Normal distribution: Definition, pdf, properties with applications

Spread the love

One of the most important tools in statistics is the normal distribution. It aids in determining specific data features and also serves as a foundation for employing other statistical techniques for decision-making. As a result, in this article, we look at the Normal distribution and its application in real life.

In probability, normal distribution is the most important continuous distribution in statistics because its common in natural phenomena. It is also known as Gaussian distribution and always symmetric about mean. There are also various probability distributions such as Bernoulli, Binomial , Negative Binomial, Geometric, Hypergeomettric, Poisson, Logarithmic series, Power series, Gamma, Beta, Uniform , Exponential distribution etc. But from all this distribution normal is the best mesure.

normal distribution
Normal curve( Bell shaped)

What is Normal Distribution?

A normal distribution is a bell-shaped curve with a single peak for a continuous random variable. A useful continuous probability distribution is the normal distribution. The normal curve is a mathematical curve that is theoretical. For normal distribution, a normal curve is employed. A normal curve should be turned into a standard normal curve for practical purposes, and a given variable should be translated into a standard normal variate.
Normal distribution has two parameter μ, σ^2.

μ=population mean and

σ^2=population variance

The normal curve is always in bell shape and mean, median and mode is always equal. That means mean=median=mode.
bell curve
Bell curve

History of the Normal Distribution

In 1733, De Moivre found the normal distribution as the limiting case of the Binomial model. It has been attributed to Gauss, who first mentioned it in 1809, due to a historical error. Various attempts were made in the 18th and 19th centuries to establish the normal model as the underlying law governing all continuous random variables—hence the name Normal. Because of the incorrect premises, their initiatives failed. Nonetheless, in statistical analysis, the normal model has become the most important probability model. Gaussian distribution is another name for it.

The PDF, CDF, CF, MGF of normal distribution 

The probability density function (pdf) is,

    \[ f(x;\mu ,\sigma  )=\frac{1}{\sigma \sqrt{2\pi }}e^{-\frac{1}{2}(\frac{x-\mu }{\sigma })^{2}}; where \mu =mean,\sigma =std. \]


The c.d.f of the normal distribution is,

    \[ F(x)=\int_{0}^{\infty }f(x)dx \]


 CDF (cumulative density function) 



 Mean, Median, Mode




 Skewness, Kurtosis


 MGF (moment generating function)




Why normal distribution is so common in natural phenomena?

There exist numerous natural events whose distribution follow normal curve. Human characteristics such as weight, height, strength, body temperature, or intelligence are among those. This explanation stems from the fact that numerous independent elements (factors) impact a characteristic such as height, where these factors may work in favor or against height by 50% chance. For example, factors such as dietary habits, genes, and life style may have positive or negative contribution on human height. Figure 1 shows a normal distribution for height of adults in a homogeneous race.
Bell shape
      Figure : Height of Adults in a homogenous race and effect of independent factors on it.
In the above Figure, mean population height is 5’7’’. For an individual human being, each contributing factor shifts the mean population height toward left or right of 5’7’’ with a probability of 0.5. The difference between number of factors that contribute in favor or against taller height results in the final height of a person. Assuming independency and equal importance among these factors, the probability of a person’s height being in a particular range is found by binomial distribution.


Properties of the Normal Distribution

For a specific μ = 3 and a σ ranging from 1 to 3, the probability density function (P.D.F.) is as shown –
normal curve
The properties are following –


  • The distribution is symmetric about the point x = μ and has a characteristic bell-shaped curve with respect to it. Therefore, its skewness is equal to zero i.e. the curve is neither inclined to the right (negatively skewed) nor to the left (positively skewed).
  • The mean, median and the mode of a normal distribution, all coincide with each other and are equal to μ.
  • The Standard Deviation for this distribution is equal to σ.
     Mean Deviation: σ√2
First Quartile: μ – 0.675σ and the Third Quartile: μ + 0.675σ
Thus, Quartile Deviation: 0.675

Application of Normal Distribution in Business

  •      The so-called central limit theorem states that the normal distribution possesses a striking quality.
  •      Even though a variable is not normally distributed, a simple transformation of the variable can occasionally bring it into normal form.
  •      Many sample distributions, such as the student’s t, Student’s F, and others, tend to be normal.
  •      The assumption behind sampling theory and significance tests is that samples were selected from a normal population with mean and variance.
  •      The normal distribution has a lot of uses in statistical quality control.
  •      The normal distribution serves as a good approximation for many discrete distributions as n grows larger (such as Binomial, Poisson, etc.).
  •      The normal distribution has a number of mathematical properties that make it widely used and relatively simple to adjust.

The Binomial, Poisson, and Normal Distributions and Their Relationships

The Binomial, Poisson, and Normal distributions are all inextricably linked. The following are the relationships:

The Poisson and normal distributions have a connection. When n is big and the probability p of an event occurring is near to zero, the binomial distribution tends to become a Poisson distribution with np remaining a finite constant.

The binomial and normal distributions have a similar relationship. Under the following circumstances, the normal distribution is a limiting form of binomial distribution:

  •      n, the number of trials is very large, i.e., n → ∞; and
  •      Neither p nor q is very small.

Practical Applications of the Standard Normal Model

The standard normal distribution could help you figure out which subject you are getting good grades in and which subjects you have to exert more effort into due to low scoring percentages. Once you get a score in one subject that is higher than your score in another subject, you might think that you are better in the subject where you got the higher score. This is not always true. You can only say that you are better in a particular subject if you get a score with a certain number of standard deviations above the mean. The standard deviation tells you how tightly your data is clustered around the mean; It allows you to compare different distributions that have different types of data — including different means. For example, if you get a score of 90 in Math and 95 in English, you might think that you are better in English than in Math. However, in Math, your score is 2 standard deviations above the mean. In English, it’s only one standard deviation above the mean. It tells you that in Math, your score is far higher than most of the students (your score falls into the tail). Based on this data, you actually performed better in Math than in English!..(source)


In the natural and behavioral sciences, the normal distribution is a useful model for quantitative phenomena. A normal distribution has been found to roughly follow a number of psychological test scores and physical phenomena such as photon counts. While the underlying reasons of these events are frequently unknown, the usage of the normal distribution in cases where several minor effects are compounded together to produce a score or variable that can be observed is logically appropriate. The normal distribution also appears in many fields of statistics: for example, even if the distribution of the population from which the sample was taken is not normal, the sampling distribution of the mean is approximately normal. Furthermore, among all distributions with known mean and variance, the normal distribution maximizes information entropy, making it the natural choice as the underlying distribution for data summarized in terms of sample mean and variance.

You cannot copy content of this page