what is kurtosis 
kurtosis is a statistical measure used to describe a characteristic of a dataset when normally distributed data is plotted on a graph it generally takes the form of a bell this is called the bell curve the plotted data that are farthest from the mean of the data usually form the tails on each side of the curve kurtosis indicates how much data resides in the tails jessica olah investopediaunderstanding kurtosiskurtosis is a measure of the combined weight of a distribution s tails relative to the center of the distribution curve the mean for example when a set of approximately normal data is graphed via a histogram it shows a bell peak with most of the data residing within three standard deviations plus or minus of the mean however when high kurtosis is present the tails extend farther than the three standard deviations of the normal bell curved distribution kurtosis is sometimes confused with a measure of the peakedness of a distribution however kurtosis is a measure that describes the shape of a distribution s tails in relation to its overall shape a distribution can be sharply peaked with low kurtosis and a distribution can have a lower peak with high kurtosis thus kurtosis measures tailedness not peakedness distributions with a large kurtosis have more tail data than normally distributed data which appears to bring the tails in toward the mean distributions with low kurtosis have fewer tail data which appears to push the tails of the bell curve away from the mean for investors high kurtosis of the return distribution curve implies that there have been many price fluctuations in the past positive or negative away from the average returns for the investment so an investor might experience extreme price fluctuations with an investment with high kurtosis this phenomenon is known as kurtosis risk formula and calculation of kurtosisthere are several different methods for calculating kurtosis the simplest way is to use the excel or google sheets formula for instance assume you have the following sample data 4 5 6 3 4 5 6 7 5 and 8 residing in cells a1 through a10 on your spreadsheet the spreadsheets use this formula for calculating kurtosis n n 1 n 1 n 2 n 3 x i x s 4 3 n 1 2 n 2 n 3 begin aligned frac n n 1 n 1 n 2 n 3 times big sum frac x i bar x s big 4 frac 3 n 1 2 n 2 n 3 end aligned n 1 n 2 n 3 n n 1 sxi x 4 n 2 n 3 3 n 1 2 however we ll use the following formula in google sheets which calculates it for us assuming the data resides in cells a1 through a10 kurt a1 a10 begin aligned text kurt a1 a10 end aligned kurt a1 a10 the result is a kurtosis of 0 1518 indicating the curve has lighter tails and is platykurtic calculating kurtosis by hand is a lengthy endeavor and takes several steps to get to the results we ll use new data points and limit their number to simplify the calculation the new data points are 27 13 17 57 113 and 25 it s important to note that a sample size should be much larger than this we are using six numbers to reduce the calculation steps a good rule of thumb is to use 30 of your data for populations under 1 000 for larger populations you can use 10 first you need to calculate the mean add up the numbers and divide by six to get 42 next use the following formulas to calculate two sums s2 the square of the deviation from the mean and s4 the square of the deviation from the mean squared note that these numbers do not represent standard deviation they represent the variance of each data point s2 y i y 2 s4 y i y 4 where y i ith variable of the sample y mean of the sample begin aligned text s2 sum y i bar y 2 text s4 sum y i bar y 4 textbf where y i text ith variable of the sample bar y text mean of the sample end aligned s2 yi y 2s4 yi y 4where yi ith variable of the sampley mean of the sample to get s2 use each variable subtract the mean and then square the result add all of the results together 27 42 2 15 2 225 13 42 2 29 2 841 17 42 2 25 2 625 57 42 2 15 2 225 113 42 2 71 2 5 041 25 42 2 17 2 289 225 841 625 225 5 041 289 7 246 begin aligned 27 42 2 15 2 225 13 42 2 29 2 841 17 42 2 25 2 625 57 42 2 15 2 225 113 42 2 71 2 5 041 25 42 2 17 2 289 225 841 625 225 5 041 289 7 246 end aligned 27 42 2 15 2 225 13 42 2 29 2 841 17 42 2 25 2 625 57 42 2 15 2 225 113 42 2 71 2 5 041 25 42 2 17 2 289225 841 625 225 5 041 289 7 246 to get s4 use each variable subtract the mean and raise the result to the fourth power add all of the results together 27 42 4 15 4 50 625 13 42 4 29 4 707 281 17 42 4 25 4 390 625 57 42 4 15 4 50 625 113 42 4 71 4 25 411 681 25 42 4 17 4 83 521 50 625 707 281 390 625 50 625 25 411 681 83 521 26 694 358 begin aligned 27 42 4 15 4 50 625 13 42 4 29 4 707 281 17 42 4 25 4 390 625 57 42 4 15 4 50 625 113 42 4 71 4 25 411 681 25 42 4 17 4 83 521 50 625 707 281 390 625 50 625 25 411 681 83 521 26 694 358 end aligned 27 42 4 15 4 50 625 13 42 4 29 4 707 281 17 42 4 25 4 390 625 57 42 4 15 4 50 625 113 42 4 71 4 25 411 681 25 42 4 17 4 83 52150 625 707 281 390 625 50 625 25 411 681 83 521 26 694 358 so our sums are s2 7 246 s4 26 694 358 begin aligned text s2 7 246 text s4 26 694 358 end aligned s2 7 246s4 26 694 358 now calculate m2 and m4 the second and fourth moments of the kurtosis formula m2 s2 n 7 246 6 1 207 67 begin aligned text m2 frac text s2 n frac 7 246 6 1 207 67 end aligned m2 ns2 67 246 1 207 67 m4 s4 n 26 694 358 6 4 449 059 67 begin aligned text m4 frac text s4 n frac 26 694 358 6 4 449 059 67 end aligned m4 ns4 626 694 358 4 449 059 67 we can now calculate kurtosis using a formula found in many statistics textbooks that assumes a perfectly normal distribution with a kurtosis of zero k m4 m2 2 3 where k kurtosis m4 fourth moment m2 second moment begin aligned k frac text m4 text m2 2 3 textbf where k text kurtosis text m4 text fourth moment text m2 text second moment end aligned k m22m4 3where k kurtosism4 fourth momentm2 second moment so the kurtosis for the sample variables is 4 449 059 67 1 458 466 83 3 05 begin aligned frac 4 449 059 67 1 458 466 83 3 05 end aligned 1 458 466 834 449 059 67 3 05 types of kurtosisthere are three categories of kurtosis that a set of data can display mesokurtic leptokurtic and platykurtic all measures of kurtosis are compared against a normal distribution curve investopediathe first category of kurtosis is mesokurtic distribution this distribution has a kurtosis similar to that of the normal distribution meaning the extreme value characteristic of the distribution is similar to that of a normal distribution therefore a stock with a mesokurtic distribution generally depicts a moderate level of risk the second category is leptokurtic distribution any distribution that is leptokurtic displays greater kurtosis than a mesokurtic distribution this distribution appears as a curve with long tails outliers 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 a stock with a leptokurtic distribution generally depicts a high level of risk but the possibility of higher returns because the stock has typically demonstrated large price movements while a leptokurtic distribution may be skinny in the center it also features fat tails the final type of distribution is platykurtic distribution these types of distributions have short tails fewer outliers platykurtic distributions have demonstrated more stability than other curves because extreme price movements rarely occurred in the past this translates into a less than moderate level of risk kurtosis vs skewnesskurtosis and skewness are both statistical measures used to describe the shape of a probability distribution but they focus on different aspects kurtosis measures the tailedness of a distribution skewness measures the asymmetry of a distribution skewness indicates the direction and degree to which the data deviates from a symmetrical bell curve a distribution with zero skewness is perfectly symmetrical meaning the left and right sides of the distribution are mirror images positive skewness means that the right tail is longer or fatter than the left suggesting that the data has a tendency to have higher values negative skewness indicates that the left tail is longer or fatter implying a tendency towards lower values while skewness focuses on the balance of data around the mean kurtosis focuses on the distribution s peak and the weight of its tails a dataset can have high kurtosis with many outliers but still be symmetric and thus have zero skewness on the other hand a dataset can be skewed with either positive or negative skewness but has low kurtosis indicating fewer extreme values using kurtosiskurtosis is used in financial analysis to measure an investment s risk of price volatility kurtosis measures the amount of volatility that an investment s price has experienced regularly high kurtosis of the return distribution implies that an investment will yield occasional extreme returns be mindful that this can swing both ways meaning high kurtosis indicates either large positive returns or extreme negative returns for example imagine a stock had an average price of 25 85 per share if the stock s price swung widely and often enough the bell curve would have heavy tails high kurtosis this means that there is a lot of variation in the stock price an investor should anticipate wide price swings often on the other hand a portfolio with a low kurtosis value indicates a more stable and predictable return profile which may indicate lower risk in this light investors may intentionally seek investments with lower kurtosis values when building safer less volatile portfolios kurtosis can also be used to strategically implement an investment allocation approach for example a portfolio manager who specializes in value investing may prefer to invest in assets with a negative kurtosis value as a negative kurtosis value indicates a flatter distribution with more frequent small returns conversely a portfolio manager who specializes in momentum investing may prefer to invest in assets with a positive kurtosis value with peaked distributions of less frequent but larger returns kurtosis vs other commonly used measurementskurtosis risk differs from more commonly used measurements alpha measures excess return relative to a benchmark index while kurtosis measures the nature of the peak or flatness of the distribution alpha measures the skewness or asymmetry of the distribution beta measures the volatility of a stock compared to the broader market each security or investment has a single beta that indicates whether or not that security is more or less volatile compared to a market benchmark again beta measures the degree of asymmetry of the distribution while kurtosis measures the peak or flatness of the distribution r squared measures the percent of movement a portfolio or fund has that can be explained by a benchmark though r squared is used in regression analysis to assess the goodness of fit of a regression model kurtosis is used in descriptive statistics to describe the shape of a distribution lastly the sharpe ratio compares return to risk the sharpe ratio is used by investors to better understand whether the level of returns they are receiving is commensurate with the level of risk incurred while kurtosis analyzes the distribution of a dataset the sharpe ratio more commonly is used to evaluate investment performance 