Normal distribution: Definition, pdf, properties with applications

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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)
Normal distribution have two parameter

 µ=population mean and

 =population variance

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

The PDF, CDF, CF, MGF of normal distribution 

The probability density function (pdf) is,

    pdf normal

The cumulative density function (cdf) is,

    cdf normal


 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.
normal 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 distribution
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

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)

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.


One thought on “Normal distribution: Definition, pdf, properties with applications


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