# Introduction to Normal Distribution

**Introduction**

The **Normal Distribution**, also sometimes known as *Gaussian Distribution* is a family of beautiful bell-shaped curves. It is indeed the most important concept in statistics. A **Normal Curve** depicts the distribution for any experiment repeated for a large number of times.

As the name suggests, the normal distribution is so normal that we can find it anywhere. Just name it! Be the size of micro-organisms or the giant planets. Monetary income, or Student grades. When we take measurements for any case a large number of times, it leads us to a bell-like symmetric Normal Distribution.

**Definition**

The average of many observations of any random variable with finite mean and variance is also a random variable. The distribution for the resultant random variable converges to a **Normal Distribution**.

Note that every Normal Distribution generally has two parameters **μ** and **σ**, which we will discuss. These parameters define the shape of a distribution.

**Properties of Normal Curve**

**Unimodal**

As the name suggests, unimodal distributions have only one mode. We can say any distribution that has a single peak is called a Unimodal Distribution.

**Symmetrical**

A distribution where the shape of the curve to the left of the peak mirrors the shape of the curve to the right of the peak.**Skewness**is generally considered as a measure of*asymmetry*.

Such a distribution is unique because the**mean**,**median**, and**mode**all lie at a single point.

**Asymptotic Tail**

On both sides, the tails of a normal curve may seem to be meeting the x-axis or having y=0 at some value of x. But in reality, they always approach the x-axis but never touch it.**Kurtosis**is one of the measures that describe the shape of a distribution’s tail compared to its peakedness in the center.

**Types of Normal Curves**

The values **μ** and **σ** tell us about the mean and standard deviation of the distribution. Mean is a measure of centrality, which balances the distribution. Whereas Standard deviation acts as a measure of spread, which decides how flat our distribution is. Normal Distributions are classified into types, based on the values that **μ** and **σ** can take.

**Standard Normal Distribution**

It is the simplest and commonly used case. A standard normal distribution has**μ**=0 and**σ**=1.

**General Normal Distribution**

All the general cases of a normal distribution are also a version of standard normal distribution. In all the cases, the distribution is stretched by some factor of**σ**and shifted by some value**μ**.

**The Empirical Rule**

The *Empirical Rule* is also known sometimes as the “*68-95-99.7 rule*” or the “*three-sigma rul*e“. It states that for any observable normal distribution, almost all the data lies 3 standard deviations away from the mean.

According to the empirical rule, 68.27% of the data can be observed within the range (**μ**±**σ**). In other words, the area under the curve in pink color is 68.27% of the total area under the curve. Similarly, 95.45% and 99.73% is the area under the curve having limits (**μ** ± 2***σ**) and (**μ** ± 3***σ**) respectively.

We can often use the *empirical rule* for rough estimates to test whether or not a given distribution is normal or not by simply checking whether how much proportion of the data lies away from 3 standard deviations from the mean, or away from (**μ** ± 3***σ**).

**Applications**

The idea of a **Normal distribution** is so important yet simple, that it has become a universal basis law for almost all statistical methods. Some commonly used examples are *Regression analysis*, *Analysis of variance,* and many *Parameter estimation* methods as well.

*Central Limit Theorem* is one of the most important applications of a Normal Distribution. It also plays an essential part in *Hypothesis Testing* where the assumption is made that the data follows a normal distribution.