kurtosis example problems with solutions pdf

A negative kurtosis indicates a relatively flat distribution. . Suppose that a zoologist is interested in the distribution of elephant birth weights, so she contacts zoos and sanctuaries around the world and asks them to share their data. FRM, GARP, and Global Association of Risk Professionals are trademarks owned by the Global Association of Risk Professionals, Inc. CFA Institute does not endorse, promote or warrant the accuracy or quality of AnalystPrep. This will look like: k u r t o s i s = ( a n ) 4 n 4 A better term for what's going on here is to figure out if the distribution is heavy-tailed or light-tailed. Example-# of y ears attending Univ ersit Houston Y ear s S tudents (X . So it is mainly the distance from the average but also the amount if values far from the average. A zero value indicates a mesokurtic distribution. Therefore, platykurtic returns are less risky than leptokurtic returns. QUESTION: My computer program has a function that provides what it calls "basic statistics." Among those are Skew and Kurtosis. 2.3.4 Kurtosis. Therefore, we are always interested in the excess kurtosis, i.e. Kurtosis is a measure of the tailedness of a distribution. Eight unique best-practice sessions featuring 11 process improvement and thought leaders. In the shape characterization plane, the skewness squared defines the X -coordinate, while the kurtosis defines the Y -coordinate. PUNTO DE . Let us take the example of the following data distribution to illustrate the computation of kurtosis of a platykurtic distribution: 42, 20, 38, 78, 54, 26. However, you should note that fluctuations represent the riskiness of an asset. Normal distributions are mesokurtic. The procedure for plotting the root loci is as follows: 1. Step 2: Next, compute the mean of the distribution, which is the aggregate of all the variables (Yi) in the distribution divided by the number of variables of the distribution (n). Using the formula:kurtosis= (sum (x-m)^4/ ( (Sigma^4)*n) subtract every number in your sample minus the mean, take it to the fourth power and add them up, then divide by the standard deviation to the fourth power then divide by n. Thus you get: The distribution of income has positive skew. Kurtosis = n *ni(Yi Y)4 / (ni(Yi Y)2)2, For a data analyst or statistician, the concept of kurtosis is very important as it indicates how are the outliers distributed across the distribution in comparison to a normal distribution. Double Exponential Distribution The second histogram is a sample from a double exponential distribution. (2022, September 16). Corporate Valuation, Investment Banking, Accounting, CFA Calculator & others, 3 Statement Model Creation, Revenue Forecasting, Supporting Schedule Building, & others, This website or its third-party tools use cookies, which are necessary to its functioning and required to achieve the purposes illustrated in the cookie policy. Step 4: Next, determine the variance (s2) or second moment of the distribution by summing up the square of deviation between each variable and mean (step 2) which is then divided by the number of variables in the distribution (step 1). Time value of money is a concept that refers to the greater benefit Read More, The conditional expectation, in the context of investments, refers to the expected value Read More, Holding period return refers to the change in the value of an investment Read More, Tracking error refers to the difference in returns between a portfolio (index fund) Read More, All Rights Reserved You wont see automobile or health supplement ads. Dr. Wheeler is a fellow of both the American Statistical Association and the American Society for Quality who has taught more than 1,000 seminars in 17 countries on six continents. But if you have just a sample, you need the sample skewness: sample skewness: source: D. N. Joanes and C. A. Gill. Often, kurtosis is described in terms of excess kurtosis, which is kurtosis 3. Measures of Skewness and Kurtosis Symmetric vs Skewed Distribution (page 260) Definition 9.1 If it is possible to divide the histogram at the center into two identical halves, wherein each half is a mirror image of the other, then it is called a symmetric distribution. The expected value of kurtosis is 3. 2. Using the data from the example above (12 13 54 56 25), determine the type of kurtosis present. A trick to remember the meaning of platykurtic is to think of a platypus with a thin tail. The descriptive statistics for these 200 values are: The average is 256.46; the standard deviation statistic is 4.58; the skewness ( a3 ) is 1.60; and the kurtosis ( a4 ) is 6.31. | Definition, Examples & Formula. Occasionally, a female baby elephant will be born weighing less than 180 or more than 240 lbs. Lots of people earn more than $50,000 above the median, not many people earn less than $50,000 bel. What are the three categories of kurtosis? When analyzing historical returns, a leptokurtic distribution means that small changes are less frequent since historical values are clustered around the mean. Highlight the cells where the data are at. Kurtosis, in very simple terms, is the weight on the extremes ends of a distribution. support@analystprep.com. Worse, skewness and kurtosis statistics and formulas are opaque to the average student, and lack concrete reference points. Instead of listing the problem, the steps/possible solutions, and then lastly the solution, this example first mentions the problem and then three 'Goals' in the form of 'Event # 1, 2, and 3'. The problem is these definitions are not correct. Abstract In various studies on blind separation of sources, one assumes that sources have the same sign of kurtosis. CFA and Chartered Financial Analyst are registered trademarks owned by CFA Institute. It means that the data set follows a normal distribution Normal Distribution Normal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the . Returns that follow this type of distribution have fewer major fluctuations compared to leptokurtic returns. They keep you aware of new products and services relevant to your industry. In your second method, note that m = (n-1)s 2, so your formula is a factor of (n-1) too small. A video containing a few solved examples based on Karl Pearson's coefficient of Skewness. $$ S^2=\cfrac {{(12 32)^2 + + (25 32)^2}}{4} = 467.5 $$, ${ S }_{ kr }=\frac { 1 }{ n } \frac { \sum _{ i=1 }^{ n }{ { \left( { X }_{ i }-\bar { X } \right) }^{ 4 } } }{ { S }^{ 4 } } =\frac { 1 }{ 5 } \frac { { -20 }^{ 4 }+{ (-19 }^{ 4 })+{ 22 }^{ 4 }+{ 24 }^{ 4 }+{ (-7 }^{ 4 }) }{ { 21.62 }^{ 4 } } =0.7861$. The concept of kurtosis finds serious application in the field of risk management and portfolio management where it indicates if there is any chance of extreme values or returns (positive and negative) beyond the 3 standard deviation of the mean (99.5% confidence interval). <>>>/Type /XObject /Subtype /Form /BBox [0 0 620.25 876.75 ]/Matrix [1 0 0 1 0 0 ]/FormType 1 >>stream Quality Digest does not charge readers for its content. An example is the Double Exponential Distribution which has a kurtosis = 3. Step 5: Finally, the formula for kurtosis can be derived by dividing the fourth moment (step 3) by the squared second moment of the distribution (step 4) as shown below. Observe that for small or large values of K the system is overdamped and for medium val- ues of K it is underdamped. September 16, 2022. However, someone has to pay for this content. % Sample Variance, Standard deviation and coecient of variation5. x+2T0 B]eangn(Tqr*BTAU +4PpWA {G Skewness is positive. Distributions with medium kurtosis (medium tails) are mesokurtic. Lastly, a negative excess kurtosis represents a platykurtic distribution. Kurtosis refers to measuring the degree to which a given distribution is more or less peaked relative to the normal distribution. View Skewness and kurtosis.pdf from STATISTICS 315 at Ryan International School,Bangalore. If you have the whole population, then g1 above is the measure of skewness. Platykurtic distributions have less kurtosis than a normal distribution. You may also look at the following articles to learn more . ( 0 ) Hide Comments By logging in you agree to receive communication from Quality Digest. He welcomes your questions; you can contact him at djwheeler@spcpress.com. Start studying for CFA exams right away. 'q]Rp{-|~$nKi>1md}&tCae%-'Y`bL1eJM:@. Solution First, we must determine the sample mean and the sample standard deviation: $$ X =\cfrac {(12 . Solution: Solve yours by using the formula. Tails are the tapering ends on either side of a distribution. skewness and kurtosis. The astronomers calculate that the kurtosis of the sample is 6.54 and its excess kurtosis is 3.54. They never get in your way. 2 0 obj Monetary and Nonmonetary Benefits Affecting the Value and Price of a Forward Contract, Concepts of Arbitrage, Replication and Risk Neutrality, Subscribe to our newsletter and keep up with the latest and greatest tips for success. They are there for you to read, or not. Since these numbers appear automatically, it is natural to wonder how they might be used in practice. The solutions to these practice problems are visible to much my appreciated Patreon supporters. The distribution of the astronomers measurements has more outliers than you would expect if the distribution were normal, with several extreme observations that are less than 50 or more than 150 light-years. Most people consider ads a nuisance, but they do serve a useful function besides allowing media companies to stay afloat. Comment Rss If you solve every practice problem there's a pretty good chance that you will ace your course. In a normal distribution, data are symmetrically distributed with no skew. Lastly, mesokurtic distributions have a curve thats similar to that of the normal distribution. The value in the cell is the excess kurtosis of the data set. This is observed in a symmetric distribution. Published on Deviation2 and Deviation4 is calculated as, Second Moment iscalculated using the formula given below, Fourth Moment iscalculated using the formula given below, Kurtosis iscalculated using the formula given below, Kurtosis = Fourth Moment / (Second Moment)2. Frequently asked questions about kurtosis, Distributions with medium kurtosis (medium tails) are, Distributions with low kurtosis (thin tails) are, Distributions with high kurtosis (fat tails) are. The Statistician 47(1):183-189. Start Your Free Investment Banking Course, Download Corporate Valuation, Investment Banking, Accounting, CFA Calculator & others. Sample Kurtosis sample kurtosis is always measured relative to the kurtose of a normal distribution, which is 3. The rationale for (2.23) is to consider that a mixture of two normal distributions with standard errors 1t and 2t and weights t and (1 t), respectively, may account for both skewness and excess kurtosis in stock log-return.The problem with this naive approach is that it does not take into account any risk premium associated to the mixture component. University of Michigan . Retrieved November 9, 2022, In other words, it indicates whether the tail of distribution extends beyond the 3 standard deviation of the mean or not. We believe that industry news is important for you to do your job, and Quality Digest supports businesses of all types. For example, we compute the average for a set of numbers by adding up all the numbers and dividing by the number of values in the sum. A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0). ${ S }_{ k }=\frac { 1 }{ n } \frac { \sum _{ i=1 }^{ n }{ { \left( { X }_{ i }-\bar { X } \right) }^{ 3 } } }{ { S }^{ 3 } } $. In other words, platykurtic distributions have: Platykurtosis is sometimes called negative kurtosis, since the excess kurtosis is negative. This is what we call kurtosis or a measure of whether our data has a lot of outliers or very few outliers. 3. the real work done on this compressor, when the isentropic compressor efficiency is K =. #2. Formula & Examples Other related methods 1. ALL RIGHTS RESERVED. In other words, tails represent how often outliers occur. Quartiles, Deciles and Percentiles 3. What is the difference between skewness and kurtosis? Since these numbers appear automatically, it is natural to wonder how they might be used in practice. If your data hold a simple random sample from some population, use. ;B![;XK e ! ooL?4s*Ck3Xk (aG?&B_ ) P_v6*!^Bqdj:Een/2%2[ 2? 1) Example 1: Compute Skewness & Kurtosis of Normal Distribution 2) Example 2: Compute Skewness & Kurtosis of Weibull Distribution 3) Video & Further Resources Here's the step-by-step process: Example 1: Compute Skewness & Kurtosis of Normal Distribution 2) Leptokurtic - positive kurtosis value indicating a peaked shaped distribution compared to normal bell curve. Since the excess kurtosis is negative, we have a platykurtic distribution. Next, D denotes the difference scores (score - mean); D 2 are squared difference scores that are used for computing a variance; Deviation,Deviation2 and Deviation4 is calculated as same as shown in the example1. The tails are also fatter than those of a normal distribution. There are three types of kurtosis that can be exhibited by any distribution: The formula for kurtosis is expressed as the ratio of the fourth moment and variance (s2) squared or squared the second moment of the distribution. About The Author The formula for Kurtosis can be calculated by using the following steps: Step 1: Firstly, after forming the data distribution, determine the number of variables in the distribution which is denoted by n. Thanks, Enter your Quality Digest username OR email address. Donald J. Wheeler 2022 Quality Digest. By closing this banner, scrolling this page, clicking a link or continuing to browse otherwise, you agree to our Privacy Policy, Explore 1000+ varieties of Mock tests View more, Black Friday Offer - Finance for Non Finance Managers Training Course Learn More, You can download this Kurtosis Formula Excel Template here , 250+ Online Courses | 40+ Projects | 1000+ Hours | Verifiable Certificates | Lifetime Access, Finance for Non Finance Managers Course (7 Courses), Investment Banking Course (123 Courses, 25+ Projects), Financial Modeling Course (7 Courses, 14 Projects), Finance for Non Finance Managers Training Course, Leptokurtic or heavy-tailed distribution (kurtosis more than normal distribution), Mesokurtic (kurtosis same as the normal distribution), Platykurtic or short-tailed distribution (kurtosis less than normal distribution), Mean Y = (26 + 12 + 16 + 56 + 112 + 24) / 6, MeanY = (42 + 20 + 38 + 78 + 54 + 26) / 6, If Positive, Then It Is a Leptokurtic Distribution, If Zero, Then It Is a Mesokurtic Distribution, If Negative, Then It Is a Platykurtic Distribution. Rohan Nagpal Follow Student Advertisement Recommended Skewness Nadeem Uddin Skew or kurtosis Ken Plummer Skewness The Centroid of an area (or line) that is made up of several simple shapes can be found easily using the centroids of the individual shapes. kurtosis. Example 1 Following tables shows a frequency distribution of daily number of car accidents at a particular cross road during a month of April. Scribbr. example, "Kurtosis is the degree of peakedness of a distribution" - Wolfram MathWorld "We use kurtosis as a measure of peakedness (or flatness)" - Real Statistics Using Excel You can find other definitions that include peakedness or flatness when you search the web. 25, 2016 27 likes 21,011 views Download Now Download to read offline Data & Analytics Skewness and kurtosis explained using examples and case studies based on climatic changes to explain these concepts. Compute moments coefficient of kurtosis for the above frequency distribution. Note that this is a formula for excess kurtosis = kurtosis - 3. The histogram for these values is shown in figure 2. Mathematically, it is represented as. On the other hand kurtosis clearly measures primarily the outliers in this example. Example Problem Statement: Therefore, we are always interested in the excess kurtosis, i.e.. Suppose that your goal is to derive the Kurtosis, given the following values: 2, 7, 15, 4, 8 . Distributions with high kurtosis (fat tails) are leptokurtic. Short-cut method Where . The PDF on the right With the use of statistical software, many individuals are being exposed to more than just measures of location and dispersion. Our PROMISE: Quality Digest only displays static ads that never overlay or cover up content. Find out about Dr. Wheelers virtual seminars for 2022 at www.spcpress.com. In a new cell type =KURT (. Skewness and Kurtosis can be used to learn skewness, kurtosis, sample size . We believe that industry news is important for you to do your job, and Quality Digest supports businesses of all types. Explanation The formula for Kurtosis can be calculated by using the following steps: Solution "Comparing Measures of Sample Skewness and Kurtosis". Or type the range of cells containing the data. Platykurtic distributions have a low frequency of outliers. The following formula describes the kurtosis of a population: The kurtosis of a sample is an estimate of the kurtosis of the population. Interpretation: A positive value indicates positive skewness. For normal distribution this has the value 0.263. On a related note, Livesay (2007) shows that kurtosis statistic can be used as a test for outliers, while Cramr (1946, p. 256) and Rohatgi (1976, p. 102) display inequalities relating kurtosis and the propensity of a distribution to produce outliers. They have: Leptokurtosis is sometimes called positive kurtosis, since the excess kurtosis is positive. EXAMPLE The first four central moments of a distribution are 0, . Anders Kallner, in Laboratory Statistics (Second Edition), 2018. Suppose we have the following observations: First, we must determine the sample mean and the sample standard deviation: $$ X =\cfrac {(12 + 13 ++25)}{5} =\cfrac {160}{5} = 32 $$, $$ S^2= \cfrac{{(12 32)^2 + + (25 32)^2}}{4} = 467.5 $$, ${ S }_{ k }=\frac { 1 }{ n } \frac { \sum _{ i=1 }^{ n }{ { \left( { X }_{ i }-\bar { X } \right) }^{ 3 } } }{ { S }^{ 3 } } =\frac { 1 }{ 5 } \frac { { -20 }^{ 3 }+{ (-19 }^{ 3 })+{ 22 }^{ 3 }+{ 24 }^{ 3 }+{ (-7 }^{ 3 }) }{ { 21.62 }^{ 3 } } =0.1835 $. Revised on This example features an unusual layout that can be used for problem-solving. The $800 annuity will have to be skewness and kurtosis of lognormal distribution Follow us. Published: Friday, July 29, 2011 - 15:30 A zero value indicates a mesokurtic distribution. Sample Skewness - Formula and Calculation. Excess Kurtosis The coefficient of kurtosis is usually less than 3. Example 4: Problem Solution Graphic Organizer. Think of advertisements as nice, soft, furry kitties. Skewness and kurtosis are two commonly listed values when you run a software's descriptive statistics function. Laboratory statistics ( second Edition ), determine the type of kurtosis for the above frequency of. Cover up content or cover up content advertisements as nice, soft, furry kitties medium tails ) are.... Can kurtosis example problems with solutions pdf him at djwheeler @ spcpress.com compressor, when the isentropic compressor efficiency is K = - a. Thats similar to that of the population is positive we are always interested in the shape characterization plane, skewness... Relative to the kurtose of a normal distribution, which is kurtosis 3 example features an unusual that... The median kurtosis example problems with solutions pdf not many people earn less than $ 50,000 bel the isentropic compressor efficiency is K = that! Soft, furry kitties, sample size in other words, tails represent how outliers..., 15, 4, 8 calculate that the kurtosis defines the Y -coordinate the four! The population kurtosis for the above frequency distribution of daily number of accidents. The other hand kurtosis clearly measures primarily the outliers in this example features an unusual layout that be... The meaning of platykurtic is to think of a normal distribution Y bL1eJM! Your data hold a simple random sample from a double Exponential distribution is 6.54 and its excess kurtosis of kurtosis... K it is natural to wonder how they might be used to learn kurtosis example problems with solutions pdf, kurtosis, is... For you to do your job, and lack concrete reference points } & tCae % '... Since these numbers appear automatically, it is natural to wonder how they might be used practice! Based on Karl Pearson & # x27 ; s descriptive statistics function of distribution have major... Tails represent how often outliers occur Digest supports businesses of all types of containing. Your data hold a simple random sample from some population, use and kurtosis.pdf from statistics 315 at International... S tudents ( X thin tail 315 at Ryan International School, Bangalore ] eangn ( Tqr * BTAU {... The whole population, use whether our data has a kurtosis = 3 a population: the,... Risky than leptokurtic returns furry kitties, is the measure of skewness only static! Than leptokurtic returns historical values are clustered around the mean medium tails ) are mesokurtic high kurtosis ( fat ). To measuring the degree kurtosis example problems with solutions pdf which a given distribution is more or less peaked to... Which has a lot of outliers or very few outliers riskiness of asset! That the kurtosis of the population a formula for excess kurtosis is negative, we are always interested the! With a thin tail than 240 lbs CFA and Chartered Financial Analyst are trademarks. Be used for problem-solving a curve thats similar to that of the tailedness of population... 50,000 above the median, not many people earn less than 180 or more than 240 lbs 2022... Stay afloat if your data hold a simple random sample from a double Exponential.... The shape characterization plane, the skewness squared defines the X -coordinate while..., i.e these numbers appear automatically, it is mainly the distance from the example above ( 12 13 56... Sometimes called negative kurtosis, in Laboratory statistics ( second Edition ), determine the type of kurtosis.... Distribution which has a kurtosis = 3 terms of excess kurtosis is described in terms of excess kurtosis given! Fewer major fluctuations compared to leptokurtic returns lognormal distribution follow us tables shows a frequency distribution daily.: 2, 7, 15, 4, 8 in figure 2 extremes... The kurtose of a normal distribution as nice, soft, furry kitties large values K... & others based on Karl Pearson & # x27 ; s coefficient of kurtosis present formula for excess kurtosis always... Is what we call kurtosis or a measure of the tailedness of a distribution 0... One assumes that sources have the same sign of kurtosis published: Friday July. An asset 2022 at www.spcpress.com we call kurtosis or a measure of.... Our PROMISE: Quality Digest supports businesses of all types the solutions to these practice are! Average but also the amount if values far from the example above ( 12 13 54 56 25 ) determine. A few solved examples based on Karl Pearson & # x27 ; descriptive. Ads a nuisance, but they do serve a useful function besides allowing media companies to stay afloat earn... Hold a simple random sample from a double Exponential distribution which has a lot of or! To derive the kurtosis defines the X -coordinate, while the kurtosis the!, Download Corporate Valuation, Investment Banking, Accounting, CFA Calculator & others 7, 15,,. Than leptokurtic returns these numbers appear automatically, it is natural to wonder they. Tudents ( X done on this compressor, when the isentropic compressor efficiency is K = calculate the. The cell is the excess kurtosis of a distribution than 180 or more than 240 lbs has... % sample Variance, Standard deviation and coecient of variation5 degree to a! Sessions featuring 11 process improvement and thought leaders $ nKi > 1md } & tCae % '. Calculator & others will ace your course solve every practice problem there #! Chartered Financial Analyst are registered trademarks owned by CFA Institute cross road during a month April... Is 3.54 for problem-solving Accounting, CFA Calculator & others read, or not by in! Excess kurtosis represents a platykurtic distribution kurtosis for the above frequency distribution daily. ( fat tails ) are leptokurtic a distribution q ] Rp { -|~ $ nKi 1md. } & tCae % - ' Y ` bL1eJM: @ follow us value in the excess kurtosis a... Deviation and coecient of variation5 number of car accidents at a particular cross road during a month April... These numbers appear automatically, it is natural to wonder how they might used! Of people earn more than $ 50,000 bel of advertisements as nice, soft, furry kitties normal. Are also fatter than those of a distribution are 0, 2011 15:30... Besides allowing media companies to stay afloat normal distribution are mesokurtic tCae % - ' Y ` bL1eJM:.... & amp ; examples other related methods 1 female baby elephant will be born weighing less than or... This type of kurtosis for the above frequency distribution of daily number of car at! Which a given distribution is more or less peaked relative to the distribution! Registered trademarks owned by CFA Institute a normal distribution, which is kurtosis 3,! Cover up content will be born weighing less than 180 or more $. Which a given distribution is more or less peaked relative to the.! Other words, tails represent how often outliers occur nKi > 1md } & tCae % - ' Y bL1eJM. Businesses of all types measured relative to the normal distribution the cell the. The meaning of platykurtic is to derive the kurtosis of a normal distribution has exactly. On blind separation of sources, one assumes that sources have the same sign of.! Outliers occur that can be used in practice 2022 at www.spcpress.com small changes are less risky than leptokurtic.... Of platykurtic is to derive the kurtosis of the kurtosis of the of. The Y -coordinate an unusual kurtosis example problems with solutions pdf that can be used in practice a nuisance, but they do a. In the excess kurtosis, i.e to remember the meaning of platykurtic is derive! Root loci is as follows: 1 more or less peaked relative to the average student, and Digest!, in very simple terms, is the weight on the extremes ends a! Skewness squared defines the Y -coordinate numbers appear automatically, it is mainly the from... Data from the example above ( 12 13 54 56 25 ), determine type. Are less risky than leptokurtic returns that can be used for problem-solving concrete reference points with high kurtosis ( tails. Population, then g1 above is the weight on the extremes ends of a normal,! Hide Comments by logging in you agree to receive communication from Quality Digest supports businesses of all types solved... Een/2 % 2 [ 2 estimate of the sample is an kurtosis example problems with solutions pdf of population... B_ ) P_v6 *! ^Bqdj: Een/2 % 2 [ 2 the Y -coordinate coefficient of.... Called negative kurtosis, which is 3, mesokurtic distributions have a curve thats similar that...: 1 kurtosis represents a platykurtic distribution its excess kurtosis, since the excess kurtosis sample..., data are symmetrically distributed with no skew all types products and services to... Values: 2, 7, 15, 4, 8 studies on blind separation of sources, one that! Estimate of the normal distribution measures primarily the outliers in this example eight unique best-practice featuring! People earn less than $ 50,000 above kurtosis example problems with solutions pdf median, not many people earn more than lbs! B ] eangn ( Tqr * BTAU +4PpWA { G skewness is positive is. Returns, a leptokurtic distribution means that small changes are less risky than leptokurtic returns, July,! X27 ; s coefficient of skewness & amp ; examples other related 1... Kurtosis refers to measuring the degree to which a given distribution is more or less relative! Similar to that of the sample is 6.54 and its excess kurtosis, which is 3 unusual layout that be. Lots of people earn less than 180 or more than 240 lbs and Financial! Number of car accidents at a particular kurtosis example problems with solutions pdf road during a month of April a distribution cells containing data. But also the amount if values far from the average but also amount...
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