# Finding kurtosis of signal

I have a signal (Y) with 200000 samples. I plotted probability density function (PDF) of the signal and the Gaussian distribution that has the same values of Mean and Standard deviation of original signal. The kurtosis of signal (Y) is 3.12. Looking to the signal shows that Normal distribution using same values of Mean and Standard deviation has higher peak than signal (Y). Can I say this signal has Leptokurtic distribution because it has of 3.12 grater than 3? If yes why PDF plotting shows higher peak for the normal distribution?

I used to find kurtosis

k = kurtosis(Y)


For Normal distribution

Gaussian=(1/sqrt(2*pi*var(Y)))*exp(-(U-mean(Y)).^2/(2*var(Y)));


Should any signal that has kurtosis > 3 will have Leptokurtic distribution

• You want to test it statistically. You can test for normality if you are interested in it, not only for excess kurtosis – gioxc88 Jan 21 '18 at 20:10
• @Hard Core. Thank you. Can you explain more? – Math Jan 21 '18 at 21:59

It's better to avoid outdated terminology like "leptokurtic distribution". Why is kurtosis of this distribution important to you? Your plot seems quite close to a normal density ... but we don't know your ultimate goal, so cannot judge what is important for you. Look in this few posts about kurtosis to see if you are really interested: Should we teach kurtosis in an applied statistics course? If so, how?, what is the meaning of 'tail' of kurtosis? and How is the kurtosis of a distribution related to the geometry of the density function?

If you want a test of normality you could look into Is normality testing 'essentially useless'?

For testing kurtosis, I would try bootstrapping (with 20000 observations that could work well). A visual version of that is a Cullen and Frey graph, for an example see Is my data gamma distributed?, which can include on the plot bootstrapped values of kurtosis, and can be very informative.

Higher kurtosis does not imply a higher peak, as was noted in 1945. But people have ignored this fact for decades and decades, writing in books, articles, and web pages that higher kurtosis means a higher peak. They still make this mistake.

"Peakedness" in general is also incorrect as a descriptor of kurtosis.

With regard to the OP, the differences in kurtosis are so small as to hardly warrant a mention.

References:

Kaplansky, I. (1945), "A Common Error Concerning Kurtosis," Journal of the American Statistical Association, 40, 259. But the “height” misinterpretation also seems to persist.

Westfall, P.H. (2014). "Kurtosis as Peakedness, 1905 – 2014. R.I.P." The American Statistician, 68, 191–195.