# When is half normal distribution useful?

Julian Faraway, in his book on linear model with R, used half normal plot to find hat(a value in hat matrix) with too large value.

That's the only example where I saw half normal distribution is used. But other than that, I can't come up with real application of it though I guess it might be useful for approximating some positive numbers whose mean is near zero.

What are real applications of it?

• Because the square root of a $\chi^2(1)$ variable is half normal, most applications involving chi squared values with 1 degree of freedom can be interpreted as applications of a half-normal model.
– whuber
Commented Dec 27, 2011 at 21:10
• The half normal is used in the EGARCH specification for asymmetric volatility modelling.
– user11978
Commented Jun 14, 2012 at 3:41

In Bayesian statistics the half normal,with a sufficiently large variation parameter, can be used as a noninformative prior distribution on the SD of a standard distribution. This is suggested in, for example:

Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian analysis, 1, 515-534.

• It's true that you use it, but if you do it's typically NOT noninformative, since then you could also use a more typical/easier noninformative prior like gamma or the uniform. The half-t makes more sense as a weakly informative prior, which has a distinct tendency but wide enough tails to let itself be overwhelmed by the data in the face of strong evidence.
– Erik
Commented Nov 13, 2012 at 12:00
• i have came across a function that uses a single value as a paramter for the half normal distribution in a Baysian Market Media Mix context - stats.stackexchange.com/questions/599740/… - and i simply cannot figure out how that is supposed to work. Maybe you can explain? Commented Dec 22, 2022 at 9:17
• @Nneka Often that's actually a half-normal centered at zero. Then the half-normal can be defined by either it's SD (would it have been a full normal) or the mean or median of the half-normal. Commented Dec 23, 2022 at 15:19
• @RasmusBååth does that mean that i can actually define the half-normal centered at zero if i only have the mean? How could i do that ? Commented Jan 2, 2023 at 10:44
• also here quora.com/… it says that the mean of a half normal distribution centred at zero is the min(X)? Commented Jan 2, 2023 at 13:30

In quality control, there is something called a moving-range statistic which is the absolute value of successive differences. The half-normal serves as the basis of the chart as discussed in the following article:

http://www.tandfonline.com/doi/abs/10.1080/08982119508904612#preview