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