Understanding the Effects of Skewness and Kurtosis on PDFs - Explained by Josh

[member 428835]

hey pf!

i was wondering if someone could either direct me to a source or help supply a proof on why skewness and kurtosis, from their definitions as higher order moments, graphically affect the pdf in the "skewed" and "flat" way.

let me know if I've been unclear.

thanks!

josh

 

[FactChecker]

Other than examining the contributors to the integral (or summation) formulas of the moments I don't know what more can be said. The skewness is positive or negative if the contributors tend to be more from the "greater than the mean" or from the "less than the mean" respectively. Likewise the the kurtosis is large or small if the the contributors tend to be more from the "farther from the mean" or from the "closer to the mean" respectively.

 

[abitslow]

You seem to be asking, as x increases, why does xⁿ (n>1) increase faster.
I don't have much use for either metric, as a matter of fact, kurtosis isn't a single parameter, see the relevant wikipedia articles. I like to graph the data vs their probability, that graph really IS useful, imho.
I look at std dev as the simplest way (before computers) to measure the average distance of the population from the mean. If the mean is 0, then -1 is as far away as +1, but adding them gives you a sum of zero, so we make them all positive by squaring them, summing that, then taking the square root. Now, this is just a crude explanation, and the actual formal mathematics is far more elegant. (Its like claiming that electrons orbit atoms like the Earth orbits the Sun). So, what happen if you cube a difference? well the SIGN comes back that is -1³ = -1 and +1³ = +1, so summing the cube up will certainly go negative if most samples of the population are less than the mean, and positive if most are greater, but read the wiki article for qualifications to that. And the 4th moment is just the 2nd squared, so it weighs the larger differences even more.

 

[FactChecker]

I don't have much use for either metric

I like them as good single-number indicators of the shape of the PDF. Skew is a good way to indicate if one tail of a PDF is "fatter" than the other. And kurtosis is a good way to indicate if the PDF has a thin peak with fat tails (large kurtosis) or a fat peak with thin tails (small kurtosis).

Its like claiming that electrons orbit atoms like the Earth orbits the Sun)

There is a good reason to distinguish a circular orbit from an elliptical orbit. I see skew and kurtosis as valuable in a similar way.

 

[blue_raver22]

Hi Josh,

Thank you for your question. Skewness and kurtosis are statistical measures that describe the shape of a distribution. Skewness measures the symmetry of the distribution, while kurtosis measures the peakedness or flatness of the distribution.

To understand how these measures affect the probability distribution function (PDF), we need to look at their definitions and how they are calculated. Skewness is calculated by taking the third moment of the distribution and dividing it by the cube of the standard deviation. This essentially measures the degree of asymmetry in the distribution. If the value is positive, it indicates a right-skewed distribution, meaning the tail of the distribution is longer on the right side. If the value is negative, it indicates a left-skewed distribution, with a longer tail on the left side.

On the other hand, kurtosis is calculated by taking the fourth moment of the distribution and dividing it by the square of the standard deviation. This measures the peakedness or flatness of the distribution. A positive value indicates a distribution that is more peaked than a normal distribution, while a negative value indicates a flatter distribution.

Now, let's consider how these measures affect the PDF. The PDF is a representation of the probability of a random variable taking on different values. In a normal distribution, the PDF is symmetrical and bell-shaped. When we have a skewed distribution, the PDF will be shifted towards the longer tail. This is because the values on the longer tail occur less frequently, resulting in a lower probability. This is why a right-skewed distribution will have a longer right tail, and a left-skewed distribution will have a longer left tail.

In terms of kurtosis, a higher value indicates a distribution that is more peaked than a normal distribution. This means that the values are more concentrated around the mean, resulting in a narrower and taller PDF. On the other hand, a lower kurtosis value indicates a flatter distribution, with values more spread out, resulting in a wider and shorter PDF.

I hope this helps to explain how skewness and kurtosis affect the PDF. If you need further clarification or assistance, please don't hesitate to ask. Best of luck with your research!

Best,