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Normal Distribution : An Ultimate Guide

Aashi
Aashi Last Updated : 08 Jun, 2021
6 min read

This article was published as a part of the Data Science Blogathon

Introduction

The normal distribution is an important class of Statistical Distribution that has a wide range of applications. This distribution applies in most Machine Learning Algorithms and the concept of the Normal Distribution is a must for any Statistician, Machine Learning Engineer, and Data Scientist.

So, In this article, we will explore all the concepts about Normal Distribution in a detailed manner.

Table of Contents

1. What is the Normal distribution?

2. Why is Normal Distribution Important?

3. Parameters of Normal Distribution

  • Mean
  • Standard Deviation

4. Properties of Normal Distribution

  • Symmetricity
  • Measures of Central Tendencies are equal
  • Empirical Rule
  • Skewness and Kurtosis
  • The area under the curve

5. Distributions Functions for the Normal Curve

  • Probability Density function (PDF)
  • Cumulative Density Function (CDF)

6. Applications of Normal Distribution

What is Normal Distribution?

The Normal distribution is also known as Gaussian or Gauss distribution. Many groups follow this type of pattern. That’s why it’s widely used in business, statistics, and in government bodies like the FDA:

  • Heights of people.
  • Measurement errors.
  • Blood pressure.
  • Points on a test.
  • IQ scores.
  • Salaries.

Let’s understand the concept with the help of the following example:

For Example,

The Normal distribution curve is seen in some of the competitive tests like the SAT, UPSC, JEE-Advanced, and GRE, etc. Here the normal distribution indicated that the bulk of students will score the average marks (grade=C), while smaller numbers of students will score the grades (B or D), and a smaller number of students score an F or an A grade.

All these types of inferences are taken from the empirical formula of Normal Distribution, which we will discuss later in this article.

Why is Normal Distribution Important?

There are several reasons why the normal distribution is crucial in statistics. Some of those are as follows:

1. The statistical hypothesis test assumes that the data follows a normal distribution.

2. Both linear and non-linear regression assumes that the residual follows the normal distribution.

3. Moreover, the central limit theorem states that as the sample size increases the distribution of the mean follows normal distribution irrespective of the distribution of the original variable

4. Apart from this most of the statistical software programs support some of the probability functions for normal distribution as well.

 

Parameters of Normal Distribution

There are two main parameters of a normal distribution- the mean and standard deviation. With the help of these parameters, we can decide the shape and probabilities of the distribution wrt our problem statement. As the parameter value changes, the shape of the distribution changes.

1. Mean

  • Researchers used the mean or average value as a measure of central tendency. It can be used to describe the distribution of variables that are measured as ratios or intervals.
  • The mean determines the location of the peak, and most of the data points are clustered around the mean in a normal distribution graph.
  • If we change the value of the mean, then the curve of normal distribution moves either to the left or right along the X-axis.

Normal distribution | means and standard deviation

Image Source: Google Images

 

2. Standard Deviation

  • The standard deviation measures how the data points are dispersed relative to the mean.
  • It determines how far the data points are away from the mean and represents the distance between the mean and the data points.
  • The standard deviation defines the width of the graph. As a result, changing the value of standard deviation tightens or expands the width of the distribution along the x-axis.
  • Usually, a smaller standard deviation wrt to the mean results in a steep curve while a larger standard deviation results in a flatter curve.

Normal distribution | same means- different distribution

Image Source: Google Images

 

Properties of Normal Distribution