# Normal Distribution Formula  ## Normal Distribution Formula

Normal distribution is a distribution that is symmetric i.e. positive values and the negative values of the distribution can be divided into equal halves and therefore, mean, median and mode will be equal. It has two tails one is known as the right tail and the other one is known as the left tail.

The 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 essential to standardize the observations to be able to compare that. It is done with the help of the z-score formula. 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

For eg:
Source: Normal Distribution Formula (wallstreetmojo.com)

Where

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

### Explanation

A distribution is normal when it follows a . 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 symmetric, which means the positive values and the negative values of the distribution can be divided into equal halves. Another essential characteristic of the variable being 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

You can download this Normal Distribution Formula Excel Template here – Normal Distribution Formula Excel Template

#### Example #1

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 normal, then let us interpret for the weight of the students in the class.

When a , then 68% of it lies within 1 standard deviation, 95% lies within 2 standard deviations, and 99% lies with 3 standard deviations.

Given,

• The mean return for the weight will be 65 kgs
• The 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 normal, then let us interpret it for the weight of the students in the class.

Given,

• The mean return for the weight will be 65 kgs
• The 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 normal, then let us interpret it for the weight of the students in the class.

Given,

• The mean return for the weight will be 65 kgs
• The 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 an essential statistical concept as most of the random variables in finance follow such a curve. It plays an important part in constructing portfolios. Apart from finance, a lot of real-life parameters are found to be following such a 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 the same distribution. It helps to normalize marks in an exam if most students scored below the passing marks by setting a limit of saying only those failed who scored below two standard deviations.

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