Mathematically speaking, kurtosis is the standardized fourth moment of a distribution. Moments are a set of measurements that tell you about the shape of a distribution. Values that fall above or below these ranges are suspect, but SEM is a fairly robust analytical method, so small deviations may not represent major violations of assumptions. Other types of analyses may have lower acceptable skew or kurtosis values so researchers should investigate their planned analysis to determine data screening guidelines. Our kurtosis examples illustrate what platykurtic, mesokurtic and leptokurtic distributions tend to look like. If the coefficient of skewness is less than 0 i.e., then the graph is said to be negatively skewed with the majority of data values greater than mean.

But you can have any shape of the peak whatsoever and have positive excess kurtosis. Note how the mass, or weight has moved around to the tails above. Any distribution with kurtosis ≈3 (excess ≈0) is called mesokurtic. An extreme positive kurtosis indicates a distribution where more of the values are located in the tails of the distribution rather than around the mean. With low kurtosis, a distribution can be extremely peaked as well, and with infinite kurtosis, it can be completely normal or flat with no deviation.

In EXCEL the “excess kurtosis” is calculated by the function KURT which gives the population kurtosis minus 3 (kurtois-3). Therefore, in EXCEL zero indicates a perfect tailedness and positive values a leptokurtic distribution. The problem may first be approached by plotting frequency histograms of empirical variables.

define kurtosis

A distribution or dataset is symmetric if it looks the same to the left and right of the center point. It is difficult to discern different types of kurtosis from the density plots because the tails are close to zero for all distributions. But differences in the tails are easy to see in the normal quantile-quantile plots .

The flat tails indicate the small outliers in a distribution. In the finance context, the platykurtic distribution of the investment returns is desirable for investors because there is a small probability that the investment would experience extreme returns. A distribution with positive excess kurtosis is called leptokurtic, or leptokurtotic. In terms of shape, a leptokurtic distribution has fatter tails. Examples of leptokurtic distributions include the Student’s t-distribution, Rayleigh distribution, Laplace distribution, exponential distribution, Poisson distribution and the logistic distribution.


For this measure, higher kurtosis corresponds to greater extremity of deviations , and not the configuration of data near the mean. Before residualization, the skewness and kurtosis coefficients of nearly all of the variables were large and significant. After removing the effects of age and gender, nearly all of the variables still showed significant skewness and kurtosis, but the absolute values of the two coefficients decreased notably. Therefore, the successive procedures of residualizing the dichotomous variables, and then nonlinearly transforming their distributions, has shifted the distributions significantly toward normality.

A positively skewed kurtosis is indicated by the term leptokurtic. It is characterized by huge tails on either side with large outliers. For investors, this could mean that the result would be an extreme of positive or negative. Thus, this graph could indicate a risky pattern for investors to make investment on either side of the distribution. Any distribution that is leptokurtic displays greater kurtosis than a mesokurtic distribution.

An extreme positive kurtosis indicates a distribution where more of the numbers are located in the tails of the distribution instead of around the mean. In negatively skewed, the mean of the data is less than the median (a large number of data-pushed on the left-hand side). Negatively Skewed Distribution is a type of distribution where the mean, median, and mode of the distribution are negative rather than positive or zero.

The log transformation proposes the calculations of the natural logarithm for each value in the dataset. It helps to understand where the most information is lying and analyze the outliers in a given data. In this article, we’ll learn about the shape of data, the importance of skewness, and kurtosis. The types of skewness and kurtosis and Analyze the shape of data in the given dataset.

  • A leptokurtic distribution is fat-tailed, meaning that there are a lot of outliers.
  • The discrepancy arises because the ADC assumes a single Gaussian displacement distribution in the radial direction.
  • With low kurtosis, a distribution can be extremely peaked as well, and with infinite kurtosis, it can be completely normal or flat with no deviation.
  • Kurtosis refers to the degree of presence of outliers in the distribution.
  • Two frequency distributions with the same mean and the same standard deviation but different kurtosis .

Where the probability mass is concentrated in the tails of the distribution. Leptokurtic distributions are statistical distributions with kurtosis over three. Timothy Li is a consultant, accountant, and finance manager with an MBA from USC and over 15 years of corporate finance experience. Timothy has helped provide CEOs and CFOs with deep-dive analytics, providing beautiful stories behind the numbers, graphs, and financial models. Skewness | Definition, Examples & Formula Skewness is a measure of the asymmetry of a distribution.

Most of the values are concentrated on the left side of the graph. Where the xi are the ordered observations (x1 ≤ x2 ≤ … ≤ xn) and coefficients wi are optimal weights for a population assumed to be normally distributed. Statistic W may be viewed as the square of the correlation coefficient (i.e. the coefficient of determination) between the abscissa and ordinate of the normal probability plot described above. Large values of W indicate normality , whereas small values indicate lack of normality. Shapiro & Wilk did provide critical values of W for sample sizes up to 50. To answer these kinds of questions we need not just a qualitative description of kurtosis, but a quantitative measure.

What does kurtosis mean?

These are normality tests to check the irregularity and asymmetry of the distribution. To calculate skewness and kurtosis in R language, moments package is required. Sometimes, the Measure of Kurtosis is characterized as a measure of peakedness that is mistaken. A distribution having a relatively high peak is called leptokurtic. The normal distribution which is neither very peaked nor very flat-topped is also called mesokurtic. The histogram in some cases can be used as an effective graphical technique for showing both the skewness and kurtosis of the data set.

Therefore, software packages use a more complicated formula. Note that \(M_2\) is simply the population-variance formula. DisclaimerAll content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional. This is the simplified prediction equation of the kurtosis of the surface elevation from frequency spectra assuming a narrow-band, unidirectional wave train. Thus, an accurate estimation of kurtosis is of paramount interest for freak wave prediction.

define kurtosis

The cokurtosis between pairs of variables is an order four tensor. For a bivariate normal distribution, the cokurtosis tensor has off-diagonal terms that are neither 0 nor 3 in general, so attempting to “correct” for an excess becomes confusing. It is true, however, that the joint cumulants of degree greater than two for any multivariate normal distribution are zero. NoteAlthough a population’s probability distribution can have a kurtosis of exactly 3, real data is almost always at least slightly platykurtic or leptokurtic. Occasionally, a female baby elephant will be born weighing less than 180 or more than 240 lbs.

Statistics, Descriptive

But, if the data have low mode or various modes, Pearson’s first coefficient is not preferred, and Pearson’s second coefficient may be superior, as it does not rely on the mode. If the values of a specific independent variable are skewed, depending on the model, skewness may violate model assumptions or may reduce the interpretation of feature importance. So, given that someone tells you that there is high kurtosis, all you can legitimately infer, in the absence of any other information, is that there are rare, extreme data points . Other than the rare, extreme data points, you have no idea whatsoever as to what is the shape of the peak without actually drawing the histogram , and zooming in on the location of the majority of the data points. When you zoom in on the bulk of the data, which is, after all, what is most commonly observed, you can have any shape whatsoever – pointy, inverted U, flat, sinusoidal, bimodal, trimodal, etc. « Back to Glossary IndexIf your data demonstrates kurtosis, it is referring to the shape of your data distribution.

define kurtosis

Kurtosis is typically measured with respect 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. The kurtosis of a mesokurtic distribution is neither high nor low, rather it is considered to be a baseline for the two other classifications. Division by the standard deviation will help you scale down the difference between mode and mean. Now understand the below relationship between mode, mean and median.

What are the three categories of kurtosis?

Kurtosis measures the heaviness of a distribution’s tails relative to a normal distribution. Uniform distributions, like the distribution of students’ ages, are the extreme cases of platykurtic distributions because outliers are so rare that they’re completely absent. There are no students younger than 14 or older than 18 years. Excess kurtosis is the tailedness of a distribution relative to a normal distribution. There are several other considerations when performing HARDI. The possible angular resolution is increased with higher b values, but at the expense of signal strength.

Thus, a kurtosis of more than 3.0 will have positive excess kurtosis and be leptokurtic, with relatively fat tails and skinny center. If there is instead a kurtosis of less than 3.0, you have negative excess kurtosis, resulting in a platykurtic distribution with smaller tails and a broader center. Kurtosis describes how much of a probability distribution falls in the tails versus its center. Positive or negative excess kurtosis will then change the shape of the distribution, accordingly.

Finding Kurtosis in Excel & SPSS

For example, look at the following normal distributions – both with the same variance but different means. They are identical, though situated at different places on the x-axis due to the difference in means. The normal define kurtosis distribution, as you are probably tired of hearing, has a familiar bell-shaped distribution. But what you may not have noticed is that this bell shape is identical for all normal distributions with the same variance.

(C–F) The maps of the metrics obtained with a diffusion kurtosis imaging sequence at a 3-Tesla MR scanner, named fractional anisotropy , mean kurtosis , axial kurtosis , and radial kurtosis , are shown. But what’s not fine is that “kurtosis” refers to either kurtosis or excess kurtosis in standard textbooks and software packages without clarifying which of these two is reported. The BFI and kurtosis of unidirectional wave in the experimental flume exceed 1.0 and 4.0 depends on given condition but these are not observed in the ocean measurement. The value of the kurtosis from linear or second-order theory barely reaches values larger than 3.15 for most severe sea state conditions. Wave directionality reduces such nonlinear enhancement of kurtosis based on quasi-wave interactions.

The types of kurtosis are determined by the excess kurtosis of a particular distribution. The excess kurtosis can take positive or negative values, as well as values close to zero. Because it’s even flatter than test 3, it has a stronger negative excess kurtosis.