The normal distribution has a unique position in probability theory, and it can be used as approximation to most of the other distributions. Discrete distributions occurring in practice including binomial, Poisson, hypergeometric, etc can also be approximated by normal distribution. You will notice in the subsequent Articles that theory of estimation of population parameters and testing of hypotheses on the basis of sample statistics have also been developed using the concept of normal distribution as most of the sampling distributions tend to normality for large samples. Therefore, study of normal distribution is very important.
The concept of normal distribution was initially discovered by English mathematician Abraham De Moivre (1667-1754) in 1733. De Moivre obtained this continuous distribution as a limiting case of binomial distribution. His work was further refined by Pierre S. Laplace (1749-1827) in 1774. But the contribution of Laplace remained unnoticed for long till it was given concrete shape by Karl Gauss (1777-1855) who first made reference to it in 1809 as the distribution of errors in Astronomy. That is why the normal distribution is sometimes called Gaussian distribution. Though, normal distribution can be used as approximation to most of the other distributions like approximation to (i) binomial distribution and (ii) Poisson distribution.
So further Moving We will see what are the main characteristics of NORMAL DISTRIBUTATION:
1. The curve of the normal distribution is in bell shaped shown in the above fig.
2. The curve of the distribution is completely symmetrical about x=µ that if we hold the curve at x=µ ,both the parts of the curve are the Minor images of each other.
3. For Normal Distribution ,Mean=Median=Mode
4. F(x) ,being the probability can never be negative and hence no portion of the curve lies below x-axis.
5. Though x-axis becomes closer and closer to the normal curve as the magnitude of the value of x goes towards ¥ or ¥ , yet it never touches it.
6. Normal Curve has only one mode.
7. Central moments of Normal distribution are µ1=0 , µ2=(sigma)2 i.e. the distribution is symmetrical and curve is always mesokurtic.
8. Note: Not only µ1 & µ3 but all the odd order central moments are zero for a normal distribution
9. For Normal Curve :
a. Q3 -Median=Median-Q1
i. This states that first & third quartile are equidistant from the median.
10. Mean deviation is approximately equal to of the standard Deviation.
A continuous random variable X is said to follow normal distribution with parameters m (-¥ < m < ¥ ) and s 2 (>0) if it takes on any real value and its probability density function is given by f(x)=(Probability Density Function) , (-¥ < X< ¥)
In my next Article more detailed discussion would be held on terminologies which are mentioned in this article.