Normal Distribution

The normal distribution is probably the most popular probability distribution out there. It is a bell-shaped continuous distribution that is symmertric about the mean. The probability density function for the normal distribution looks as follows.

Some salient features of a normal distribution are as follows:

1. It is symmetric about the mean.

2. It follows a bell curve. Every bell curve need not be a normal distribution but, every normal distribution is a bell curve.

3. The mean, median and mode are all equal.

4. The total area under the curve is 1.

5. 68.26% of the data lies between one standard deviation from the mean.

6. 95.44% of the data lies between two standard deviations of the mean.

7. 99.73% of the data lies between three standard deviations of the mean.

An example of normal distribution is, as follows.

Take the quiz marks of students in a classroom. Most of the students tend to get a C. The second most popular grade is B and D. Getting A and F are the least likely. The plot for the marks distribution in this class would look like this.

Let me introduce a new variable named z here. z is the difference between each element in the data and the mean divided by the standard deviation. It looks like this.

An interesting feature about z is that :

E(z) = 0 (Expected value = 0)

V(Z) = 1 (Variance =1)

The process of converting the values in a column into their corresponding z values is called standardisation. Also, in a data that follows normal distribution, even the z values follow a normal distribution. So, we can say that when, X~N(mu, sigma²) this implies that the corresponding z values follow, Z~N(0,1²).