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Normal and Standard Normal Distribution

If you are new to distributions, then let us understand first about “what is a distribution?”


Distribution means summarizing the data by showing possible values that your data can have and how frequently those data values can occur.


  • When you roll out a dice, there are only 6 possible values (1, 2, 3, 4, 5, 6) that can come up.
  • When you flip a coin, there are only 2 possible values (Heads, Tails) that can come up.

Also for a dice or for a coin, each outcome has an equally-likely chance of occurrence.

When we throw a coin 10 times, 

  • we may get Heads 4 times and Tails 6 times
  • Or, we may get Heads 5 times and Tails 5 times
  • Or, we may get Heads 6 times and Tails 4 times

But the chances of coming Heads as 10 times and Tails as 0 times is very very low.

So, we can plot these outcomes and their frequency through charts and these outcome-frequency representation is called as Distribution.



Now let us extend this information and understand what is a Normal Distribution?


Normal Distribution:

Normal distribution (also called as Gaussian distribution) is a probability distribution which is symmetric about the mean and the distribution looks like a Bell shaped curve.

Bell shaped curve denotes that most of the data lies in the center and least at the edges.




In our day to day life, we have infinite number of examples where the distribution of the variable looks like a bell shape.


  • Weight of people in the world (Consider any specific age group 25 - 30)
  • Height (Consider any specific age group 15 - 20)
  • Employee salary
  • Rolling a Dice
  • Student’s Marks in exam


All these are examples where if we plot the data on the chart, the distribution will look like a bell shaped curve.


Characteristics of Normal Distribution:

  • Symmetric in nature
  • Mean, Mode and Median of the distribution are equal
  • Distribution is unimodal (has single mode)
  • Normal Distribution follows Empirical Rule


What is the Empirical Rule or 68-95-99.7% rule?

It states that for a normal distribution, 

  • 68% of the data lies within 1 standard deviation around mean
  • 95% of the data lies within 2 standard deviation around mean
  • 99.7% of the data lies within 3 standard deviation around mean
  • and the remaining proportion are the outliers which exist in the data.


What is Standard Normal Distribution?

It is the special case of Normal Distribution where the mean of the distribution is 0 and standard deviation is 1 and rest characteristics are the same.


Characteristics of Standard Normal Distribution:

  • Mean is 0 and Standard Deviation is 1
  • Symmetric in nature
  • Mean, Mode and Median of the distribution are equal
  • Distribution is unimodal (has single mode)
  • Follows Empirical Rule


Important Points for Interview:

  • All the standard Normal Distributions are Normal Distributions but all normal distributions are not Standard Normal Distributions
  • All Normal Distributions are symmetric in nature but all symmetric distributions are not normal.

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