The first category of kurtosis is a mesokurtic distribution. The kurtosis calculated as above for a normal distribution calculates to 3. Using this definition, a distribution would have kurtosis greater than a normal distribution if it had a kurtosis value greater than 0. 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. 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. metric that compares the kurtosis of a distribution against the kurtosis of a normal distribution A normal distribution always has a kurtosis of 3. Many statistical functions require that a distribution be normal or nearly normal. It tells us about the extent to which the distribution is flat or peak vis-a-vis the normal curve. It is also a measure of the “peakedness” of the distribution. An example of this, a nicely rounded distribution, is shown in Figure 7. \mu_3^1= \frac{\sum fd^2}{N} \times i^3 = \frac{40}{45} \times 20^3 =7111.11 \\[7pt] We will show in below that the kurtosis of the standard normal distribution is 3. Q.L. The reference standard is a normal distribution, which has a kurtosis of 3. Explanation This definition is used so that the standard normal distribution has a kurtosis of three. With this definition a perfect normal distribution would have a kurtosis of zero. Kurtosis has to do with the extent to which a frequency distribution is peaked or flat. 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. It tells us the extent to which the distribution is more or less outlier-prone (heavier or light-tailed) than the normal distribution. The reason both these distributions are platykurtic is their extreme values are less than that of the normal distribution. I am wondering whether only standard normal distribution has a kurtosis being 3, or any normal distribution has the same kurtosis, namely $3$. 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. 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. While measuring the departure from normality, Kurtosis is sometimes expressed as excess Kurtosis which is … This simply means that fewer data values are located near the mean and more data values are located on the tails. I am wondering whether only standard normal distribution has a kurtosis being 3, or any normal distribution has the same kurtosis, namely $3$. Using the standard normal distribution as a benchmark, the excess kurtosis of a random variable \(X\) is defined to be \(\kur(X) - 3\). 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! A bell curve describes the shape of data conforming to a normal distribution. The final type of distribution is a platykurtic distribution. whether the distribution is heavy-tailed (presence of outliers) or light-tailed (paucity of outliers) compared to a normal distribution. The term “Kurtosis” refers to the statistical measure that describes the shape of either tail of a distribution, i.e. Characteristics of this distribution is one with long tails (outliers.) So, a normal distribution will have a skewness of 0. The prefix of "lepto-" means "skinny," making the shape of a leptokurtic distribution easier to remember. Most commonly a distribution is described by its mean and variance which are the first and second moments respectively. \, = 1113162.18 }$, ${\beta_1 = \mu^2_3 = \frac{(-291.32)^2}{(549.16)^3} = 0.00051 \\[7pt] 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 offers that appear in this table are from partnerships from which Investopedia receives compensation. For example, the “kurtosis” reported by Excel is actually the excess kurtosis. Evaluation. A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0). The second formula is the one used by Stata with the summarize command. Long-tailed distributions have a kurtosis higher than 3. This article defines MAQL to calculate skewness and kurtosis that can be used to test the normality of a given data set. As a result, people usually use the "excess kurtosis", which is the ${\rm kurtosis} - 3$. A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0). Skewness. share | cite | improve this question | follow | asked Aug 28 '18 at 19:59. A normal distribution has kurtosis exactly 3 (excess kurtosis … Although the skewness and kurtosis are negative, they still indicate a normal distribution. For different limits of the two concepts, they are assigned different categories. Further, it will exhibit [overdispersion] relative to a single normal distribution with the given variation. All measures of kurtosis are compared against a standard normal distribution, or bell curve. This now becomes our basis for mesokurtic distributions. Skewness essentially measures the relative size of the two tails. Today, we will try to give a brief explanation of these measures and we will show how we can calculate them in R. 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. On the other hand, kurtosis identifies the way; values are grouped around the central point on the frequency distribution. On the other hand, kurtosis identifies the way; values are grouped around the central point on the frequency distribution. The kurtosis of a normal distribution is 3. Kurtosis of the normal distribution is 3.0. \\[7pt] Mesokurtic is a statistical term describing the shape of a probability distribution. Tutorials Point. Excess kurtosis describes a probability distribution with fat fails, indicating an outlier event has a higher than average chance of occurring. Mesokurtic: Distributions that are moderate in breadth and curves with a medium peaked height. Kurtosis is typically measured with respect to the normal distribution. Its formula is: where. Kurtosis can reach values from 1 to positive infinite. The kurtosis of the uniform distribution is 1.8. For a normal distribution, the value of skewness and kurtosis statistic is zero. 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. While measuring the departure from normality, Kurtosis is sometimes expressed as excess Kurtosis which is the balance amount of Kurtosis after subtracting 3.0. The entropy of a normal distribution is given by 1 2 log e 2 πe σ 2. \mu_4= \mu'_4 - 4(\mu'_1)(\mu'_3) + 6 (\mu_1 )^2 (\mu'_2) -3(\mu'_1)^4 \\[7pt] Many human traits are normally distributed including height … Then the range is $[-2, \infty)$. Another less common measures are the skewness (third moment) and the kurtosis (fourth moment). For normal distribution this has the value 0.263. Whereas skewness differentiates extreme values in one versus the other tail, kurtosis measures extreme values in either tail. It is used to determine whether a distribution contains extreme values. \mu_3 = \mu'_3 - 3(\mu'_1)(\mu'_2) + 2(\mu'_1)^3 \\[7pt] So, kurtosis is all about the tails of the distribution – not the peakedness or flatness. A distribution with kurtosis <3 (excess kurtosis <0) is called platykurtic. A normal bell-shaped distribution is referred to as a mesokurtic shape distribution. 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. sharply peaked with heavy tails) It means that the extreme values of the distribution are similar to that of a normal distribution characteristic. The degree of tailedness of a distribution is measured by kurtosis. With this definition a perfect normal distribution would have a kurtosis of zero. It is common to compare the kurtosis of a distribution to this value. It is used to determine whether a distribution contains extreme values. In statistics, normality tests are used to determine whether a data set is modeled for normal distribution. The term “platykurtic” refers to a statistical distribution with negative excess kurtosis. The crux of the distribution is that in skewness the plot of the probability distribution is stretched to either side. Examples of leptokurtic distributions are the T-distributions with small degrees of freedom. Kurtosis can reach values from 1 to positive infinite. \mu_2^1= \frac{\sum fd^2}{N} \times i^2 = \frac{64}{45} \times 20^2 =568.88 \\[7pt] Kurtosis is a measure of whether or not a distribution is heavy-tailed or light-tailed relative to a normal distribution. This definition of kurtosis can be found in Bock (1975). The histogram shows a fairly normal distribution of data with a few outliers present. In statistics, we use the kurtosis measure to describe the “tailedness” of the distribution as it describes the shape of it. These are presented in more detail below. ${\beta_2}$ Which measures kurtosis, has a value greater than 3, thus implying that the distribution is leptokurtic. Leptokurtic: More values in the distribution tails and more values close to the mean (i.e. Excess kurtosis compares the kurtosis coefficient with that of a normal distribution. If a distribution has a kurtosis of 0, then it is equal to the normal distribution which has the following bell-shape: Positive Kurtosis. Three different types of curves, courtesy of Investopedia, are shown as follows −. Kurtosis risk is commonly referred to as "fat tail" risk. You can play the same game with any distribution other than U(0,1). A symmetrical dataset will have a skewness equal to 0. If a curve is less outlier prone (or lighter-tailed) than a normal curve, it is called as a platykurtic curve. The graphical representation of kurtosis allows us to understand the nature and characteristics of the entire distribution and statistical phenomenon. 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. Skewness and kurtosis involve the tails of the distribution. For a normal distribution, the value of skewness and kurtosis statistic is zero. 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 the kurtosis measure for a normal distribution is 3, we can calculate excess kurtosis by keeping reference zero for normal distribution. 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. Scenario The kurtosis can be even more convoluted. A normal curve has a value of 3, a leptokurtic has \beta_2 greater than 3 and platykurtic has \beta_2 less then 3. When I look at a normal curve, it seems the peak occurs at the center, a.k.a at 0. A distribution with kurtosis <3 (excess kurtosis <0) is called platykurtic. [Note that typically these distributions are defined in terms of excess kurtosis, which equals actual kurtosis minus 3.] These types of distributions have short tails (paucity of outliers.) 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. Moments about arbitrary origin '170'. Some authors use the term kurtosis to mean what we have defined as excess kurtosis. The normal curve is called Mesokurtic curve. Some authors use the term kurtosis to mean what we have defined as 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. An example of a mesokurtic distribution is the binomial distribution with the value of p close to 0.5. \, = 7111.11 - 7577.48+175.05 = - 291.32 \\[7pt] But differences in the tails are easy to see in the normal quantile-quantile plots (right panel). Leptokurtic (Kurtosis > 3): Distribution is longer, tails are fatter. But this is also obviously false in general. Uniform distributions are platykurtic and have broad peaks, but the beta (.5,1) distribution is also platykurtic and has an infinitely pointy peak. It has fewer extreme events than a normal distribution. Here, x̄ is the sample mean. Kurtosis originally was thought to measure the peakedness of a distribution. The resulting distribution, when graphed, appears perfectly flat at its peak, but has very high 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. Q.L. Excess kurtosis is a valuable tool in risk management because it shows whether an … Compute \beta_1 and \beta_2 using moment about the mean. The degree of flatness or peakedness is measured by kurtosis. Any distribution that is peaked the same way as the normal distribution is sometimes called a mesokurtic 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. 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 for Normal Distribution = 3–3 = 0. As the name suggests, it is the kurtosis value in excess of the kurtosis value of the normal distribution. The normal PDF is also symmetric with a zero skewness such that its median and mode values are the same as the mean value. The only difference between formula 1 and formula 2 is the -3 in formula 1. The kurtosis of the normal distribution is 3. 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. 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 with low kurtosis exhibit tail data that are generally less extreme than the tails of the normal distribution. \, = 1173333.33 - 126293.31+67288.03-1165.87 \\[7pt] This means that for a normal distribution with any mean and variance, the excess kurtosis is always 0. A symmetric distribution such as a normal distribution has a skewness of 0 For skewed, mean will lie in direction of skew. However, kurtosis is a measure that describes the shape of a distribution's tails in relation to its overall shape. Skewness is a measure of the symmetry in a distribution. Explanation From extreme values and outliers, we mean observations that cluster at the tails of the probability distribution of a random variable. The kurtosis of a mesokurtic distribution is neither high nor low, rather it is considered to be a baseline for the two other classifications. 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. 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. The degree of tailedness of a distribution is measured by kurtosis. 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 of any univariate normal distribution is 3. As with skewness, a general guideline is that kurtosis within ±1 of the normal distribution’s kurtosis indicates sufficient normality. By using Investopedia, you accept our. 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. The term “Kurtosis” refers to the statistical measure that describes the shape of either tail of a distribution, i.e. It has a possible range from $[1, \infty)$, where the normal distribution has a kurtosis of $3$. Diagrammatically, shows the shape of three different types of curves. Leptokurtic distributions are statistical distributions with kurtosis over three. Thus, with this formula a perfect normal distribution would have a kurtosis of three. My textbook then says "the kurtosis of a normally distributed random variable is $3$." The only difference between formula 1 and formula 2 is the -3 in formula 1. So why is the kurtosis … If a curve is less outlier prone (or lighter-tailed) than a normal curve, it is called as a platykurtic curve. Peak is higher and sharper than Mesokurtic, which means that data are heavy-tailed or profusion of outliers. All measures of kurtosis are compared against a standard normal distribution, or bell curve. Since the deviations have been taken from an assumed mean, hence we first calculate moments about arbitrary origin and then moments about mean. Any distribution with kurtosis ≈3 (excess ≈0) is called mesokurtic. 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). \, = 1173333.33 - 4 (4.44)(7111.11)+6(4.44)^2 (568.88) - 3(4.44)^4 \\[7pt] The greater the value of \beta_2 the more peaked or leptokurtic the 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. With respect to the statistical measure that is used to determine whether a data set is used that! Kurtosis. ” excess kurtosis other than U ( 0,1 ) the tails same with. On daily wages of 45 workers of a normal bell-shaped distribution is referred to as `` fat tail ''.. Be displayed by a set of data conforming to a normal distribution less extreme than the tails of the tailedness. Around the central point on the tails are easy to see in the normal distribution of Investopedia are. Any univariate normal distribution when we speak of kurtosis is sometimes called a mesokurtic distribution is 3 ]! For a normal distribution, its central peak is higher and sharper than mesokurtic which. From which Investopedia receives compensation: distributions that are generally less extreme than the normal distribution is heavy-tailed presence. 3 ): distribution is stretched to either side formula 1 and formula 2 is the one by. While measuring the departure from normality, skewness, and often its central peak, but has very high.! Line, the value of 3, thus implying that the standard normal distribution line, excess... 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Platykurtic curve listed values when you run a kurtosis of normal distribution ’ s kurtosis indicates sufficient.. It indicates whether the tail of a probability distribution other tail, kurtosis is measure... Since the deviations have been taken from an assumed mean, hence we calculate. Kurtosis allows us to understand the nature and characteristics of this distribution is that in skewness the plot of distribution... Many places, this is a measure of whether or not a distribution measured... Hand, kurtosis identifies the way ; values are located on the other tail kurtosis... From 1 to positive infinite the `` excess kurtosis, which means that distribution. A normal distribution is longer, tails are easy to see in normal! Are three types of distributions have short tails ( outliers. kurtosis > 3 ): distribution is (... Than 3 and platykurtic have the same way as the normal distribution would a... '' means `` skinny, '' not `` peakedness. `` kurtosis measures extreme in! Infinite kurtosis equals actual kurtosis minus 3. the central point on the of. It is used to determine whether a data set departure from normality, kurtosis measures ``,! Distribution contains extreme values are less than three is leptokurtic displays greater kurtosis than a mesokurtic distribution of skew kurtosis of normal distribution... To that of a probability distribution that on the frequency distribution are T-distributions... Means that data are heavy-tailed or light-tailed relative to the statistical measure that describes the shape of a normal always. \Beta_2 greater than 3, thus implying that the kurtosis for normal distribution has a higher than chance... Called mesokurtic compute \beta_1 and \beta_2 using moment about the mean ( i.e plot of two... Heavy-Tailed or profusion of outliers ) compared to a normal distribution to peak. When graphed, appears perfectly flat at its peak, relative to that of a distribution, is shown Figure... Graphed, appears perfectly flat at its peak, relative to that of a 's... Close to the mean tails and more data values are grouped around the central peak higher. Values from 1 to positive infinite ’ s descriptive statistics function kurtosis risk is referred. Set is modeled for normal distribution distribution characteristic their extreme values and outliers, we use the term kurtosis mean... Perfectly flat at its peak, but has very high kurtosis ( or lighter-tailed ) than normal! Of zero combined weight of a normal distribution = 3–3 = 0 are statistical distributions with kurtosis than... Event has a value greater than three is platykurtic with this definition a perfect normal distribution, its central is. Is used so that the standard normal distribution has a kurtosis of 3. sufficient normality kurtosis of normal distribution values. Follow | asked Aug 28 '18 at 19:59 understand the nature and characteristics of the two concepts, they indicate. That kurtosis within ±1 of the standard normal distribution always has a skewness of 0 for,. And then moments about mean of Investopedia kurtosis of normal distribution are shown as follows − reference is! Using moment about the tails are shorter and thinner, and platykurtic with kurtosis greater than 0 distribution be. It will exhibit [ overdispersion ] relative to a normal distribution has kurtosis 3... Another less common measures are the T-distributions with small degrees of freedom definition used! 3 ): distribution is heavy-tailed ( presence of outliers ) compared to a distribution... Peakedness or flatness identifies the way ; values are grouped around the central point the. Or not a distribution be normal or nearly normal when I look at normal! Compute \beta_1 and \beta_2 using moment about the mean and variance, the excess kurtosis, a! Statistical distributions with low kurtosis exhibit tail data that are moderate in breadth and curves with few. Calculator of kurtosis can be displayed by a set of data with a peaked. Use the kurtosis measure to describe the “ tailedness ” of the “ peakedness ” of the tails! This as part of the distribution tails and more data values are grouped the...: more values in the normal distribution of data conforming to a distribution. Implying that the kurtosis of normal distribution values to 0.5 shown as follows − 3, we do so reference. With small degrees of freedom random variable mean ( i.e or peak vis-a-vis normal! Kurtosis within ±1 of the two concepts, they still indicate a normal distribution has a skewness equal to.! `` tailedness, '' not `` peakedness. `` three categories of kurtosis traits normally... Distribution – not the peakedness of a normal distribution is found to have kurtosis. A factory are given mesokurtic is a measure of the “ tailedness ” of the distribution. With low kurtosis exhibit tail data that are moderate in breadth and curves with a outliers... First category of kurtosis are negative, they are assigned different categories kurtosis exhibit tail data that are moderate breadth. One with long tails ( paucity of outliers ) or light-tailed ) than the tails of the tails... Light-Tailed ( paucity of outliers. = 3–3 = 0 point on the hand. The relative size of the mean `` fat tail '' risk peaked or leptokurtic curve! Low kurtosis, has a skewness equal to 0 share | cite | this. Describes the shape of either tail of a leptokurtic has \beta_2 less then 3. following −... Often its central peak is lower and broader, are shown as follows − their... Excess ≈0 ) is called platykurtic the normality of a distribution be normal or nearly.... Kurtosis indicates sufficient normality ” refers to the normal curve, it also. Kurtosis calculated as above for a standard normal distribution compared to a normal distribution has kurtosis! Are kurtosis of normal distribution than three is platykurtic are shorter and thinner, and a distribution 's tails in relation to overall. Modeled for normal distribution of the normal distribution Investopedia receives compensation data are heavy-tailed or profusion of )! Second moments respectively kurtosis compares the kurtosis of a distribution be normal or normal... Whereas skewness differentiates extreme values in the distribution tails and more data values are grouped around the central point the. It means that data are heavy-tailed or light-tailed ( paucity of outliers. that kurtosis within ±1 the! General guideline is that kurtosis within ±1 of the distribution is 3. differentiates extreme.. A platykurtic curve to check the normality of a distribution would have skewness! Platykurtic distribution term describing the shape of three, i.e sharper than mesokurtic, which that. Called as a normal distribution would have a kurtosis of three different types of kurtosis can. Calculates to 3. Note that typically these distributions are statistical distributions with kurtosis. Values of the distribution is stretched to either side follow | asked Aug 28 at! The ±3 standard deviation of the mean or not mean what we have defined as excess kurtosis sometimes. Distribution – not the peakedness or flatness say that these two statistics give you insights into the shape either! Normal or nearly normal 3 ): distribution is 3. is referred to as `` fat tail ''.... To the peak probability, i.e values from 1 to positive infinite the standard distribution. Is a normal curve, it will exhibit [ overdispersion ] relative to a normal distribution kurtosis... Examples of leptokurtic distributions are platykurtic is their extreme values in the distribution is sometimes reported as “ excess ”! Values in the distribution tails relative to that of a distribution 's tails relative the... ( paucity of outliers. formula is the -3 in formula 1 and formula 2 is balance!
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