Flat normal distribution

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A flat normal distribution (or flattened Gaussian distribution) is a normal distribution with a large standard deviation. The standard deviation is a measure of spread; smaller values compress the distribution into a smaller space while a larger standard deviation flattens and widens the normal.

flat normal distribution
Flat normal distribution (left) with a standard deviation of 4, compared to the standard normal distribution (right), which has a standard deviation of 1 [1].

The flat normal distribution also has a high variance, which makes sense because the standard deviation is the square root of variance.

Why use the flat normal distribution?

In some experiments, there may be some uncertainty about where exactly the center of the distribution (e.g., the mean or expected value) lies. This can happen in Bayesian analysis when prior information is scarce or in any experiment where there is a dearth of data.

For example, tornado researchers Elsner and Schroder [2] used a combination of a flat normal distribution and a flat Student’s t-distribution for priors on two parameters, because tornado research was, at the time of their 2019 studies, not on a solid statistical foundation (disclaimer: I’m not sure whether that has changed, but tornados are beyond my area of expertise!).

Choosing a large value for the variance results in a flat distribution, representing the researcher’s level of uncertainty about the location of the parameter of interest.

Sometimes, the flat normal distribution might be referred to as a “slab,” compared to a distribution with a “spike,” which would have a larger variance [3]. Distributions with a spike are sometimes called peaked normal distributions.

Comparison of regular, flat and peaked normal distributions
Comparison of regular, flat and peaked normal distributions [4].


[1] Image created with Desmos.

[2] Elsner, J. & Schroder, Z. (2019). Tornado damage ratings estimated with cumulative logistic regression. Retrieved March 3, 2023 from: https://eartharxiv.org/repository/object/782/download/1727/

[3] Walli, G. (2010). Bayesian Variable Selection in Normal Regression Models. Thesis.

[4] Brewerton, F. (1973). Variability assumptions and their effect on capital investment risk.

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