Do You Understand Normal Distribution

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Analytics Vidhya

4 min readJun 15, 2021

The Normal or Gaussian Distribution is Quite Famous in Statistics . As a Data scientist you might came across this distribution . It has Lots of properties, which people use to estimate some probability or sometime get some information.

In this article we will go through some questions over Normal Distribution in order to see if we understand it well enough

Here are some Question before reading this article further try to answer these question and check if you are right or not

What do you mean by Distribution?
Can you tell me what is Normal Distribution ?
What are the Parameter we required in order to get the Gaussian Distribution?
What is the PDF of Normal Distribution?
Can you explain the PDF and CDF plot of Normal Distribution?
Can you tell me what is empirical rule?
How we can get Standard Normal Variate from Normal Distribution ?

After Reading this Blog you will able to answer all the above questions

Distribution is — the possible value a variable can take and How frequently they are occurring.

For example :

You throw a die 5 time, what is the possible value a die can take :{1,2,3,4,5,6} , How frequently they are occurring : lets say {1:2,2:1,3:0,4:1,5:0,6:1}

We mainly have 2 kind of distribution

continuous probability distributions
discrete probability distribution

i will not go depth of it.

Normal Distribution or Gaussian Distribution

is a Continuous Probability Distribution , where it take Real valued random variable .

In order to get a Normal distribution we need 2 thing / parameter

mean (Average value)
variance (measure of spread / How much the data vary from mean)
[mean and variance should be finite]

Notation

Now we can use the mean and variance to get the PDF(Probability Density Function).The formula for Normal Distribution PDF is

PDF at a given point gives the probability density not the probability at that point itself

you don’t have to remember the formula although some time people ask about it.

When you have some mean and standard deviation you can easily plot Normal Distribution .Its look something like the below picture . Its bell curved shape.

Some properties of Gaussian distribution (from the pdf we can see)

It is symmetric (both side same from the mean)
As you move from mean it will reduce exponentially we can see from the formula
Depending upon the mean and variance the plot will vary from one another . As variance increasing the plot is getting fatter and as variance decreasing the plot is getting thinner

Now The CDF of a continuous random variable can be expressed as the integral of its probability density function (PDF) . We have our PDF of normal Distribution above . so we can take that and plot CDF

it is ranges from 0–1

The cumulative distribution function (CDF) calculates the cumulative probability for a given x-value .

if i take value 2 from x-axis , i can read the CDF like there are equal or less than 80% data from the whole data present at random variable 2.

Empirical rule or 68–95–99.7 rule

It is a very simple and informative rule : the rule say 68% of the data present at 1 standard deviation , 95% of the data present at 2 standard deviation and 97.7% of data present at 3 standard deviation.