Kurtosis is sometimes referred to as the "fourth moment" of a data distribution, in the way that skewness is often called the "third moment". A kurtosis number is produced when a data set is analyzed by many statistical software packages such as JMP, Mintab, Excel, SPSS etc. It may be positive or negative (or zero for a perfectly normalized distribution.) It is important to understand what kurtosis is, as well as to be able to estimate when it is significant.
Kurtosis Formula
For a population, kurtosis used to be defined as:
(Σ(x – mean)^4) ÷ (Σ(x – mean)^2.(1/n))^2
Where mean = the population mean.
If this formula is used, the kurtosis for a normalized Gaussian distribution is 3. For this reason, it has become common for software packages such as JMP, Excel and others to subtract three from the result of this formula, to define kurtosis as
(Σ(x – mean)^4) ÷ (Σ(x – mean)^2.(1/n))^2 - 3
A slightly different formula is used for calculating kurtosis based on a sample from a population.
Significant Kurtosis - Standard Error Of Kurtosis
Having calculated the kurtosis value, and assuming that the second version of the formula is used (because Excel, JMP and others use this), then it is possible to evaluate the kurtosis value returned by the software. A positive value means that the distribution is "higher" than a normal distribution, and a negative value is flatter.
There will almost always be a non-zero value returned, so it is useful to know when the kurtosis is significant. Tabachnick & Fidell suggest that the standard error of kurtosis may be estimated as
√(24 / n)
where n is the sample size.
Kurtosis is deemed to be significant when the kurtosis value supplied by the software is greater than two standard error of kurtosis.
Kurtosis Types
Distributions may often be spoken of as leptokurtic, platykurtic, or mesokurtic. Leptokurtic distributions have a positive kurtosis value and have a high peak and fatter tails. Platykurtic distributions have negative kurtosis values and flat peaks with thinner tails. Normal distributions are mesokurtic, with a zero or very low absolute value of kurtosis.
Kurtosis Calculation and Evaluation Summary
Low kurtosis is important, since it is a condition on which many parametric statistical tests are based. Non-Gaussian distributions may become normalized using the Central Limit Theorem, and a check for normality is usually needed before proceeding with a parametric test. It is sometimes known as the fourth standardized moment, in the way that skewness is sometimes called the third standardized moment.
Kurtosis Calculation References
Tabachnick, B.G., & Fidell, L.S. (1996). Using multivariate statistics (3rd Ed). New York: Harper Collins.
Microsoft [Computer Software]. (1996). Excel. Redmond, WA: Microsoft Corporation.
Martin Bell - Martin holds a B.Sc. degree in chemical engineering, and an M.Sc. degree in electronics and computing. He has spent more than 25 years ...
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