Editing Kurtosis (section)

== Applications ==
The sample kurtosis is a useful measure of whether there is a problem with outliers in a data set. Larger kurtosis indicates a more serious outlier problem, and may lead the researcher to choose alternative statistical methods.

[[D'Agostino's K-squared test]] is a [[goodness-of-fit]] [[normality test]] based on a combination of the sample skewness and sample kurtosis, as is the [[Jarque–Bera test]] for normality.

For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please see [[Variance#Distribution of the sample variance|variance]].

Pearson's definition of kurtosis is used as an indicator of intermittency in [[turbulence]].{{r|Sandborn1959}} It is also used in magnetic resonance imaging to quantify non-Gaussian diffusion.<ref>{{cite journal |last1=Jensen |first1=J. |last2=Helpern |first2=J. |last3=Ramani |first3=A. |last4=Lu |first4=H.|first5=K. |last5=Kaczynski |title=Diffusional kurtosis imaging: The quantification of non-Gaussian water diffusion by means of magnetic resonance imaging |journal=Magn Reson Med |date=19 May 2005 |volume=53 |issue=6 |pages=1432–1440 |doi = 10.1002/mrm.20508 |pmid=15906300 |s2cid=11865594 |url=https://onlinelibrary.wiley.com/doi/full/10.1002/mrm.20508}}</ref>

A concrete example is the following lemma by He, Zhang, and Zhang:<ref name=He2010>{{cite journal | last1 = He | first1 = Simai | last2 = Zhang | first2 = Jiawei | last3 = Zhang | first3 = Shuzhong | year = 2010 | title = Bounding probability of small deviation: A fourth moment approach | journal = [[Mathematics of Operations Research]] | volume = 35 | issue = 1| pages = 208–232 | doi = 10.1287/moor.1090.0438 | s2cid = 11298475 }}</ref>
Assume a random variable {{mvar|X}} has expectation <math>\operatorname{E}[X] = \mu</math>, variance <math>\operatorname{E}\left[(X - \mu)^2\right] = \sigma^2</math> and kurtosis <math display="inline">\kappa = \tfrac{1}{\sigma^4} \operatorname{E}\left[(X - \mu)^4\right]. </math>
Assume we sample <math>n = \tfrac{2\sqrt{3} + 3}{3} \kappa \log\tfrac{1}{\delta}</math> many independent copies. Then
<math display="block">
  \Pr\left[\max_{i=1}^n X_i \le \mu\right] \le \delta
    \quad\text{and}\quad
  \Pr\left[\min_{i=1}^n X_i \ge \mu\right] \le \delta.
</math>

This shows that with <math>\Theta(\kappa\log\tfrac{1}\delta)</math> many samples, we will see one that is above the expectation with probability at least <math>1-\delta</math>.
In other words: If the kurtosis is large, we might see a lot values either all below or above the mean.

===Kurtosis convergence===
Applying [[band-pass filter]]s to [[digital image]]s, kurtosis values tend to be uniform, independent of the range of the filter. This behavior, termed ''kurtosis convergence'', can be used to detect image splicing in [[forensic analysis]].{{r|Pan2012}}

=== Seismic signal analysis ===
Kurtosis can be used in [[geophysics]] to distinguish different types of [[Seismology|seismic signals]]. It is particularly effective in differentiating seismic signals generated by human footsteps from other signals.<ref>{{Cite journal |last1=Liang |first1=Zhiqiang |last2=Wei |first2=Jianming |last3=Zhao |first3=Junyu |last4=Liu |first4=Haitao |last5=Li |first5=Baoqing |last6=Shen |first6=Jie |last7=Zheng |first7=Chunlei |date=2008-08-27 |title=The Statistical Meaning of Kurtosis and Its New Application to Identification of Persons Based on Seismic Signals |journal=Sensors |volume=8 |issue=8 |pages=5106–5119 |doi=10.3390/s8085106 |doi-access=free |pmid=27873804 |pmc=3705491 |bibcode=2008Senso...8.5106L |issn=1424-8220}}</ref> This is useful in security and surveillance systems that rely on seismic detection.

=== Weather prediction ===
In [[meteorology]], kurtosis is used to analyze weather data distributions. It helps predict extreme weather events by assessing the probability of outlier values in historical data,<ref>{{Cite web |last=Supraja |date=2024-05-27 |title=Kurtosis in Practice: Real-World Applications and Interpretations |url=https://www.analyticsinsight.net/tech-news/kurtosis-in-practice-real-world-applications-and-interpretations |access-date=2024-11-11 |website=Analytics Insight |language=en}}</ref> which is valuable for long-term climate studies and short-term weather forecasting.