The Importance of Normal distribution :

1- Normal distribution is very useful because:
Many things actually are normally distributed, or very close to it. Forimages (2) example, height and intelligence are approximately normally distributed; measurement errors also often have a normal distribution
• The normal distribution is easy to work with mathematically. In many practical cases, the methods developed using normal theory work quite well even when the distribution is not normal.
• There is a very strong connection between the size of a sample N and the extent to which a sampling distribution approaches the normal form. Many sampling distributions based on large N can be approximated by the normal distribution even though the population distribution itself is definitely not normal.

2- Normal distribution as an Approximation to the Binomial:

The normal may also be viewed as a limiting case to the binomial , so we may
use it to approximate the value of a binomial for large n. However, because
binomial only takes values at integers, whereas the normal is a continuous
curve, we will represent an integer value with a unit-long interval centered at
that integer.