Normal Distribution Formula

What is a Normal Distribution?

The normal distribution is the most significant probability distribution in statistics as it is suitable for various natural phenomena such as heights, measurement of errors, blood pressure, and IQ scores follow the normal distribution. The other names for the normal distribution are Gaussian distribution and the bell curve.


The normal distribution is a probability distribution that outlines how the values of a variable are distributed. Most of the observations clusters around the central peak and probabilities for values far away from mean fall off equally in both the directions in a normal distribution. Extreme values in both the tails of the distribution are uniformly unexpected.


The spread of the normal distribution is managed by the standard deviation . The smaller the standard deviation value in a normal distribution formula, the more concentrated the data.


The normal probability distribution formula is given as:


\[P (x) = \frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x - \mu)^{2}}{2 \sigma^{2}}}\]


In the above normal probability distribution formula.

μ is the mean of the data.

σ is the standard deviation of data.

x is the normal random variable.

If (μ) = 0 and standard normal deviation is equal to 1, then distribution is said to be a normal distribution.


What is the Probability Distribution Formula?

There are generally two types of probability distribution formulas in a probability distribution. The two types of probability distribution formulas are normal probability distribution formulas (also known as the Gaussian distribution formulas) and binomial distribution formulas.


The value that assigns a probability to each outcome of a random experiment is considered as a probability distribution equation. In other words, it assigns the probability for each value of the random variable.


Probability Distribution Formula

The two types of probability distribution formulas are:

  • The normal probability distribution formula.

  • The binomial Probability Distribution Formula.

Normal Probability Distribution Formula

The normal distribution is also known as the Gaussian distribution and it denotes the equation or graph which are bell-shaped.


The normal probability distribution formula is given by:


\[P (x) = \frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x - \mu)^{2}}{2 \sigma^{2}}}\]


In the above normal probability distribution formula.

μ is the mean of the data.

σ is the standard deviation of data.

x is the normal random variable.


Binomial Probability Distribution Formula

These are the probabilities that appear when the event consists of n repeated trials and the results of each trial may or may not appear.


The binomial probability distribution formula is stated below:

P ( r out of n) = n!/ r!(n-r)!

pr (1-p) n-r = n Cr

pr (1-p) n-r


In the above binomial distribution formula,

N is the total number of events

r is the total number of successful events

p is the probability of success on each trial.

nCr = n!/r!(n-r)!

1- p = probability of failure.


What is a Lognormal Distribution?

In probability theory, a log-normal distribution is considered as the continuous probability distribution of a random variable whose logarithm follows the pattern of normal distribution. It represents phenomena whose relative growth rate is independent of the size, which is true for most of the natural phenomena including the size of tissue and blood pressure, income distribution and then the length of the chess.


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Lognormal Distribution Formula

Some of the lognormal distribution formulas are given below:

The lognormal distribution formula for mean is given as

m = eμ + σ²/2

Which implies that μ can be calculated from m:

m = In m – 1/2 σ²

The above both equations are derived from the mean of the normal distribution.


The lognormal distribution formula for the median is given as

Med [X] = eμ

The above equation is derived by placing the cumulative distribution values equal to 0.5 and solving the resulting equation

The lognormal distribution formula for mode is given as

Mode [X] = eμ - σ²

The above equation is derived by placing the probability distribution function values equal to 0 as the mode denotes the global maximum of distribution.

The lognormal distribution formula for variance is given as:

Var  [X] = (eσ² -1) e2μ + σ²,

Which can also be represented as (eσ² -1) m2 , where m denotes the mean of the distribution.


Gaussian Distribution Formula

Gaussian distribution is a common continuous probability distribution. The Gaussian Distribution Formulas are significant in statistics and are primarily used in natural and social science to denote real-valued random variables.


The probability density function formula for Gaussian distribution formula is given as:


\[f( x,\mu, \sigma) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x - \mu)^{2}}{2 \sigma^{2}}}\]


In the above Gaussian Distribution Formula,

μ is the mean of the data.

σ is the standard deviation of data.

x is the normal random variable.


Standard Normal Distribution

The standard normal distribution is a type of normal distribution. It mostly appears when a normal random variable has a mean value equal to 0 and value of standard deviation is equal to 1. The random variable of a standard normal distribution is considered as a standard score or z-score. Each normal random variable such as X can easily be converted into a z-score using the normal distribution z formula.


\[z = \frac{(X - \mu)}{\sigma}\]


In the above normal distribution z formula,

X is a normal random variable.

Μ is mean of data.

σ is the standard deviation of the data.


Solved Examples

1. Calculate the probability of normal distribution with the population mean 2, standard deviation 3 or random variable 5.


Solution:

x = 5

Mean = μ = 2

Standard Deviation = σ = 3


We will solve the questions with the help of the above normal probability  distribution formula:

\[P (x) = \frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x - \mu)^{2}}{2 \sigma^{2}}}\]

\[P (x) = \frac{1}{\sqrt{2 \times 3.14 \times 3^{2}}} e^{-\frac{(5 - 2)^{2}}{2 \times 3^{2}}}\]

= \[\frac{1}{\sqrt{56.52}} e^{-\frac{9}{2 \times 9}}\]

= \[\frac{1}{7.518} e^{-\frac{1}{2}}\]

= \[\frac{0.6065}{7.518}\]

= \[0.0807\]


2. An average electric bulb lasts for 300 days with a standard deviation of 50 days. Assume that bulb life is normally distributed, what is the probability that the electric bulb will last at most 365 days.


Solution : 

Given, Mean score = 300 days

Standard deviation = 50 days

We need to determine the cumulative probability that bulb life is less than or equals to 365 days.


We have the following information:

The  normal random variable value is given as 365 days.

The mean is equivalent to 300 days.

The standard deviation is equivalent to 50 days.

Hene, the answer of the question is P (x < 365) = 0.90.

It states that there is a 90% probability that an electric bulb will burn out within 365 days.


Quiz Time

1. The shape of the normal curve is

  1. Bell shape

  2. Flat

  3. Circular

  4. Spiked

2. The area under a standard normal curve is

  1. 0

  2. 1

  3. Not defined

FAQ (Frequently Asked Questions)

1. What are the Common Properties for all Forms of Normal Distribution?

Ans: In Spite of different shapes, all forms of normal distribution have the following characteristics properties.

  • The mean, median, and mode values are equal.

  • Half of the population is less than mean and half is greater than mean.

  • They are all symmetric. The normal distribution cannot model skewed distribution.

  • The Empirical rule regulates the proportion of values that lies within a certain distance from the mean. 

  • The total area under the curve is 1. 

  • The bell curve is symmetric at the center ( i.e. around the mean, μ).

2. Why is the Normal Distribution Important?

Ans: The normal distribution uses a continuous probability distribution that is symmetrical on a shape on both sides of the mean, so the right side of the image is a mirror image of the left side. The area under the normal distribution curve denotes probability and the total area under the curve is equal to one. The normal distribution is the most important probability distribution in statistics because various continuous data and psychology in nature exhibit this bell -shaped curve when collected and graphed.


For example, if we randomly sample 100 individuals, we would expect to see a normal distribution curve of various continuous variables such as IQ, height, weight, and blood pressure.