Kurtosis in statistics is used to describe the distribution of the data set and depicts to what extent the data set points of a particular distribution differ from the data of a normal distribution. When I look at a normal curve, it seems the peak occurs at the center, a.k.a at 0. Kurtosis is sometimes reported as “excess kurtosis.” Excess kurtosis is determined by subtracting 3 from the kurtosis. This definition is used so that the standard normal distribution has a kurtosis of three. Whereas skewness differentiates extreme values in one versus the other tail, kurtosis measures extreme values in either tail. On the other hand, kurtosis identifies the way; values are grouped around the central point on the frequency distribution. Because kurtosis compares a distribution to the normal distribution, 3 is often subtracted from the calculation above to get a number which is 0 for a normal distribution, +ve for leptokurtic distributions, and –ve for mesokurtic ones. Here you can get an Excel calculator of kurtosis, skewness, and other summary statistics.. Kurtosis Value Range. \, = 1173333.33 - 126293.31+67288.03-1165.87 \\[7pt] All measures of kurtosis are compared against a standard normal distribution, or bell curve. \, = 1173333.33 - 4 (4.44)(7111.11)+6(4.44)^2 (568.88) - 3(4.44)^4 \\[7pt] Here, x̄ is the sample mean. A distribution can be infinitely peaked with low kurtosis, and a distribution can be perfectly flat-topped with infinite kurtosis. Since the deviations have been taken from an assumed mean, hence we first calculate moments about arbitrary origin and then moments about mean. \mu_4^1= \frac{\sum fd^4}{N} \times i^4 = \frac{330}{45} \times 20^4 =1173333.33 }$,${\mu_2 = \mu'_2 - (\mu'_1 )^2 = 568.88-(4.44)^2 = 549.16 \\[7pt] KURTOSIS. This means that for a normal distribution with any mean and variance, the excess kurtosis is always 0. In this view, kurtosis is the maximum height reached in the frequency curve of a statistical distribution, and kurtosis is a measure of the sharpness of the data peak relative to the normal distribution. Kurtosis originally was thought to measure the peakedness of a distribution. The normal curve is called Mesokurtic curve. Leptokurtic (Kurtosis > 3): Distribution is longer, tails are fatter. The degree of tailedness of a distribution is measured by kurtosis. The normal distribution is found to have a kurtosis of three. Skewness and kurtosis involve the tails of the distribution. The normal distribution has excess kurtosis of zero. For investors, high kurtosis of the return distribution implies the investor will experience occasional extreme returns (either positive or negative), more extreme than the usual + or - three standard deviations from the mean that is predicted by the normal distribution of returns. Dr. Wheeler defines kurtosis as: The kurtosis parameter is a measure of the combined weight of the tails relative to the rest of the distribution. There are two different common definitions for kurtosis: (1) mu4/sigma4, which indeed is three for a normal distribution, and (2) kappa4/kappa2-square, which is zero for a normal distribution. Thus, with this formula a perfect normal distribution would have a kurtosis of three. It is common to compare the kurtosis of a distribution to this value. Leptokurtic - positive excess kurtosis, long heavy tails When excess kurtosis is positive, the balance is shifted toward the tails, so usually the peak will be low , but a high peak with some values far from the average may also have a positive kurtosis! Any distribution that is leptokurtic displays greater kurtosis than a mesokurtic distribution. The only difference between formula 1 and formula 2 is the -3 in formula 1. Kurtosis is a measure of the combined weight of a distribution's tails relative to the center of the distribution. A normal curve has a value of 3, a leptokurtic has \beta_2 greater than 3 and platykurtic has \beta_2 less then 3. In statistics, normality tests are used to determine whether a data set is modeled for normal distribution. Excess kurtosis compares the kurtosis coefficient with that of a normal distribution. There are three types of kurtosis: mesokurtic, leptokurtic, and platykurtic. Explanation The normal PDF is also symmetric with a zero skewness such that its median and mode values are the same as the mean value. statistics normal-distribution statistical-inference. If the curve of a distribution is more outlier prone (or heavier-tailed) than a normal or mesokurtic curve then it is referred to as a Leptokurtic curve. Kurtosis is a measure of whether or not a distribution is heavy-tailed or light-tailed relative to a normal distribution. When a set of approximately normal data is graphed via a histogram, it shows a bell peak and most data within + or - three standard deviations of the mean. The prefix of "platy-" means "broad," and it is meant to describe a short and broad-looking peak, but this is an historical error. The crux of the distribution is that in skewness the plot of the probability distribution is stretched to either side. The kurtosis can be even more convoluted. These are presented in more detail below. However, when high kurtosis is present, the tails extend farther than the + or - three standard deviations of the normal bell-curved distribution. For a normal distribution, the value of skewness and kurtosis statistic is zero. Kurtosis is measured by moments and is given by the following formula −. For normal distribution this has the value 0.263. A distribution with kurtosis <3 (excess kurtosis <0) is called platykurtic. Using this definition, a distribution would have kurtosis greater than a normal distribution if it had a kurtosis value greater than 0. Peak is higher and sharper than Mesokurtic, which means that data are heavy-tailed or profusion of outliers. Skewness. The second category is a leptokurtic distribution. The kurtosis of the uniform distribution is 1.8. You can play the same game with any distribution other than U(0,1). Comment on the results. Three different types of curves, courtesy of Investopedia, are shown as follows −. 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. \mu_3 = \mu'_3 - 3(\mu'_1)(\mu'_2) + 2(\mu'_1)^3 \\[7pt] The reason both these distributions are platykurtic is their extreme values are less than that of the normal distribution. Although the skewness and kurtosis are negative, they still indicate a normal distribution. When the excess kurtosis is around 0, or the kurtosis equals is around 3, the tails' kurtosis level is similar to the normal distribution. Though you will still see this as part of the definition in many places, this is a misconception. Many statistical functions require that a distribution be normal or nearly normal. If a distribution has positive kurtosis, it is said to be leptokurtic, which means that it has a sharper peak and heavier tails compared to a normal distribution. Examples of leptokurtic distributions are the T-distributions with small degrees of freedom. It tells us about the extent to which the distribution is flat or peak vis-a-vis the normal curve. Explanation If a distribution has a kurtosis of 0, then it is equal to the normal distribution which has the following bell-shape: Positive Kurtosis. It tells us the extent to which the distribution is more or less outlier-prone (heavier or light-tailed) than the normal distribution. 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