Normal Distributions

Normal Distributions

Normal distribution is also known as Gaussian distribution. It is one of the most commonly found probability distribution when studying theory of probability; more specifically when studying continuous probability distribution. Normal distribution function is a function in which the output is the probability of the occurrence of an event is such that it lies between two real numbered values. This can be illustrated with the help of the following example.

Consider a group of students that have taken a test. The distribution of marks of the students would be a normal distribution. The corresponding probability distribution of the probability of marks obtained by a particular student would also be normally distributed.

This type of distribution (abbreviated as dist’n) is very important in statistics. When the variable in question is real valued as well as random, then this type of dist’n is used when otherwise the dist’n is now known. It finds application in the fields of social sciences and natural sciences.

It is because of the central limit theorem that this dist’n is very useful. According to this theorem, if the conditions influencing the random variable are mild then the probability distribution is normally distributed around the mean. Even of the original dist’n is not normal, it would still give us a dist’n that is centered around the mean. In general the curve of a normal distribution would look as follows:



As we can see in the above picture, the shape of the curve resembles that of a bell. Therefore a normal dist’n curve is sometimes also called a bell curve. However this is not the only dist’n that is bell shaped, there are others as well, such as: Cauchy’s, Student’s, logistic etc. The normal dist’n function can be given by the following formula:
f(x,μ,σ)=1/(σ√2π)*e^(-(x-μ)^2/(2σ^2 ))

Where,
μ is the mean of the dist’n. It is also sometimes called the expected value of the dist’n. It can also be the median or the mode of the data set.
σ is the standard deviation of the data set. That makes σ^2 the variance of the distribution.

A random variable that follows this type of Gaussian distribution is said to be normally distributed. It is also sometimes called a normal deviate. In this dist’n if we have μ=0 and σ=1, then the dist’n is called a standard normal dist’n or a unit normal dist’n and the random variable which follows this dist’n is said to be standard normal deviate.

The value of a normal dist’n is practically zero when the x value goes beyond 3 standard deviations on either side of the mean. That is why this dist’n becomes useless when there are many outliers in the data set.

The most simple case of a normal dist’n that is called the standard normal dist’n can be defined by the probability density function as follows:
φ(x)= 1/√2π e^(-1/2 x^2 )

The total area under this curve is said to be 1.