Gaussian Distribution

 

If Probability Density Function (PDF) of a distribution is like bell shaped curve then it is called Gaussian Distribution.

Fig1. Bell shaped Probability Density Function

Gaussian distribution is also known as Normal or Gauss or Laplace-Gauss distribution.

There are two parameters of Gaussian distribution:

    (i) Mean (μ)

    (ii) Variance (${\displaystyle \sigma ^{2}}$)

If we know mean and standard deviation of a distribution, then we can draw its Gaussian distribution because it is symmetric about it's mean and on both side there is equal deviation.

Let, x follows normal distribution with mean μ and variance then, probability of occurrence of x will be:

For example, let x be the mean of the distribution then it will lie at the top of the graph. Since 50% of data are on the left and 50% of the data are on the right side then probability of occurrence will be 0.5.

Now, let's see the distribution of values within each standard deviation:

Fig2. Gaussian distribution graph

68.2% of values lies within 1 standard deviation.

95.4% of values lies within 2 standard deviation.

99.7% of values lies within 3 standard deviation.

 

If we have data points with large numbers such as 8568, 2309,5671,897..., then it's mean and standard deviation will be very large. To avoid this we standardize our data between 1 and -1 and then find mean and standard deviation. Which makes our calculation very easy.

Characteristic properties of Gaussian distribution:

•  Gaussian distribution is symmetric about the mean
•  In Gaussian distribution mean=median=mode
•  50% of values less than the mean and 50% greater than the mean  
•  It is not skewed
•  Gaussian distribution have zero kurtosis  

Things that follows Gaussian or Normal distribution are:

• Height of the population
• Rolling of dice
• Tossing the coin
• Birth weight, etc.