< List of probability distributions

A **zero-modified normal distribution** is a normal distribution modified to put extra probability mass at 0. In other words, it’s a kind of mixture distribution where part of the population comes from a normal distribution but the rest of the population is all zeros [1].

The probability density function of a zero-modified normal random variable `Y`

, denoted h`(`

, is given by [3]:*y*;*μ*,*σ*,*p*)

Note that the mean(*μ)* and standard deviation (*σ) *in the PDF are the mean and standard deviation of the normal part of the mixture distribution — not the whole distribution.

## Uses of the Zero-Modified Normal Distribution

In chemistry, the zero-modified normal distribution is occasionally used to model concentrations when some observations are below a certain detection limit, However, while USEPA [2] recommends this strategy for some situations, Helsel [4] strongly advises against it. A zero-modified lognormal (delta distribution) may be more appropriate as chemical concentrations are bounded below at 0 [5, 6].

## References

[1] EnvironmentalStats for SPlus.

[2] USEPA. (2009). *Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance*. EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C.

[3] R Documentation. The Zero-Modified Normal Distribution. Retrieved June 26, 2022 from: https://search.r-project.org/CRAN/refmans/EnvStats/html/ZeroModifiedNormal.html

[4] Helsel, D.R. (2012). *Statistics for Censored Environmental Data Using Minitab and R*. Second Edition. John Wiley and Sons, Hoboken, NJ, Chapter 1.

[5] Gilliom, R.J., and D.R. Helsel. (1986). Estimation of Distributional Parameters for Censored Trace Level Water Quality Data: 1. Estimation Techniques. *Water Resources Research* **22**, 135-146.

[6] Owen, W., and T. DeRouen. (1980). Estimation of the Mean for Lognormal Data Containing Zeros and Left-Censored Values, with Applications to the Measurement of Worker Exposure to Air Contaminants. *Biometrics* **36**, 707-719.