Introduction to Normal Distribution in Statistics

A continuous random variable has an infinite number of values that can be represented by an interval on the number line.  It’s probability distribution  is called a continuous probability distribution.  In this article, we will be understanding the most important continuous probability distribution in statistics, the normal distribution.  
A normal distribution is a continuous probability distribution for a random variable, x.  The graph of a normal distribution is called the normal curve.  A normal distribution has the following properties. 
1. The mean, median and mode are equal. 
2. The normal curve is bell-shaped and is symmetric about the mean.
3. The total area under the normal curve is equal to 1.
4. The normal curve approaches, but never touches the x-axis as it extends farther and farther away from the mean.
5. Between m - s and m + s (in the center of the curve) the graph curves downward.  The graph curves upward to the left of m - s and to the right of m + s.  The points at which the curve changes from curving upward to curving downward are called inflection points. 
6. A normal distribution can have any mean and any positive standard deviation.  These two parameters, m and s completely determine the shape of a normal curve.  The mean gives the location of the line of symmetry and the standard deviation describes how much the data are spread out. 
See the line of symmetry for each?  That’s the mean.  However, if it is fatter, then the standard deviation is greater.  That’s the difference.  

Understanding Mean & Standard Deviation
Which normal curve has a greater mean?
Which normal curve has a greater standard deviation
The line of symmetry of curve A occurs at x = 15.  The line of symmetry of curve B occurs at x = 12.  So, curve A has a greater mean.
Curve B is more spread out than curve A, so curve B has a greater standard deviation. 

The Empirical Rule
In a normal distribution with mean  m and standard deviation s, you can approximate areas under the normal curve as follows:
1. About 68% of the area lies between m - s and m + s
2. About 95% of the area lies between 
m - 2s and m + 2s
3. About 99.7% of the area lies between 
m - 3s and m + 3s