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A **normal distribution**, denoted *Ν* (μ, σ^{2}) 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**:

See also:

## 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, σ^{2}) 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) [2]

For the standard normal, the equation becomes

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

## References

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

[2] 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 [1].

The term can also refer to normal variance mixture distributions described by Romanowski [2] 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

[1] Stats Means Business: Statistics and Business Analytics for Business, Hospitality and Tourism

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

[3[ Environmental Statistics with S-PLUS p.192

## Lineo normal distribution

The **Lineo-normal distribution** is a member of the modified normal distributions constructed by Romanowski [1].

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) [2]

with *a* = 1.

The above formula is a mixture distribution obtained by giving a power function distribution with density (*a* + 1)*t ^{a}*, 0 ≤

*t*≤ 1 and

*a*≥ 1 to the standard normal distribution

*N*(0,

*σ*)/

^{2}*σ*.

^{2}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 [3].

## Properties of the Lineo-Normal Distribution

- Symmetric about 0.
- Variance: σ
^{2}(*a*+ 1)/(*a*+ 2) = σ^{2}(1 + 1)/(1 + 2) = 2σ^{2}/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

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

[2] 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.

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

## Radico-Normal distribution

The **Radico-normal distribution** is a member of the modified normal distributions constructed by Romanowski [1].

It is a special case of the modified normal cumulative distribution function (CDF) [1]

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 [2].

## Properties of the Radico-Normal Distribution

- Symmetric about 0.
- Variance: σ
^{2}(*a*+ 1)/(*a*+ 2) = σ^{2}(½ + 1)/( ½ + 2) = σ^{2}(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

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

[2] 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 [1].

The probability density function (PDF) 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.