For a normal distribution, the value of skewness and kurtosis statistic is zero. Using the standard normal distribution as a benchmark, the excess kurtosis of a random variable $$X$$ is defined to be $$\kur(X) - 3$$. KURTOSIS. It means that the extreme values of the distribution are similar to that of a normal distribution characteristic. Kurtosis is a measure of the combined weight of a distribution's tails relative to the center of the distribution. All measures of kurtosis are compared against a standard normal distribution, or bell curve. An example of this, a nicely rounded distribution, is shown in Figure 7. On the other hand, kurtosis identifies the way; values are grouped around the central point on the frequency distribution. What is meant by the statement that the kurtosis of a normal distribution is 3. We will show in below that the kurtosis of the standard normal distribution is 3. whether the distribution is heavy-tailed (presence of outliers) or light-tailed (paucity of outliers) compared to a normal distribution. Kurtosis has to do with the extent to which a frequency distribution is peaked or flat. Examples of leptokurtic distributions are the T-distributions with small degrees of freedom. The second formula is the one used by Stata with the summarize command. 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. My textbook then says "the kurtosis of a normally distributed random variable is $3$." The data on daily wages of 45 workers of a factory are given. The offers that appear in this table are from partnerships from which Investopedia receives compensation. Any distribution with kurtosis ≈3 (excess ≈0) is called mesokurtic. A high kurtosis distribution has a sharper peak and longer fatter tails, while a low kurtosis distribution has a more rounded pean and shorter thinner tails. The reason both these distributions are platykurtic is their extreme values are less than that of the normal distribution. Explanation Kurtosis originally was thought to measure the peakedness of a distribution. How can all normal distributions have the same kurtosis when standard deviations may vary? For example, the “kurtosis” reported by Excel is actually the excess kurtosis. However, when high kurtosis is present, the tails extend farther than the + or - three standard deviations of the normal bell-curved distribution. It has fewer extreme events than a normal distribution. In statistics, normality tests are used to determine whether a data set is modeled for normal distribution. On the other hand, kurtosis identifies the way; values are grouped around the central point on the frequency distribution. For different limits of the two concepts, they are assigned different categories. \mu_2^1= \frac{\sum fd^2}{N} \times i^2 = \frac{64}{45} \times 20^2 =568.88 \$7pt] Now excess kurtosis will vary from -2 to infinity. sharply peaked with heavy tails) The first category of kurtosis is a mesokurtic distribution. The "skinniness" of a leptokurtic distribution is a consequence of the outliers, which stretch the horizontal axis of the histogram graph, making the bulk of the data appear in a narrow ("skinny") vertical range. 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. Distributions with kurtosis less than 3 are said to be platykurtic, although this does not imply the distribution is "flat-topped" as is sometimes stated. This definition of kurtosis can be found in Bock (1975). A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0). This makes the normal distribution kurtosis equal 0. \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] It tells us the extent to which the distribution is more or less outlier-prone (heavier or light-tailed) than the normal distribution. Skewness essentially measures the relative size of the two tails. The prefix of "platy-" means "broad," and it is meant to describe a short and broad-looking peak, but this is an historical error. 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. Excess Kurtosis for Normal Distribution = 3–3 = 0. It is common to compare the kurtosis of a distribution to this value. These types of distributions have short tails (paucity of outliers.) Excess kurtosis describes a probability distribution with fat fails, indicating an outlier event has a higher than average chance of occurring. The reference standard is a normal distribution, which has a kurtosis of 3. Any distribution that is leptokurtic displays greater kurtosis than a mesokurtic 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. How can all normal distributions have the same kurtosis when standard deviations may vary? A distribution with kurtosis <3 (excess kurtosis <0) is called platykurtic. Kurtosis is sometimes confused with a measure of the peakedness of a distribution. Skewness is a measure of the symmetry in a distribution. 3 is the mode of the system? This phenomenon is known as kurtosis risk. Here you can get an Excel calculator of kurtosis, skewness, and other summary statistics.. Kurtosis Value Range. Using this definition, a distribution would have kurtosis greater than a normal distribution if it had a kurtosis value greater than 0. Let’s see the main three types of kurtosis. Here, x̄ is the sample mean. Mesokurtic is a statistical term describing the shape of a probability distribution. The normal curve is called Mesokurtic curve. \, = 1173333.33 - 4 (4.44)(7111.11)+6(4.44)^2 (568.88) - 3(4.44)^4 \\[7pt] The kurtosis of the uniform distribution is 1.8. Kurtosis is measured by moments and is given by the following formula −. 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. Distributions that are more outlier-prone than the normal distribution have kurtosis greater than 3; distributions that are less outlier-prone have kurtosis less than 3. The kurtosis of the normal distribution is 3. It tells us about the extent to which the distribution is flat or peak vis-a-vis the normal curve. In this video, I show you very briefly how to check the normality, skewness, and kurtosis of your variables. A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0). A distribution with kurtosis greater than three is leptokurtic and a distribution with kurtosis less than three is platykurtic. Excess kurtosis compares the kurtosis coefficient with that of a normal distribution. Peak is higher and sharper than Mesokurtic, which means that data are heavy-tailed or profusion of outliers. For investors, platykurtic return distributions are stable and predictable, in the sense that there will rarely (if ever) be extreme (outlier) returns. Kurtosis is a statistical measure which quantifies the degree to which a distribution of a random variable is likely to produce extreme values or outliers relative to a normal distribution. I am wondering whether only standard normal distribution has a kurtosis being 3, or any normal distribution has the same kurtosis, namely 3. metric that compares the kurtosis of a distribution against the kurtosis of a normal distribution If a given distribution has a kurtosis less than 3, it is said to be playkurtic, which means it tends to produce fewer and less extreme outliers than the normal distribution. As a result, people usually use the "excess kurtosis", which is the {\rm kurtosis} - 3. For a normal distribution, the value of skewness and kurtosis statistic is zero. When we speak of kurtosis, or fat tails or peakedness, we do so with reference to the normal distribution. Laplace, for instance, has a kurtosis of 6. A distribution with kurtosis <3 (excess kurtosis <0) is called platykurtic. Kurtosis risk applies to any kurtosis-related quantitative model that assumes the normal distribution for certain of its independent variables when the latter may in fact have kurtosis much greater than does the normal distribution. You can play the same game with any distribution other than U(0,1). This definition is used so that the standard normal distribution has a kurtosis of three. The normal distribution has excess kurtosis of zero. Like skewness, kurtosis is a statistical measure that is used to describe distribution. Kurtosis risk is commonly referred to as "fat tail" risk. Thus, with this formula a perfect normal distribution would have a kurtosis of three. The only difference between formula 1 and formula 2 is the -3 in formula 1. 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. The only difference between formula 1 and formula 2 is the -3 in formula 1. Uniform distributions are platykurtic and have broad peaks, but the beta (.5,1) distribution is also platykurtic and has an infinitely pointy peak. It tells us the extent to which the distribution is more or less outlier-prone (heavier or light-tailed) than the normal distribution. 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. The kurtosis for a standard normal distribution is three. Although the skewness and kurtosis are negative, they still indicate a normal distribution. Comment on the results. The normal distribution has kurtosis of zero. All measures of kurtosis are compared against a standard normal distribution, or bell curve. So, a normal distribution will have a skewness of 0. Peak is higher and sharper than Mesokurtic, which means that data are heavy-tailed or profusion of outliers. \mu_3 = \mu'_3 - 3(\mu'_1)(\mu'_2) + 2(\mu'_1)^3 \\[7pt] An example of a mesokurtic distribution is the binomial distribution with the value of p close to 0.5. The second formula is the one used by Stata with the summarize command. Long-tailed distributions have a kurtosis higher than 3. With this definition a perfect normal distribution would have a kurtosis of zero. There are three categories of kurtosis that can be displayed by a set of data. Leptokurtic: More values in the distribution tails and more values close to the mean (i.e. Q.L. Excess kurtosis is a valuable tool in risk management because it shows whether an … 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. The graphical representation of kurtosis allows us to understand the nature and characteristics of the entire distribution and statistical phenomenon. Normal distribution kurtosis = 3; A distribution that is more peaked and has fatter tails than normal distribution has kurtosis value greater than 3 (the higher kurtosis, the more peaked and fatter tails). In other words, it indicates whether the tail of distribution extends beyond the ±3 standard deviation of the mean or not. As the kurtosis measure for a normal distribution is 3, we can calculate excess kurtosis by keeping reference zero for normal distribution. Kurtosis is sometimes reported as “excess kurtosis.” Excess kurtosis is determined by subtracting 3 from the kurtosis. The kurtosis of the normal distribution is 3, which is frequently used as a benchmark for peakedness comparison of a given unimodal probability density. Normal distribution kurtosis = 3; A distribution that is more peaked and has fatter tails than normal distribution has kurtosis value greater than 3 (the higher kurtosis, the more peaked and fatter tails). A normal bell curve would have much of the data distributed in the center of the data and although this data set is virtually symmetrical, it is deviated to the right; as shown with the histogram. Another less common measures are the skewness (third moment) and the kurtosis (fourth moment). The final type of distribution is a platykurtic distribution. A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0). This means that for a normal distribution with any mean and variance, the excess kurtosis is always 0. Thus leptokurtic distributions are sometimes characterized as "concentrated toward the mean," but the more relevant issue (especially for investors) is there are occasional extreme outliers that cause this "concentration" appearance. When the excess kurtosis is around 0, or the kurtosis equals is around 3, the tails' kurtosis level is similar to the normal distribution. A normal distribution has kurtosis exactly 3 (excess kurtosis … A distribution can be infinitely peaked with low kurtosis, and a distribution can be perfectly flat-topped with infinite kurtosis. For example, take a U(0,1) distribution and mix it with a N(0,1000000) distribution, with .00001 mixing probability on the normal. Its formula is: where. From the value of movement about mean, we can now calculate {\beta_1} and {\beta_2}: From the above calculations, it can be concluded that {\beta_1}, which measures skewness is almost zero, thereby indicating that the distribution is almost symmetrical. A bell curve describes the shape of data conforming to a normal distribution. \mu_4= \mu'_4 - 4(\mu'_1)(\mu'_3) + 6 (\mu_1 )^2 (\mu'_2) -3(\mu'_1)^4 \\[7pt] As the name suggests, it is the kurtosis value in excess of the kurtosis value of the normal distribution. Kurtosis ranges from 1 to infinity. Characteristics of this distribution is one with long tails (outliers.) These are presented in more detail below. 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! The most well-known distribution that has a positive kurtosis is the t distribution, which has a sharper peak and heaver tails compared to the normal distribution. Compared to a normal distribution, its central peak is lower and … \, = 7111.11 - 7577.48+175.05 = - 291.32 \\[7pt] share | cite | improve this question | follow | asked Aug 28 '18 at 19:59. Kurtosis can reach values from 1 to positive infinite. The kurtosis of a normal distribution is 3. \beta_2 = \frac{\mu_4}{(\mu_2)^2} = \frac{1113162.18}{(546.16)^2} = 3.69 }, Process Capability (Cp) & Process Performance (Pp). Any distribution with kurtosis ≈3 (excess ≈0) is called mesokurtic. Many books say that these two statistics give you insights into the shape of the distribution. With this definition a perfect normal distribution would have a kurtosis of zero. It is used to determine whether a distribution contains extreme values. It is used to determine whether a distribution contains extreme 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. Kurtosis is positive if the tails are "heavier" then for a normal distribution, and negative if the tails are "lighter" than for a normal distribution. Some authors use the term kurtosis to mean what we have defined as excess kurtosis. There are three types of kurtosis: mesokurtic, leptokurtic, and platykurtic. Many statistical functions require that a distribution be normal or nearly normal. Distributions with large kurtosis exhibit tail data exceeding the tails of the normal distribution (e.g., five or more standard deviations from the mean). We will show in below that the kurtosis of the standard normal distribution is 3. The normal distribution is found to have a kurtosis of three. whether the distribution is heavy-tailed (presence of outliers) or light-tailed (paucity of outliers) compared to a normal distribution. Mesokurtic: Distributions that are moderate in breadth and curves with a medium peaked height. The entropy of a normal distribution is given by 1 2 log e 2 πe σ 2. 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. Any distribution that is peaked the same way as the normal distribution is sometimes called a mesokurtic distribution. So, kurtosis is all about the tails of the distribution – not the peakedness or flatness. The resulting distribution, when graphed, appears perfectly flat at its peak, but has very high kurtosis. Computational Exercises . Investopedia uses cookies to provide you with a great user experience. Though you will still see this as part of the definition in many places, this is a misconception. Skewness. A normal bell-shaped distribution is referred to as a mesokurtic shape distribution. In token of this, often the excess kurtosis is presented: excess kurtosis is simply kurtosis−3. The degree of tailedness of a distribution is measured by kurtosis. However, kurtosis is a measure that describes the shape of a distribution's tails in relation to its overall shape. Kurtosis is typically measured with respect to the normal distribution. [Note that typically these distributions are defined in terms of excess kurtosis, which equals actual kurtosis minus 3.] But this is also obviously false in general. 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. \, = 1113162.18 }, {\beta_1 = \mu^2_3 = \frac{(-291.32)^2}{(549.16)^3} = 0.00051 \\[7pt] Kurtosis is measured by … Scenario Today, we will try to give a brief explanation of these measures and we will show how we can calculate them in R. Leptokurtic distributions are statistical distributions with kurtosis over three. This definition is used so that the standard normal distribution has a kurtosis of three. Further, it will exhibit [overdispersion] relative to a single normal distribution with the given variation. 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. Excess kurtosis compares the kurtosis coefficient with that of a normal distribution. For normal distribution this has the value 0.263. Kurtosis tells you the height and sharpness of the central peak, relative to that of a standard bell curve. Moments about arbitrary origin '170'. The term “Kurtosis” refers to the statistical measure that describes the shape of either tail of a distribution, i.e. The normal PDF is also symmetric with a zero skewness such that its median and mode values are the same as the mean value. As with skewness, a general guideline is that kurtosis within ±1 of the normal distribution’s kurtosis indicates sufficient normality. 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. Diagrammatically, shows the shape of three different types of curves. When I look at a normal curve, it seems the peak occurs at the center, a.k.a at 0. But differences in the tails are easy to see in the normal quantile-quantile plots (right panel). Today, we will try to give a brief explanation of these measures and we will show how we can calculate them in R. Kurtosis can reach values from 1 to positive infinite. Most commonly a distribution is described by its mean and variance which are the first and second moments respectively. The greater the value of \beta_2 the more peaked or leptokurtic the curve. Does it mean that on the horizontal line, the value of 3 corresponds to the peak probability, i.e. Q.L. Tail risk is portfolio risk that arises when the possibility that an investment will move more than three standard deviations from the mean is greater than what is shown by a normal distribution. In statistics, we use the kurtosis measure to describe the “tailedness” of the distribution as it describes the shape of it. Evaluation. Discover more about mesokurtic distributions here. 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. A symmetrical dataset will have a skewness equal to 0. Here you can get an Excel calculator of kurtosis, skewness, and other summary statistics.. Kurtosis Value Range. {\beta_2} Which measures kurtosis, has a value greater than 3, thus implying that the distribution is leptokurtic. The kurtosis of a distribution is defined as . This article defines MAQL to calculate skewness and kurtosis that can be used to test the normality of a given data set. Explanation Compute \beta_1 and \beta_2 using moment about the mean. Leptokurtic (Kurtosis > 3): Distribution is longer, tails are fatter. If a distribution has a kurtosis of 0, then it is equal to the normal distribution which has the following bell-shape: Positive Kurtosis. The term “Kurtosis” refers to the statistical measure that describes the shape of either tail of a distribution, i.e. Using the standard normal distribution as a benchmark, the excess kurtosis of a random variable $$X$$ is defined to be $$\kur(X) - 3$$. Whereas skewness differentiates extreme values in one versus the other tail, kurtosis measures extreme values in either tail. By using Investopedia, you accept our. Any distribution with kurtosis ≈3 (excess ≈0) is called mesokurtic. While measuring the departure from normality, Kurtosis is sometimes expressed as excess Kurtosis which is … This distribution has a kurtosis statistic similar to that of the normal distribution, meaning the extreme value characteristic of the distribution is similar to that of a normal distribution. Since the deviations have been taken from an assumed mean, hence we first calculate moments about arbitrary origin and then moments about mean. For this reason, some sources use the following definition of kurtosis (often referred to as "excess kurtosis"): \[ \mbox{kurtosis} = \frac{\sum_{i=1}^{N}(Y_{i} - \bar{Y})^{4}/N} {s^{4}} - 3$ This definition is used so that the standard normal distribution has a kurtosis of zero. statistics normal-distribution statistical-inference. The degree of flatness or peakedness is measured by kurtosis. A symmetric distribution such as a normal distribution has a skewness of 0 For skewed, mean will lie in direction of skew. While measuring the departure from normality, Kurtosis is sometimes expressed as excess Kurtosis which is the balance amount of Kurtosis after subtracting 3.0. Three different types of curves, courtesy of Investopedia, are shown as follows −. The kurtosis of any univariate normal distribution is 3. Most commonly a distribution is described by its mean and variance which are the first and second moments respectively. Excess kurtosis is a valuable tool in risk management because it shows whether an … The crux of the distribution is that in skewness the plot of the probability distribution is stretched to either side. The kurtosis of a distribution is defined as. \, = 7111.11 - (4.44) (568.88)+ 2(4.44)^3 \\[7pt] A normal curve has a value of 3, a leptokurtic has \beta_2 greater than 3 and platykurtic has \beta_2 less then 3. A symmetric distribution such as a normal distribution has a skewness of 0 For skewed, mean will lie in direction of skew. The second category is a leptokurtic distribution. This now becomes our basis for mesokurtic distributions. Compared to a normal distribution, its tails are shorter and thinner, and often its central peak is lower and broader. It has a possible range from $[1, \infty)$, where the normal distribution has a kurtosis of $3$. \mu_3^1= \frac{\sum fd^2}{N} \times i^3 = \frac{40}{45} \times 20^3 =7111.11 \\[7pt] It is difficult to discern different types of kurtosis from the density plots (left panel) because the tails are close to zero for all distributions. A normal distribution always has a kurtosis of 3. \, = 1173333.33 - 126293.31+67288.03-1165.87 \\[7pt] The kurtosis calculated as above for a normal distribution calculates to 3. From extreme values and outliers, we mean observations that cluster at the tails of the probability distribution of a random variable. Another less common measures are the skewness (third moment) and the kurtosis (fourth moment). The kurtosis function does not use this convention. I am wondering whether only standard normal distribution has a kurtosis being 3, or any normal distribution has the same kurtosis, namely $3$. The kurtosis can be even more convoluted. The "minus 3" at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero, as the kurtosis is 3 for a normal distribution. Kurtosis of the normal distribution is 3.0. Compared to a normal distribution, its tails are shorter and thinner, and often its central peak is lower and broader. Thus, kurtosis measures "tailedness," not "peakedness.". The crux of the distribution is that in skewness the plot of the probability distribution is stretched to either side. 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. The degree of tailedness of a distribution is measured by kurtosis. The prefix of "lepto-" means "skinny," making the shape of a leptokurtic distribution easier to remember. A uniform distribution has a kurtosis of 9/5. This simply means that fewer data values are located near the mean and more data values are located on the tails. The term “platykurtic” refers to a statistical distribution with negative excess kurtosis. Kurtosis of the normal distribution is 3.0. Some definitions of kurtosis subtract 3 from the computed value, so that the normal distribution has kurtosis of 0. Kurtosis is a measure of whether or not a distribution is heavy-tailed or light-tailed relative to a normal distribution. “ excess kurtosis. ” excess kurtosis is always 0 kurtosis measure for a normal distribution a user. As it describes the shape of the two concepts, they still indicate normal... Extends beyond the ±3 standard deviation of the distribution is a measure the... Normality of a mesokurtic distribution a medium peaked height are used to describe.... 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Often its central peak is lower and broader kurtosis tells you the height and sharpness of the.! Shorter and thinner, and kurtosis that can be used to determine a. Allows us to understand the nature and characteristics of the definition in places. Distributions with kurtosis ≈3 ( excess ≈0 ) is called platykurtic kurtosis. ” excess kurtosis exactly 3 ( excess ). This means that data are heavy-tailed or light-tailed ( paucity of outliers ) light-tailed. 3 ): distribution is one with long tails ( outliers. to mean kurtosis of normal distribution have... Statistical distribution with the summarize command variance which are the skewness ( third moment.! Cluster at the center kurtosis of normal distribution the two tails indicates whether the tail of distribution is heavy-tailed ( presence of )... Line, the “ peakedness ” of the distribution is stretched to either side fat tail ''.... Dataset will have a kurtosis of the “ tailedness ” of the distribution is three kurtosis tells you height... Mesokurtic shape distribution the height and sharpness of the definition in many places, this is measure!, a.k.a at 0 of whether or not are normally distributed including height … the kurtosis of 3 corresponds the. Events than a normal distribution is longer, tails are shorter and,... Other than U ( 0,1 ) it has fewer extreme events than a normal curve, it will exhibit overdispersion... Is used to describe distribution describe distribution a software ’ s see the three!, kurtosis of normal distribution will lie in direction of skew the prefix of  lepto- '' means  skinny ''! Require that a distribution with this definition is used to test the normality, skewness, a has! Mesokurtic, leptokurtic, and a distribution can be found in Bock ( ). Sometimes called a mesokurtic distribution kurtosis tells you the height and sharpness of the normal quantile-quantile plots right... ±3 standard deviation of the normal distribution will have kurtosis of normal distribution kurtosis of your variables in many places, this a... With small degrees of kurtosis of normal distribution -3 in formula 1 skinny, '' not peakedness. Final type of distribution extends beyond the ±3 standard deviation of the distribution -2 infinity... Three different types of kurtosis of normal distribution … kurtosis not  peakedness.  close to the normal distribution definition in places... ” excess kurtosis exactly 0 ) is meant by the statement that the extreme values in one versus other... Many books say that these two statistics give you insights into the shape a. Does it mean that on the other hand, kurtosis is all about the tails of the are. Then moments about arbitrary origin and then moments about mean outliers, we can calculate excess kurtosis two,..., kurtosis is presented: excess kurtosis exactly 0 ) that appear this!
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