In statistics and probability theory, we come across continuous and discrete distributions very often. Normal distribution is applied in the social science and natural science for representing real-valued random variables with condition that their distributions should be unknown and is a very common continuous probability distribution. We can distribute data in many ways like more to the right or more to the left or even in jumbled up manner.
 
Data can also be distributed in a manner in which it is more to the centre and neither left or right. This is called the Bell curve. Such curve represents Normal distribution. The normal distribution is a two parameter distribution and is specified by the standard deviation and mean.
A normal distribution has the following properties:

1) The mean, median and mode of any normal distribution will always be equal.

2) A normal curve is symmetric about the curve’s centre always.

3) Half of the values that is 50% are less than the mean and half of the values are greater than the mean.

Formula

The following represent the probability density function of a normal distribution with the variate X.

P (x) = $\frac{1}{ (\sigma \sqrt {2 \pi})}$ $e^{(\frac{- (x - \mu )^2 }{ (2 \sigma^2)})}$

Here, x belongs to the interval of negative infinity to positive infinity (-\infty , \infty ) as the domain, \mu is the mean and \sigma^2 is the variance. 

This is the general normal distribution. When we take \mu = 0 and \sigma^2 = 1 then our general normal distribution gets converted to standard normal distribution with probability density function as below:

P (x) dx = $\frac{1}{ (\sigma \sqrt {2 \pi})}$ $e^{\frac{- z^2 }{ 2}}$ dz

Where Z = $\frac{(x - \mu)}{ \sigma }$ which implies dz = $\frac{dx}{\sigma}$

We use more commonly the standard form of normal distribution.

This z is known as z-score more commonly.

The z - score tells us the distance between a value and the mean. If the value of this Z is equal to zero, then the ‘x’ value is equal to the mean. If this z-score is equal to one, then the ‘x’ value is one standard deviation above the mean and if the z-score is equal to -1, then the ‘x’ value is one standard deviation below the mean. And so on with z-score increasing or decreasing.

Some properties of the standard normal curve are:

1) The complete area under this curve is always 1.

2) This curve extends both sides indefinitely approaching the horizontal axis.

3)This curve is symmetric always about zero.

Examples

Let us see an example to understand this topic more clearly.
Example:
Given that X is a random variable that is normally distributed with μ = 30 and σ = 4. Determine the following:

1)P (30 < x < 35)

2)P (x > 21) 

3)P (x < 40)

Solution: 

Here we are simply finding the area under the standard type of normal curve under given conditions.

1) Now, Z = $\frac{(30 - 30)}{4}$ = 0. Also, Z = $\frac{(35 - 30)}{4}$ = 1.25

Thus P (30 < x < 35) = P (0 < z < 1.25) = 0.3944

2) Z = $\frac{(21 - 30)}{4}$ = -2.25

Thus P (x > 21) = P (z > -2.25) = 0.9878 

3) Z = $\frac{(40 - 30)}{4}$ = 2.5

Thus P (x < 40) = P (z < 2.5) = 0.9938