Normal Distribution Formula Step By Step Calculations

Normal Distribution Formula

The normal distribution is symmetric, i.e., one can divide the positive and negative values of the distribution into equal halves; therefore, the 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 represents as follows:

X ~ N (µ, α)

Where

N= no of observationsµ= mean of the observationsα= standard deviation

In most cases, the observations do not reveal much in their raw form. So, it is essential to standardize the observations to compare that. One can do it with the help of the z-score formulaZ-score FormulaThe Z-score of raw data refers to the score generated by measuring how many standard deviations above or below the population mean the data, which helps test the hypothesis under consideration. In other words, it is the distance of a data point from the population mean that is expressed as a multiple of the standard deviation.read more. It is required to calculate the Z-score for an observation.

The equation for Z-score calculation for the normal distribution represents as follows:

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Z= Z-score of the observationsµ= mean of the observationsα= standard deviation

Explanation

A distribution is normal when it follows a bell curveBell CurveBell Curve graph portrays a normal distribution which is a type of continuous probability. It gets its name from the shape of the graph which resembles to a bell. read more. 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 that it is symmetric, which means one can divide the positive and negative values of the distribution into equal halves. Another essential characteristic of the variable 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 within three standard deviations from the mean 99% of the time.

Examples

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 distribution is normalDistribution Is NormalNormal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays a key role in assets return calculation and in risk management strategy.read more, then 68% of it lies within 1 standard deviation, 95% lies within 2 standard deviations, and 99% lies within 3 standard deviations.

Given,

The mean return for the weight will be 65 kgsThe standard deviation will be 3.5 kgs

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

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

Example #2

Let’s continue with the same example. The mean of weight of a class of students is 65kg, and the standard of 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.

The mean return for the weight will be 65 kgThe standard deviation will be 3.5 kg

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

Upper Range =65+(3.52)= 72 Lower Range = 65-(3.52)= 58Each 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 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.

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

Upper Range = 65+(3.53)= 75.5Lower Range = 65-(3.53)= 54.5Each 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 to be following such a distribution. For example, if we try to find the height of students in a class or the weight of the students, the observations are normally distributed. Similarly, the marks of an exam also follow the same distribution. It helps to normalize marks in an exam if most students score below the passing marks by setting a limit of saying only those who failed who scored below two standard deviations.