# Normal distribution

normal distribution, denoted Ν (μ, σ2) is a symmetrical, bell-shaped distribution. It’s widely used in business and statistics because many real-life phenomena fit a bell-curve shape like heights of people, blood pressure readings, or standardized test scores like the SAT.

The normal has many variants, many of which fall under the umbrella term of modified normal.

Contents:

## What is a normal distribution?

The empirical rule, depicted above, tells you what percentage of normally distributed data falls within x standard deviations (σ) from the mean (μ):

• 68% of data falls within 1σ.
• 95% of data falls within 2 σ.
• 99.7% of data falls within 3 σ.

Standard deviation (σ) controls the spread of the distribution.

• Small standard deviations result in a tall, thin bell curve with data that is tightly clustered around μ.
• Larger standard deviations result in flatter, wider curves with data more widely spread out around μ.

Note that the notation Ν (μ, σ2) contains the variance, which is the standard deviation squared.

## Standard Normal

A standard normal distribution (shown in the image above) has the following properties:

• mean (μ) = 0
• standard deviation (σ) = 1

The standard normal is also called the unit normal.

## Properties

• Mean = mode = median = μ.
• Support (range): x ∈ ℝ (i.e., x is real valued).
• Symmetry around μ:  half of values are to the left of μ and half are to the right.
• Like all probability distributions, the total area under the curve is 1.
• Skewness = 0.
• Kurtosis = 3 (standard normal).

General Probability density function (PDF) 

For the standard normal, the equation becomes

The cumulative distribution function (CDF), which must be computed numerically, is the integral

## References

 Standard deviations from the mean image. D Wells, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

 Normal dist. Engineering Statistics Handbook. Online: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm

## Modified normal distribution

The term modified normal (or equinormal distribution) may refer to any number of distributions that are similar in appearance to a normal distribution (in other words, it’s simply a normal modified in some way).

For example, the t distribution is one example of a modified normal .

The term can also refer to normal variance mixture distributions described by Romanowski  that modify the normal distribution to fit the variation seen in real life data sets. Romanowski called these modulated normal distributions and — as they are particular instances of normal variance mixtures — can better be described as mixture distributions.

The actual term “modified normal” is usually used in a loose sense to describe the non-normal behavior of a particular distribution, rather than to describe a specific distribution. That’s because most “non normal” distributions have their own names. For example, positively skewed, unimodal distributions might better fit a gamma distribution or lognormal distribution.

An exponentially modified normal distribution is a three-parameter distribution that is a generalization of the normal distribution for skewed cases.

## References

 Romanowski, M., Green, E. Tabulation of the modified normal distribution functions. Bull. Geodesique 78, 369–377 (1965).

## Lineo normal distribution

The Lineo-normal distribution is a member of the modified normal distributions constructed by Romanowski .

He states

“The lineo-normal distributions seem to be strikingly well confirmed by types of observations and measurements”.

M. Romanowski

It is a special case of the modified normal cumulative distribution function (CDF) 

with a = 1.

The above formula is a mixture distribution obtained by giving a power function distribution with density (a + 1)ta, 0 ≤ t ≤ 1 and a ≥ 1 to the standard normal distribution N(0, σ2)/ σ2.

When a tends to infinity, the curve becomes a normal distribution. When a = 3 the curve is approximately normal. The ratio between the peak of the modified normal curve and the peak of the corresponding normal curve—with equal variance—is 1.09 for the lineo-normal distribution .

## Properties of the Lineo-Normal Distribution

• Variance: σ2(a + 1)/(a + 2) = σ2(1 + 1)/(1 + 2) = 2σ2/3.
• Kurtosis: 3(a + 2)2/{(a + 1)(a + 3)} = 3(3 + 2)2/{(1 + 1)(1 + 3)} = 3(5)2/{(2)(4)} = 75/6 = 12.5.

## References

 M. ROMANOWSKI: On the normal law of errors. Bull. Géodésique 73, 95 (1964).

 Kotz, S., and Johnson, N. L. (1985). Modified normal distributions, Encyclopedia of Statistical Sciences, 5, S. Kotz, N. L. Johnson, and C. B. Read (editors), 590-591, New York: Wiley.

 Maarek, A. & Konecny, G. Modulated Normal Distribution. Online: https://www.asprs.org/wp-content/uploads/pers/1973journal/sep/1973_sep_959-965.pdf

The Radico-normal distribution is a member of the modified normal distributions constructed by Romanowski .

It is a special case of the modified normal cumulative distribution function (CDF) 

with a = ½.

Note that when a is infinitely large, the curve is a normal distribution. When a = 3 the curve is approximately normal. The ratio between the peak of the modified normal curve and the peak of the corresponding normal curve—with equal variance—is 1.16 for the Radico-normal distribution .

## Properties of the Radico-Normal Distribution

• Variance: σ2(a + 1)/(a + 2) = σ2(½ + 1)/( ½ + 2) = σ2(1½ )/( 2½).
• Kurtosis: 3(a + 2)2/{(a + 1)(a + 3)} = 3(½ + 2)2/{(½ + 1)( ½ + 3)} = 3(2½)2/{(1½)( 3½)}  3.57.

## References

 M. ROMANOWSKI: Bull. Géodésique 73, 95 (1964).

 Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

## Zero-modified normal distribution

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 (PDF) 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].

## References

 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.

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