Normal distribution is the bell curve or also called as the normal curve. Standard normal curve is a normal curve with a mean of zero and a standard deviation of one. With the help of normal curve one can estimate the probability of occurrence of any value of a normally distributed variable.

It is a family of distributions and the total area under the curve is defined to be One. A normal curve is defined by two parameters, mean and the standard deviation.Normal probability model serves as one of the most important probability models in statistical analysis.

## Properties

Some of the important properties of normal curve are listed below:

1) The curve is symmetrical about the line X = $\mu$.

2) The distribution is symmetrical, mean, median and mode coincide.
Thus Mean = Median = Mode = $\mu$

3) The total area under normal probability curve is 1, the area to the right of the ordinate as well as to the left of the ordinate at X = $\mu$ is 0.5.

4) Distribution is unimodal, the only mode occurring at X = $\mu$.

5) Moments of odd order about the mean are said to be Zero.

6) A linear combination of independent normal variates is also a normal variate. If

$X_{1}, X_{2},......... X_{n}$ are independent normal variates with means $\mu_{1}, \mu_{2},......, \mu_{n}$

and standard deviations $\sigma_{1}, \sigma_{2}, ............., \sigma_{n}$ respectively then the linear combination a$_{1}X_{1}+ a_{2}X_{2} + ........+ a_{n}X_{n}$

where a$_{1}, a_{2}, ........, a_{n}$ are constants and is also a normal variate with
Mean = a$_{1}\mu_{1}+ a_{2}\mu_{2}+ .....+ a_{n}\mu_{n}$ and

Variance = a$_{1}^{2}\sigma_{1}^{2} + a_{2}^{2}\sigma_{2}^{2} + ......... + a_{n}^{2}\sigma_{n}^{2}$

7) Quartile deviation is almost approximately equal to the two third of sigma and it is also possible to obtain Quartile deviation equal to the fifth-sixth of the mean deviation.

8) Points of inflexion of the normal curve are at X = $\mu$ $\pm$ $\sigma$, they are equidistant from the mean at a distance of $\sigma$ and is given by

[ x = $\mu$ $\pm$ $\sigma$,   p(x) =  $\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}}$.

9) Theoretically range of the distribution is from $\infty$ to $\infty$, But practically the range is 6 times the standard deviation.

## Equation

One of the most important continuous theoretical distribution in statistics is Normal distribution. This distribution is also known as Gaussian distribution. First discovered by De- Moivre in 1773 while dealing with problems arising in the game of chance.
Normal Equation is explained below:
Consider X to be a continuous random variable following normal probability distribution with mean $\mu$ and standard deviation $\sigma$, then its probability density function is given by

p(x) = $\frac{1}{\sqrt{2 \pi}. \sigma}$ e$^-\frac{(x-\mu)^{2}}{2\sigma^{2}}$ ; - $\infty <x < \infty$

where $\sqrt{2\pi}$ and e are constants and are equal to 2.5066 and 2.7183 respectively.

## Area Under Normal Curve

One of the most fundamental properties of the normal probability curve is the area property.  Area under normal curve is used to determine the probability of occurence of a given value, the value is first standardized so that it can be converted to a Z score.

The area under the normal probability curve between the coordinates at X = $\mu$ - $\sigma$ and X = $\mu$ + $\sigma$ is 0.6826.

Area under the normal probability curve between the ordinates at X = $\mu$ - 2$\sigma$ and X = $\mu$ +

2$\sigma$ is 0.9544  and the area under the normal probability curve between the ordinates at X = $\mu$

- 3$\sigma$ and X = $\mu$ + 3$\sigma$ is 0.9973. this range covers almost the entire area which is 1.

The standard normal variate corresponding to X is Z = $\frac{X - \mu}{\sigma}$

When X = $\mu$ + $\sigma$, Z = $\frac{\mu + \sigma - \mu}{\sigma}$ = 1;

When X = $mu$ - $\sigma$, Z = $\frac{\mu - \sigma - \mu}{\sigma}$ = - 1;

When X = $\mu$ + 2 $\sigma$, Z = $\frac{\mu + 2\sigma - \mu}{\sigma}$ = 2;

When X = $mu$ - 2$\sigma$, Z = $\frac{\mu - 2\sigma - \mu}{\sigma}$ = - 2;

When X = $\mu$ + 3 $\sigma$, Z = $\frac{\mu + 3\sigma - \mu}{\sigma}$ = 3;

When X = $mu$ - 3 $\sigma$, Z = $\frac{\mu - 3 \sigma - \mu}{\sigma}$ = - 3;

Therefore the area under the standard normal probability curve
1) Between the ordinates at Z = $\pm$ 1 is 0.6826.

2) Between the ordinates at Z = $\pm$ 2 is 0.9544.

3)
Between the ordinates at Z = $\pm$ 3 is 0.9973.

## Normal Curve Standard Deviation

Consider X to be a random variable following normal distribution with mean $\mu$ and standard deviation $\sigma$, then the random variable Z is defined as follows.

Z = $\frac{X - E(X)}{\sigma_{x}}$

= $\frac{X - \mu}{\sigma}$

is called the standard normal variate.
E(Z) = E($\frac{X - \mu}{\sigma}$)

= $\frac{1}{\sigma}$ E(X - $\mu$)

$\frac{1}{\sigma }$ $[E(X - \mu)]$

= $\frac{1}{\sigma }$ $[E(X)-E(\mu)]$

= $\frac{1}{\sigma }$ $[\mu - \mu]$ = 0

Var(Z)= var($\frac{X-\mu}{\sigma}$)

= $\frac{1}{\sigma ^{2}}$ $Var(X - \mu$)

= $\frac{1}{\sigma ^{2}}$.Var(X)

Var(Z)= $\frac{1}{\sigma ^{2}}$. $\sigma ^{2}$
= 1
Therefore from above we can see that Z has mean 0 and standard deviation 1.

The probability density function of Standard normal variate Z is given by

$\phi(z)$ = $\frac{1}{\sqrt{2 \pi}}$ $e^{-\frac{z^2}{2}}$ ; - $\infty <z < \infty$

## Table

The rows contains the first two most significant digits of Z and columns contains the least significant digits of Z. The values within the table are said to be the probabilities. Thus these probabilities serve as the calculation of the area under the normal curve from the starting point of Z.

Given below is an sample of the standard normal table.