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 .
The probability density function of a zero-modified normal random variable
Y, denoted h
(y;μ,σ,p), is given by :
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  recommends this strategy for some situations, Helsel  strongly advises against it. A zero-modified lognormal (delta distribution) may be more appropriate as chemical concentrations are bounded below at 0 [5, 6].
 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.
 R Documentation. The Zero-Modified Normal Distribution. Retrieved June 26, 2022 from: https://search.r-project.org/CRAN/refmans/EnvStats/html/ZeroModifiedNormal.html
 Helsel, D.R. (2012). Statistics for Censored Environmental Data Using Minitab and R. Second Edition. John Wiley and Sons, Hoboken, NJ, Chapter 1.
 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.
 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.