Normal distribution -definition, example,properties, applications and special cases





Normal curve( Bell shaped)


Normal distribution

In probability distribution, 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 distribution, Binomial Distribution, Negative Binomial, Geometric distribution, Hypergeometric distribution, Poisson distribution, Logarithmic series distribution, Power series distribution, Gamma distribution, Beta distribution, Uniform distribution, Exponential distribution etc. But from all this distribution normal distribution is the best mesure.
Normal distribution have two parameter

 µ=population mean and



 =population variance



The normal distribution curve is always in bell shape and the distribution curve is called normal curve.

Normal curve


The PDF, CDF, CF, MGF of normal distribution 

The pdf of normal distribution is,

    

The cdf of normal distribution is,


    



 

 CDF 

 

 Mean, Median, Mode

 μ

 Varience

 

 Skewness, Kurtosis

 0

 MGF

 

 CF

 


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.
 
      Figure : Height of Adults in a homogenous race and effect of independent factors on it.


In 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.


The Standard Form of normal distribution

Since the effect of changing the μ and σ is only to shift the curve along the x-axis or just broaden it or narrow it respectively. Thus, we can define a new random variable Z that would accommodate these changes in itself as –
Z=xμσ
Z is also known as the standardized normal variable or the normal deviate. In terms of this standard variable, the Normal Distribution gets reduced to the following form –
ϕ(z)=12πexp(z22)

This distribution has the parameters equal to μ = 0 and σ2 = 1. This we can say, ZN(0,1).



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 –

The following properties follow –

  • 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


Skewness and Kurtosis

  The skewness and kurtosis coefficients always measure how different a given distribution is from a normal distribution. The skewness measures the symmetry of a distribution. The normal distribution is symmetric and has zero skewness . If the distribution of a data set has negative skewness (skewness<0), then the left tail of the distribution is longer than the right tail and if skewness>0 then the right tail of the distribution is longer than the left.

The kurtosis statistic measures the thickness of the tail ends of a distribution in relation to the tails of the normal distribution. If
  • kurtosis=3 then mesokurtic
  • kurtosis<3 then platykurtic
  • kurtosis>3 then leptokurtic
  • Normal distribution is always in a bell shape that means mesokurtic.




 

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