## What is Normal Distribution Formula?

A distribution is said to be following a normal distribution when it is symmetric i.e. positive values and the negative values of the distribution can be divided into equal halves. It is known as the bell curve as it takes the shape of the bell. One of the most important characteristics of a normal curve is, it is

If a distribution is normally distributed than the mean, median and mode of the distribution will be equal. A normal distribution has two tails one is known as the right tail and the other one is known as the left tail. For determining the normal distribution of a variable it is important to know its mean and its variance.

A normal distribution formula for the calculation can be represented as

**X ~ N (µ, α)**

Where

- N= no of observations
- µ= mean of the observations
- α= standard deviation

In most of the cases, the observations do not reveal much in its raw form. So it is very important to standardize the observations in order to be able to compare that. It is done with the help of the z-score. It is required to calculate the Z-score for an observation.

The equation for Z Score Calculation for the normal distribution is represented as follows,

**Z=(X- µ)/ α**

Where

- Z= Z-score of the observations
- µ= mean of the observations
- α= standard deviation

### Explanation

A distribution is said to be following a normal distribution when the distribution follows a bell curve. It is known as the bell curve as it takes the shape of the bell. A distribution is normal when it has certain characteristics in it. One of the most important characteristics of a normal curve is, it is symmetric which means the positive values and the negative values of the distribution can be divided into equal halves. Another very important characteristic of the variable being normally distributed is that the observations will be within 1 standard deviation of the mean 90% of the time. The observations will be two standard deviations from the mean 95% of the time and it will be within three standard deviations from the mean 99% of the time.

### Examples

#### Example #1

**Let’s understand the concept of normal distribution with the help of an example. The mean of the weights of a class of students is 65kg and the standard of the weight is .5 kg. If we assume that the distribution of the return is normally distributed than let us interpret the distribution for the weight of the students in the class**.

When a distribution is normally distributed then 68% of the distribution lies within 1 standard deviation, 95% of the distribution lies within 2 standard deviations and 99% of the distribution lies with 3 standard deviations.

Given,

- The mean return for the weight will be 65 kgs
- Standard deviation will be 3.5 kgs

So, 68% of the time the value of the distribution will be in the range as below,

**Upper Range = 65+3.5= 68.5****Lower Range = 65-3.5= 61.5****Each tail will (68%/2) = 34%**

#### Example #2

**Let’s continue with the same example. The mean of the weights of a class of students is 65kg and the standard of the weight is 3.5 kg. If we assume that the distribution of the return is normally distributed than let us interpret the distribution for the weight of the students in the class.**

When a distribution is normally distributed then 68% of the distribution lies within 1 standard deviation, 95% of the distribution lies within 2 standard deviations and 99% of the distribution lies with 3 standard deviations.

Given,

- The mean return for the weight will be 65 kgs
- Standard deviation will be 3.5 kgs

So, 95% of the time the value of the distribution will be in the range as below,

**Upper Range =65+(3.5*2)= 72****Lower Range = 65-(3.5*2)= 58****Each tail will (95%/2) = 47.5%**

#### Example #3

**Let’s continue with the same example. The mean of the weights of a class of students is 65kg and the standard of the weight is 3.5 kg. If we assume that the distribution of the return is normally distributed than let us interpret the distribution for the weight of the students in the class.**

When a distribution is normally distributed then 68% of the distribution lies within 1 standard deviation, 95% of the distribution lies within 2 standard deviations and 99% of the distribution lies with 3 standard deviation

Given,

- The mean return for the weight will be 65 kgs
- Standard deviation will be 3.5 kgs

So, 99% of the time the value of the distribution will be in the range as below,

**Upper Range = 65+(3.5*3)= 75.5****Lower Range = 65-(3.5*3)= 54.5****Each tail will (99%/2) = 49.5%**

### Relevance and Use

The normal distribution is a very important statistical concept, as most of the random variables in the world of finance are distributed normally. Normal distribution plays an important part in constructing portfolios. Apart from finance a lot of real-life parameters are found to be following a normal distribution. Like for example if we try to find the height of students in a class or the weight of the students in a class, the observations are distributed normally. Similarly, the marks of an exam also follow a normal distribution. It helps to normalize marks in an exam if most of the students scored below the passing marks by setting a limit of saying only those failed who scored below two standard deviations.

### Recommended Articles

This has been a guide to the Normal Distribution Formula. Here we will do the calculation of normal distribution along with the practical examples and downloadable excel sheet. You can learn more about financing from the following articles –

- POISSON.DIST Function
- Formula of Z Score
- Poisson Distribution Formula
- Hypergeometric Distribution
- What is Standard Normal Distribution Formula?
- Create Lognormal Distribution in Excel
- Create a Normal Distribution Graph in Excel
- What is the Binomial Distribution Formula?

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