Normal Distribution

A Simple Explanation - By Varsha Saini

Normal Distribution / Gaussian Distribution

A Random Variable (X) having mean (\mu) and standard deviation (\sigma) is said to be Normally Distributed if it has the following properties:

Properties of Normal Distribution

  • The mean, median and mode are all at the centre point.
  • There is no skewness.
  • It has a kurtosis of 3.
  • It follows Bell Curve.
  • It is symmetrical on both sides of the mean.
  • 50% data lies before the mean and the other 50% is on the right side of the mean.
  • The data near the mean is more frequent in occurrence than the data far from it.

Empirical Rule

  • If you go one standard deviation to the left and one standard deviation to the right, it covers 68% of the total data.
  • If you go two standard deviations to the left and two standard deviations to the right, it covers 95% of the total data.
  • If you go three standard deviations to the left and three to the right, it covers 99.7% of the total data.

Standard Normal Distribution

Standard Normal Distribution is a particular type of Normal Distribution where the mean is 0 and the standard deviation is 1.

How to convert Normal Distribution into Standard Normal Distribution?

Normal Distribution can be converted to Standard Normal Distribution by using a z-score. This process is also called Standardization.

z-score = 


  •   = mean
  • = standard deviation
  • = current value

Central Limit Theorem

The Normal Distribution is key to the central limit theorem. According to it, if multiple samples are taken from a population then the distribution of their sample means will follow a normal distribution as the sample size increases.

Applications of Normal Distribution

  • All kind of variables in nature is nearly normally distributed like height, weight, job satisfaction, body temperature, strength etc.
  • The central limit theorem is based on the concept of normal distribution.
  • The assumption behind statistical tests is that data follows a normal distribution.
  • If a variable is not normally distributed, it can be converted into normal distribution by a simple transformation.

You can check other probability distributions here.