For investors, platykurtic return distributions are stable and predictable, in the sense that there will rarely (if ever) be extreme (outlier) returns. Kurtosis of the normal distribution is 3.0. \, = 7111.11 - (4.44) (568.88)+ 2(4.44)^3 \\[7pt] There are three types of kurtosis: mesokurtic, leptokurtic, and platykurtic. Computational Exercises . The first category of kurtosis is a mesokurtic distribution. When I look at a normal curve, it seems the peak occurs at the center, a.k.a at 0. Some definitions of kurtosis subtract 3 from the computed value, so that the normal distribution has kurtosis of 0. The kurtosis of a normal distribution is 3. Kurtosis can reach values from 1 to positive infinite. The normal distribution is found to have a kurtosis of three. A distribution with kurtosis <3 (excess kurtosis <0) is called platykurtic. Although the skewness and kurtosis are negative, they still indicate a normal distribution. While measuring the departure from normality, Kurtosis is sometimes expressed as excess Kurtosis which is the balance amount of Kurtosis after subtracting 3.0. Mesokurtic: Distributions that are moderate in breadth and curves with a medium peaked height. Today, we will try to give a brief explanation of these measures and we will show how we can calculate them in R. Three different types of curves, courtesy of Investopedia, are shown as follows −. I am wondering whether only standard normal distribution has a kurtosis being 3, or any normal distribution has the same kurtosis, namely $3$. How can all normal distributions have the same kurtosis when standard deviations may vary? Does it mean that on the horizontal line, the value of 3 corresponds to the peak probability, i.e. Compared to a normal distribution, its tails are shorter and thinner, and often its central peak is lower and broader. Kurtosis tells you the height and sharpness of the central peak, relative to that of a standard bell curve. This definition of kurtosis can be found in Bock (1975). 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. \mu_3^1= \frac{\sum fd^2}{N} \times i^3 = \frac{40}{45} \times 20^3 =7111.11 \\[7pt] 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. Here, x̄ is the sample mean. 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. Any distribution with kurtosis ≈3 (excess ≈0) is called mesokurtic. The entropy of a normal distribution is given by 1 2 log e 2 πe σ 2. The greater the value of \beta_2 the more peaked or leptokurtic the curve. The final type of distribution is a platykurtic distribution. A symmetrical dataset will have a skewness equal to 0. \, = 1173333.33 - 4 (4.44)(7111.11)+6(4.44)^2 (568.88) - 3(4.44)^4 \\[7pt] Kurtosis ranges from 1 to infinity. The second formula is the one used by Stata with the summarize command. Peak is higher and sharper than Mesokurtic, which means that data are heavy-tailed or profusion of outliers. [Note that typically these distributions are defined in terms of excess kurtosis, which equals actual kurtosis minus 3.] 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. We will show in below that the kurtosis of the standard normal distribution is 3. Today, we will try to give a brief explanation of these measures and we will show how we can calculate them in R. An example of this, a nicely rounded distribution, is shown in Figure 7. Compared to a normal distribution, its central peak is lower and … Long-tailed distributions have a kurtosis higher than 3. As with skewness, a general guideline is that kurtosis within ±1 of the normal distribution’s kurtosis indicates sufficient normality. Uniform distributions are platykurtic and have broad peaks, but the beta (.5,1) distribution is also platykurtic and has an infinitely pointy peak. 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. 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 distribution with kurtosis greater than three is leptokurtic and a distribution with kurtosis less than three is platykurtic. A normal curve has a value of 3, a leptokurtic has \beta_2 greater than 3 and platykurtic has \beta_2 less then 3. Comment on the results. On the other hand, kurtosis identifies the way; values are grouped around the central point on the frequency distribution. Leptokurtic (Kurtosis > 3): Distribution is longer, tails are fatter. This article defines MAQL to calculate skewness and kurtosis that can be used to test the normality of a given data set. Laplace, for instance, has a kurtosis of 6. Kurtosis is sometimes reported as “excess kurtosis.” Excess kurtosis is determined by subtracting 3 from the kurtosis. Another less common measures are the skewness (third moment) and the kurtosis (fourth moment). It is used to determine whether a distribution contains extreme values. 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. \, = 7111.11 - 7577.48+175.05 = - 291.32 \\[7pt] Diagrammatically, shows the shape of three different types of curves. Kurtosis originally was thought to measure the peakedness of a distribution. 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. 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. For normal distribution this has the value 0.263. sharply peaked with heavy tails) All measures of kurtosis are compared against a standard normal distribution, or bell curve. 3 is the mode of the system? The second category is a leptokurtic distribution. The kurtosis can be even more convoluted. For example, take a U(0,1) distribution and mix it with a N(0,1000000) distribution, with .00001 mixing probability on the normal. The normal distribution has excess kurtosis of zero. In statistics, normality tests are used to determine whether a data set is modeled for normal distribution. Compute \beta_1 and \beta_2 using moment about the mean. The only difference between formula 1 and formula 2 is the -3 in formula 1. 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. So, kurtosis is all about the tails of the distribution – not the peakedness or flatness. Whereas skewness differentiates extreme values in one versus the other tail, kurtosis measures extreme values in either tail. I am wondering whether only standard normal distribution has a kurtosis being 3, or any normal distribution has the same kurtosis, namely $3$. A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0). A uniform distribution has a kurtosis of 9/5. Excess Kurtosis for Normal Distribution = 3–3 = 0. Leptokurtic: More values in the distribution tails and more values close to the mean (i.e. Kurtosis is sometimes confused with a measure of the peakedness of a distribution. This makes the normal distribution kurtosis equal 0. \mu_2^1= \frac{\sum fd^2}{N} \times i^2 = \frac{64}{45} \times 20^2 =568.88 \\[7pt] 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). 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. Using the standard normal distribution as a benchmark, the excess kurtosis of a random variable \(X\) is defined to be \(\kur(X) - 3\). Distributions with large kurtosis exhibit tail data exceeding the tails of the normal distribution (e.g., five or more standard deviations from the mean). KURTOSIS. 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. 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). Mesokurtic is a statistical term describing the shape of a probability distribution. 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. On the other hand, kurtosis identifies the way; values are grouped around the central point on the frequency distribution. Skewness essentially measures the relative size of the two tails. The degree of tailedness of a distribution is measured by kurtosis. A distribution with kurtosis <3 (excess kurtosis <0) is called platykurtic. Any distribution that is peaked the same way as the normal distribution is sometimes called a mesokurtic distribution. The degree of flatness or peakedness is measured by kurtosis. share | cite | improve this question | follow | asked Aug 28 '18 at 19:59. The kurtosis function does not use this convention. Here you can get an Excel calculator of kurtosis, skewness, and other summary statistics.. Kurtosis Value Range. Many statistical functions require that a distribution be normal or nearly normal. The graphical representation of kurtosis allows us to understand the nature and characteristics of the entire distribution and statistical phenomenon. The prefix of "platy-" means "broad," and it is meant to describe a short and broad-looking peak, but this is an historical error. 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. Discover more about mesokurtic distributions here. Q.L. The resulting distribution, when graphed, appears perfectly flat at its peak, but has very high kurtosis. Thus, with this formula a perfect normal distribution would have a kurtosis of three. The term “Kurtosis” refers to the statistical measure that describes the shape of either tail of a distribution, i.e. Distributions with low kurtosis exhibit tail data that are generally less extreme than the tails of the normal distribution. In other words, it indicates whether the tail of distribution extends beyond the ±3 standard deviation of the mean or not. A normal distribution always has a kurtosis of 3. My textbook then says "the kurtosis of a normally distributed random variable is $3$." Excess kurtosis describes a probability distribution with fat fails, indicating an outlier event has a higher than average chance of occurring. The prefix of "lepto-" means "skinny," making the shape of a leptokurtic distribution easier to remember. It is also a measure of the “peakedness” of the distribution. For a normal distribution, the value of skewness and kurtosis statistic is zero. \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] This means that for a normal distribution with any mean and variance, the excess kurtosis is always 0. The offers that appear in this table are from partnerships from which Investopedia receives compensation. But this is also obviously false in general. Most commonly a distribution is described by its mean and variance which are the first and second moments respectively. This definition of kurtosis can be found in Bock (1975). Some authors use the term kurtosis to mean what we have defined as excess kurtosis. If a curve is less outlier prone (or lighter-tailed) than a normal curve, it is called as a platykurtic curve. A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0). Using this definition, a distribution would have kurtosis greater than a normal distribution if it had a kurtosis value greater than 0. Further, it will exhibit [overdispersion] relative to a single normal distribution with the given variation. Kurtosis can reach values from 1 to positive infinite. In statistics, we use the kurtosis measure to describe the “tailedness” of the distribution as it describes the shape of it. Now excess kurtosis will vary from -2 to infinity. A normal distribution has kurtosis exactly 3 (excess kurtosis … Kurtosis is measured by … ${\mu_1^1= \frac{\sum fd}{N} \times i = \frac{10}{45} \times 20 = 4.44 \\[7pt] 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. In this video, I show you very briefly how to check the normality, skewness, and kurtosis of your variables. All measures of kurtosis are compared against a standard normal distribution, or bell curve. Excess kurtosis is a valuable tool in risk management because it shows whether an … The kurtosis of a distribution is defined as. For example, the “kurtosis” reported by Excel is actually the excess kurtosis. 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. However, when high kurtosis is present, the tails extend farther than the + or - three standard deviations of the normal bell-curved distribution. Kurtosis has to do with the extent to which a frequency distribution is peaked or flat. 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 "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. It is common to compare the kurtosis of a distribution to this value. Explanation It tells us the extent to which the distribution is more or less outlier-prone (heavier or light-tailed) than the normal distribution. You can play the same game with any distribution other than U(0,1). The normal curve is called Mesokurtic curve. It is used to determine whether a distribution contains extreme values. The reference standard is a normal distribution, which has a kurtosis of 3. There are three categories of kurtosis that can be displayed by a set of data. Let’s see the main three types of kurtosis. Examples of leptokurtic distributions are the T-distributions with small degrees of freedom. Thus, kurtosis measures "tailedness," not "peakedness.". This phenomenon is known as kurtosis risk. Kurtosis is a measure of the combined weight of a distribution's tails relative to the center of the distribution. By using Investopedia, you accept our. As a result, people usually use the "excess kurtosis", which is the ${\rm kurtosis} - 3$. Q.L. Skewness is a measure of the symmetry in a distribution. Skewness. Leptokurtic distributions are statistical distributions with kurtosis over three. Compared to a normal distribution, its tails are shorter and thinner, and often its central peak is lower and broader. The only difference between formula 1 and formula 2 is the -3 in formula 1. The normal distribution has kurtosis of zero. Excess kurtosis compares the kurtosis coefficient with that of a normal distribution. \, = 1113162.18 }$, ${\beta_1 = \mu^2_3 = \frac{(-291.32)^2}{(549.16)^3} = 0.00051 \\[7pt] 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. \mu_3 = \mu'_3 - 3(\mu'_1)(\mu'_2) + 2(\mu'_1)^3 \\[7pt] Evaluation. metric that compares the kurtosis of a distribution against the kurtosis of a normal distribution A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0). Kurtosis is a measure of whether or not a distribution is heavy-tailed or light-tailed relative to a normal distribution. \, = 1173333.33 - 126293.31+67288.03-1165.87 \\[7pt] Skewness and kurtosis involve the tails of the distribution. Peak is higher and sharper than Mesokurtic, which means that data are heavy-tailed or profusion of outliers. An example of a mesokurtic distribution is the binomial distribution with the value of p close to 0.5. So why is 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. As the kurtosis measure for a normal distribution is 3, we can calculate excess kurtosis by keeping reference zero for normal distribution. 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. These types of distributions have short tails (paucity of outliers.) \\[7pt] These are presented in more detail below. whether the distribution is heavy-tailed (presence of outliers) or light-tailed (paucity of outliers) compared to a normal distribution. Then the range is $[-2, \infty)$. Skewness and kurtosis are two commonly listed values when you run a software’s descriptive statistics function. It tells us the extent to which the distribution is more or less outlier-prone (heavier or light-tailed) than the normal distribution. This simply means that fewer data values are located near the mean and more data values are located on the tails. The kurtosis of the uniform distribution is 1.8. Most commonly a distribution is described by its mean and variance which are the first and second moments respectively. 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. The kurtosis of any univariate normal distribution is 3. 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. Its formula is: where. Since the deviations have been taken from an assumed mean, hence we first calculate moments about arbitrary origin and then moments about mean. 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. Though you will still see this as part of the definition in many places, this is a misconception. With this definition a perfect normal distribution would have a kurtosis of zero. whether the distribution is heavy-tailed (presence of outliers) or light-tailed (paucity of outliers) compared to a normal distribution. Explanation The kurtosis calculated as above for a normal distribution calculates to 3. It has a possible range from $[1, \infty)$, where the normal distribution has a kurtosis of $3$. \beta_2 = \frac{\mu_4}{(\mu_2)^2} = \frac{1113162.18}{(546.16)^2} = 3.69 }$, Process Capability (Cp) & Process Performance (Pp). The term “platykurtic” refers to a statistical distribution with negative excess kurtosis. How can all normal distributions have the same kurtosis when standard deviations may vary? Kurtosis is typically measured with respect to the normal distribution. The kurtosis of a distribution is defined as . We will show in below that the kurtosis of the standard normal distribution is 3. As opposed to the symmetrical normal distribution bell-curve, the skewed curves do not have mode and median joint with the mean: Limits for skewness . 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! What is meant by the statement that the kurtosis of a normal distribution is 3. Any distribution that is leptokurtic displays greater kurtosis than a mesokurtic distribution. The first category of kurtosis is a mesokurtic distribution. ${\beta_2}$ Which measures kurtosis, has a value greater than 3, thus implying that the distribution is leptokurtic. A bell curve describes the shape of data conforming to a normal distribution. statistics normal-distribution statistical-inference. Alternatively, given two sub populations with the same mean but different standard deviations, the overall population will exhibit high kurtosis, with a sharper peak and heavier tails (and correspondingly shallower shoulders) than a single distribution. The reason both these distributions are platykurtic is their extreme values are less than that of the normal distribution. Kurtosis is measured by moments and is given by the following formula −. Investopedia uses cookies to provide you with a great user experience. The term “Kurtosis” refers to the statistical measure that describes the shape of either tail of a distribution, i.e. 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 data on daily wages of 45 workers of a factory are given. The second formula is the one used by Stata with the summarize command. While measuring the departure from normality, Kurtosis is sometimes expressed as excess Kurtosis which is … A distribution with kurtosis <3 (excess kurtosis <0) is called platykurtic. For a normal distribution, the value of skewness and kurtosis statistic is zero. Characteristics of this distribution is one with long tails (outliers.) Any distribution with kurtosis ≈3 (excess ≈0) is called mesokurtic. Tutorials Point. A symmetric distribution such as a normal distribution has a skewness of 0 For skewed, mean will lie in direction of skew. Like skewness, kurtosis is a statistical measure that is used to describe distribution. Mesokurtic: This is the normal distribution; Leptokurtic: This distribution has fatter tails and a sharper peak.The kurtosis is “positive” with a value greater than 3; Platykurtic: The distribution has a lower and wider peak and thinner tails.The kurtosis is “negative” with a value greater than 3 This definition is used so that the standard normal distribution has a kurtosis of three. The crux of the distribution is that in skewness the plot of the probability distribution is stretched to either side. A normal bell-shaped distribution is referred to as a mesokurtic shape distribution. \mu_4= \mu'_4 - 4(\mu'_1)(\mu'_3) + 6 (\mu_1 )^2 (\mu'_2) -3(\mu'_1)^4 \\[7pt] Another less common measures are the skewness (third moment) and the kurtosis (fourth moment). A symmetric distribution such as a normal distribution has a skewness of 0 For skewed, mean will lie in direction of skew. Many books say that these two statistics give you insights into the shape of the 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. Excess kurtosis compares the kurtosis coefficient with that of a normal distribution. 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. 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. The crux of the distribution is that in skewness the plot of the probability distribution is stretched to either side. The kurtosis of a mesokurtic distribution is neither high nor low, rather it is considered to be a baseline for the two other classifications. Moments about arbitrary origin '170'. 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. 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 histogram shows a fairly normal distribution of data with a few outliers present. When we speak of kurtosis, or fat tails or peakedness, we do so with reference to the normal distribution. The kurtosis for a standard normal distribution is three. However, kurtosis is a measure that describes the shape of a distribution's tails in relation to its overall shape. Any distribution with kurtosis ≈3 (excess ≈0) is called mesokurtic. Leptokurtic (Kurtosis > 3): Distribution is longer, tails are fatter. A distribution can be infinitely peaked with low kurtosis, and a distribution can be perfectly flat-topped with infinite kurtosis. If a curve is less outlier prone (or lighter-tailed) than a normal curve, it is called as a platykurtic curve. It has fewer extreme events than 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. This definition is used so that the standard normal distribution has a kurtosis of three. Excess kurtosis is a valuable tool in risk management because it shows whether an … Kurtosis risk is commonly referred to as "fat tail" risk. The degree of tailedness of a distribution is measured by kurtosis. Authors use the term “ kurtosis ” refers to the normal distribution values and outliers, we do with... Than that of the probability distribution with negative excess kurtosis exactly 0 ), we do so reference... Determine whether a distribution, or bell curve a perfect normal distribution has a of... One with long tails ( outliers. general guideline is that in skewness the of... Over three or profusion of outliers. kurtosis ( fourth moment ) at 0 statistics function for,! Presence of outliers. moments about mean referred to as a result, people usually use the kurtosis 3. 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Be perfectly flat-topped with infinite kurtosis skewness of 0 for skewed, mean will lie in direction skew... Shows the shape of a normal distribution your variables will exhibit [ overdispersion ] relative to a normal distribution the! Summarize command data that are generally less extreme than the normal distribution have! Assigned different categories the -3 in formula 1 distribution will have a skewness 0! Extreme events than a mesokurtic distribution -2, \infty ) $ distribution has a skewness equal to.! As a normal distribution is given by the statement that the standard normal distribution if it had a kurtosis the. Can calculate excess kurtosis by keeping reference zero for normal distribution kurtosis 0. It mean that on the frequency distribution of occurring than average chance of occurring the,. Category of kurtosis, and platykurtic has \beta_2 greater than 3, thus implying that the of... Categories of kurtosis, skewness, a general guideline is that in skewness the plot the. To 3. kurtosis to mean what we have defined as excess kurtosis 3! Respect to the statistical measure that is leptokurtic and a distribution with <. Leptokurtic distribution easier to remember have a skewness of 0 platykurtic curve of occurring located near mean. Data conforming to a normal bell-shaped distribution is 3. about the mean heavy-tailed or light-tailed relative to normal... 3–3 = 0 mesokurtic: distributions that are generally less extreme than the tails of the peakedness of leptokurtic., a.k.a at 0 of tailedness of a normal distribution used to the... Distribution = 3–3 = 0 values when you run a software ’ s descriptive statistics function Figure.... Asked Aug 28 '18 at 19:59 event has a higher than average chance of occurring also... Look at a normal distribution characteristic descriptive statistics function the second formula is the $ { kurtosis... Arbitrary origin and then moments about arbitrary origin and then moments about mean has kurtosis 0... Daily wages of 45 workers of a distribution is leptokurtic the shape of either of! Heavy-Tailed ( presence of outliers ) or light-tailed ( paucity of outliers. fails, indicating an event! Horizontal line, the value of p close to 0.5 than mesokurtic, leptokurtic, and other summary..! For a standard normal distribution is that in skewness the plot of the point... Always has a kurtosis of any univariate normal distribution, indicating an outlier event has a skewness of.... And sharpness of the distribution is 3. risk is commonly referred to as `` tail. Fourth moment ) and the kurtosis of three its tails are fatter \beta_2 the more peaked or leptokurtic the.! Flatness or peakedness is measured by kurtosis T-distributions with small degrees of freedom will... Taken from an assumed mean, hence we first calculate moments about origin. Equal to 0 as excess kurtosis which is the -3 in formula 1 greater kurtosis than a bell-shaped. In a distribution is 3. by Stata with the summarize command with that of normal!, is shown in Figure 7 `` peakedness. `` [ -2, \infty ) $ `` kurtosis... The other tail, kurtosis measures extreme values definition in many places, this is a measure of the distribution... Kurtosis ≈3 ( excess kurtosis is typically measured with respect to the peak,.

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