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Last Updated: Mar 27, 2024
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Normal distribution

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Prerita Agarwal
Data Specialist @
23 Jul, 2024 @ 01:30 PM


The Normal Distribution is also known as the Gaussian Distribution and Bell-Shaped Distribution. When a random experiment is replicated, the random variable that will equal the average or total result over the replicates will have a normal distribution as the number of replicates becomes large. 

Probability Density Function

For −∞<μ<∞ and σ>0, the Normal Distribution is denoted by N(μ,σ2), and its probability density is given by


σ is the Standard Deviation, and μ is the Mean.

Let’s break the formula into smaller pieces to understand what’s happening.

Z-score - It measures how many standard deviations away a data point lies from the mean.


The exp in the above probability density formula is the square of the z-score times -½. The values that are away from the mean have a lower probability than those near the mean. The values away from the mean have a higher z-score and thus a lower probability since the exp is negative. The vice-versa is for the values closer to the mean.

Now comes the 68-95-99.7 rule. The rule states that the % of values within a band around the mean in a normal distribution having a width of two, four, and six standard deviations comprise 68%, 95%, and 99.7% of all the values.




The effects of σ and μ on the Distribution are shown in the above picture.

The above f(x) is integrated into one, which can be proved by the Gaussian integral below.


Let ‘X’ = normal distributed random variable with parameters μ and σ^2. The area inside the normal distribution curve is 1 as the probability is 1.

Thus, = 1;

writing x as (x-μ ) + μ  yields

Letting y=x-μ

The first part is symmetric about the y-axis, hence its value is 0.


Expectation E[x] = μ

Variance = σ^2

Standard Deviation = .

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Standard Normal Distribution

When the mean is set to 0, and the standard deviation is set to 1 in the General Normal Distribution, the corresponding Distribution is called the Standard Normal Distribution.

The probability density function now equals:-

We do not get a closed formula by the cumulative density function of normal distribution. Thus there are pre-computed values in tables that can be used whenever needed. The values stored in this table are only for the standard Distribution. So if we want to find the cumulative probability for the general normal Distribution, we need to standardize and then compute using the value table.



If X is a normal random variable with E(X)=μ and V(X)=σ^2

the random variable is a normal random variable with E(Z)=0 and V(Z)=1. 

That is, Z is a standard normal random variable.

The expected value of a standard normal random variable X is

Expected value


Standard Deviation = 1.


  1. Are normal distributions symmetrical?
    Normal Distributions are always symmetrical. Not all symmetrical distributions are normal.
  2. Name the two common parameters of normal Distribution.
    The two common parameters of normal Distribution are:- Mean and Average
  3. What is standard normal Distribution?
    Normal Distribution having a mean 0 and standard deviation of 1 is called Standard Normal Distribution.
  4. Can Normal Distribution be discrete?
    In some cases, Normal Distribution can be used to approximate discrete data.
  5. How many standard normal distributions are there?
    There is only one standard normal Distribution.

Key Takeaways

In this article, we have extensively discussed Normal Distribution and Standard Normal Distribution. We hope this blog has helped you enhance your knowledge regarding Normal Distribution.

If you want to learn more, check out our articles on Poisson distributionProsecutor’s fallacy.

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Topics covered
Probability Density Function
Standard Normal Distribution
Key Takeaways