Zero-modified normal distribution

< 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(y;μ,σ,p), is given by [3]:

Zero-Modified Normal Distribution
PDF for the zero-modified normal distribution.

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].


[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:

[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.