtic distribution is medium-tailed, so outliers are neither highly frequent, nor highly infrequent. A high kurtosis is a trend that investors watch closely as it could result that there will be sharper results in either directions of profits or loss. This in comparison to the normal deviation or the regular changes. Kurtosis risk is the name given to this indicator that gives investor a sign about their assets. The following exercise gives a more complicated continuous distribution that is not symmetric but has skewness 0. It is one of a collection of distributions constructed by Erik Meijer.

The data can be heavy-tailed, and the peak can be flatter, almost like punching the distribution or squishing it. If the distribution is light-tailed and the top curve steeper, like pulling up the distribution, it is called Positive Kurtosis (Leptokurtic). Division by the standard deviation will help you scale down the difference between mode and mean. Now understand the below relationship between mode, mean and median.

## What is a platykurtic distribution?

It’s important to note that a sample size should be much larger than this; we are using six numbers to reduce the calculation steps. A good rule of thumb is to use 30% of your data for populations under 1,000. Mesokurtic distributions have outliers that are neither highly frequent, nor highly infrequent, and this is true of the elephant birth weights. Occasionally, a female baby elephant will be born weighing less than 180 or more than 240 lbs. Kurtosis is used to find the presence of outliers in our data. Calculating kurtosis by hand is a lengthy endeavor, and takes several steps to get to the results.

The standard measure of a distribution’s kurtosis, originating with Karl Pearson,[1] is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not its peak;[2] hence, the sometimes-seen characterization of kurtosis as “peakedness” is incorrect. For this measure, higher kurtosis corresponds to greater extremity of deviations (or outliers), and not the configuration of data near the mean.

## Indicator Variables

High Kurtosis of the return distribution implies that an investment will yield occasional extreme returns. Be mindful that this can swing both ways, meaning high kurtosis indicates either large positive returns or extreme negative returns. The reason not to subtract 3 is that the bare fourth moment better generalizes to multivariate distributions, especially when independence is not assumed. The cokurtosis between pairs of variables is an order four tensor. For a bivariate normal distribution, the cokurtosis tensor has off-diagonal terms that are neither 0 nor 3 in general, so attempting to “correct” for an excess becomes confusing. It is true, however, that the joint cumulants of degree greater than two for any multivariate normal distribution are zero.

Excess kurtosis means the distribution of event outcomes have lots of instances of outlier results, causing fat tails on the bell-shaped distribution curve. Excess kurtosis can, therefore, be calculated by subtracting kurtosis by three. Open the special distribution simulator and select the Pareto distribution. Vary the shape parameter and note define kurtosis the shape of the probability density function in comparison to the moment results in the last exercise. For selected values of the parameter, run the experiment 1000 times and compare the empirical density function to the true probability density function. The leptokurtic distribution shows heavy tails on either side, indicating large outliers.

This form is implemented in the Wolfram Language as Kurtosis[dist]. On the other hand, a portfolio with a low kurtosis value indicates a more stable and predictable return profile, which may indicate lower risk. In this light, investors may intentionally seek investments with lower kurtosis values when building safer, less volatile portfolios. Skewness and kurtosis are both important measures of a distribution’s shape. Moments are standardized by dividing them by the standard deviation raised to the appropriate power.

Leptokurtic distributions are named by the prefix “lepto” meaning “skinny.” Kurtosis is typically measured with respect to the normal distribution. A distribution that has tails shaped in roughly the same way as any normal distribution, not just the standard normal distribution, is said to be mesokurtic.

Distributions with a large kurtosis have more tail data than normally distributed data, which appears to bring the tails in toward the mean. Distributions with low kurtosis have fewer tail data, which appears to push the tails of the bell curve away from the mean. The sample kurtosis is https://1investing.in/ a useful measure of whether there is a problem with outliers in a data set. Larger kurtosis indicates a more serious outlier problem, and may lead the researcher to choose alternative statistical methods. Where X is a random variable, μ is the mean and σ is the standard deviation.