# Kurtosis Definition

### Kurtosis Definition

This is a statistical procedure used in reporting the distribution. Unlike skewness which differentiates extreme values between one tail and another, kurtosis computes the absolute values in each tail. Large kurtosis is present in the distributions that possess tail data surpassing the tails of the normal distribution. Conversely, the distributions that exhibit less extreme tail data in comparison to the tails of the normal distribution have low kurtosis.

Investors interpret high kurtosis of the return distribution as a signal that they will face frequent and more extreme returns than the typical + or – three standard deviations from the mean that the normal distribution of returns predicts. This is called kurtosis risk.

### A Little More on What is Kurtosis

Kurtosis is computed from the combination of the tails of a distribution relative to the center of the distribution. Graphing a set of approximately normal data through a histogram displays a bell peak and a majority of data within + or – three standard deviations of the mean. These tails, however, go beyond the + or – 3 standard deviations of the normal bell-curved distribution in the presence of high kurtosis.

Kurtosis doesn’t measure the peakedness of distribution but instead describes the shape of the distribution’s tail in relation to its form. This means that distribution can have an infinite peak and a low kurtosis and an infinite kurtosis but a perfectly flat top. It measures only tailedness.

### Types of Kurtosis

A set of data can display up to three categories of kurtosis whose measures are compared against a bell curve. These categories are as follows:

Mesokurtic distribution. In this distribution, the kurtosis statistic is the same as that of the bell curve, and so the distribution’s extreme value characteristic is the same as the one belonging to a normal distribution.

Leptokurtic distribution. This has a higher kurtosis than that of a mesokurtic distribution. It has elongated tails and is skinny because of the outliers stretching the horizontal axis of the histogram graph causing the most of the data to result in a narrow vertical range. This has led to the leptokurtic distribution being viewed as a concentrated towards the mean. However, some extreme outliers lead to concentration appearance.

Platykurtic distribution. These distributions possess short tails. Uniform distributions have broad peaks although the beta (.5, 1) has an infinitely pointy peak. These two distributions are platykurtic since their extreme values are lower than that of the bell curve. Investors view these distributions as stable and predictable because they don’t often become extreme returns.

### References for Kurtosis

• Measures of multivariate skewness and kurtosis with applications, Mardia, K. V. (1970). Biometrika, 57(3), 519-530. This paper develops more measures of multivariate skewness and kurtosis through the extension of several studies on the vigorousness of the  statistic.
• On the meaning and use of kurtosis., DeCarlo, L. T. (1997). Psychological methods, 2(3), 292. This article illustrates kurtosis with known distributions and discusses the features of its interpretation and misinterpretation.
• Statistical notes for clinical researchers: assessing normal distribution (2) using skewness and kurtosis, Kim, H. Y. (2013). Restorative dentistry & endodontics, 38(1), 52-54. This paper explains that there is no existing gold standard method to test the normality of data in numerous ways although many statistical methods have been proposed.
• Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies, Mardia, K. V. (1974). Sankhyā: The Indian Journal of Statistics, Series B, 115-128. This article conducts an investigation to determine whether the size of the standard theory tests of covariance matrices is affected by kurtosis.
• Nonlinear pricing kernels, kurtosis preference, and evidence from the cross section of equity returns, Dittmar, R. F. (2002). The Journal of Finance, 57(1), 369-403. This paper examines the nonlinear pricing kernels that determine the risk factor and in which the definition of the pricing kernel is restricted by the preferences. The kernels are used to generate models of nonlinear and multifactor empirical performance.
• Comparing measures of sample skewness and kurtosis, Joanes, D. N., & Gill, C. A. (1998). Journal of the Royal Statistical Society: Series D (The Statistician), 47(1), 183-189. This article compares the measures of sample skewness and kurtosis adopted by established statistical computing packages, focuses on standard sample’s bias and mean-squared error while presenting various comparisons arising from simulation results for non-normal samples.
• The spectral kurtosis: a useful tool for characterising non-stationary signals, Antoni, J. (2006). Mechanical Systems and Signal Processing, 20(2), 282-307. This paper attempts to fill the gap of spectral kurtosis caused by the lack of a formal definition and an understood estimation procedure. It also precedes another paper presenting spectral kurtosis as having found successful applications in vibration-based condition monitoring.
• Measuring skewness and kurtosis, Groeneveld, R. A., & Meeden, G. (1984). The Statistician, 391-399. This article provides an answer on how to measure the extent of skewness on a continuous random variable and also considers several properties to be satisfied by a measure of skewness.
• Kurtosis: a critical review, Balanda, K. P., & MacGillivray, H. L. (1988). The American Statistician, 42(2), 111-119. This paper reviews the concept of kurtosis and concludes that kurtosis should be defined vaguely as the location and scale-free movement of probability mass from top of a distribution to its center as well as tails and understand that it can be formalized in many ways.
• On more robust estimation of skewness and kurtosis, Kim, T. H., & White, H. (2004). Finance Research Letters, 1(1), 56-73. This study surveys robust measures of skewness and kurtosis from the statistics data and then carry out comprehensive Monte Carlo simulations that are used in the comparison of conventional measures with the robust measures of the survey.
• Skewness and kurtosis in S&P 500 index returns implied by option prices, Corrado, C. J., & Su, T. (1996). Journal of Financial research, 19(2), 175-192. This paper investigates a method that can be used in extending the Black-Scholes model to explain the biases resulting from nonnormal skewness and kurtosis in stock return distributions.