Normal vs. LaPlace Distributions: Critical Values

[kimberley]

Hello all. With the standard caveat that my background is neither in math nor science, I've nonetheless been conducting some further independent study in various areas of statistics that are of interest to me.

With the foregoing as background, I'm trying to appreciate the material difference(s) between the Normal distribution and the Laplace distribution. My understanding thus far is that the principle difference is that an ideal Normal distribution has a Kurtosis of 3/Excess Kurtosis of 0 while a LaPlace distribution has a Kurtosis of 6/Excess Kurtosis of 3. It's also my understanding that each has a Skewness of 0 and their points of Central Tendency are their arithmetic Means.

What I haven't been able to find, however, are the Z-scores/critical values for a LaPlace distribution. By this I specifically mean the two-tailed .01, .05, .3173 levels which, for a standard normal, would be 2.576, 1.96 and 1. Typically, if I'm looking for Z-scores/critical levels for a Normal distribution I specifically use the Student's T Table to get my Z-scores for the given sample size, or use the T Inverse function found in most software. Am I not able to find any literature/tables with regard to these critical levels because they are the same for the Laplace and Normal Distributions, or am I simply not looking in the right place or missing something? Stated in the simplest terms, are 95% of all values in a standard LaPlace distribution +/- 1.96 from the arithmetic mean also? I found one cryptic reference that mentioned 3.842 and 6.635 standard deviations as the .05 and .01 levels for a LaPlace, but frankly I had difficulty following the general topic to attach any weight to the reference.



Your response would be very much appreciated. As always,


Thanks,


Kimberley

 

[EnumaElish]

Am I not able to find any literature/tables with regard to these critical levels because they are the same for the Laplace and Normal Distributions,

No!

or am I simply not looking in the right place or missing something?

Laplace is not a "staple" distribution so its tables may be difficult to find. This page tells you the inverse CDF formula F^-1. If you assume mu = 0 and b = 1/Sqrt 2, you will have the standard Laplace. To find the 95% value in a two-sided test, just evaluate F^-1(0.025). Note, sgn(x) = -1 if x < 0, sgn(x) = 1 if x > 0 and sgn(0) = 0.

Stated in the simplest terms, are 95% of all values in a standard LaPlace distribution +/- 1.96 from the arithmetic mean also?

I'd guess not necessarily.

I found one cryptic reference that mentioned 3.842 and 6.635 standard deviations as the .05 and .01 levels for a LaPlace, but frankly I had difficulty following the general topic to attach any weight to the reference.

You can verify these values by using the inverse CDF formula.

 

[tacman]

Hello Kimberley,

Thank you for your question. The Normal and Laplace distributions are both types of continuous probability distributions, but they have some key differences. As you mentioned, the Normal distribution has a kurtosis of 3 and an excess kurtosis of 0, while the Laplace distribution has a kurtosis of 6 and an excess kurtosis of 3. This means that the Laplace distribution has a more peaked and pointed shape compared to the Normal distribution.

In terms of critical values, the Laplace distribution does not have specific Z-scores like the Normal distribution. This is because the Laplace distribution does not have a standard deviation like the Normal distribution does. Instead, the Laplace distribution has a scale parameter that determines the shape and spread of the distribution. Therefore, there is no specific value that corresponds to a certain percentage of the distribution, like the 1.96 for the 95% confidence interval in the Normal distribution.

However, there are some approximate critical values that can be used for the Laplace distribution. As you mentioned, the values 3.842 and 6.635 correspond to the .05 and .01 levels, respectively. These values are based on the asymptotic normality of the Laplace distribution, which means that for large sample sizes, the Laplace distribution approximates the Normal distribution. These values are not exact, but they can be used as a rough estimate for critical values in the Laplace distribution.

I hope this helps clarify the difference between the Normal and Laplace distributions and their critical values. It's always important to check the assumptions and characteristics of the data before deciding which distribution to use in statistical analysis. Best of luck in your studies!