Normal Distribution Formula

In probability and statistics, the normal distribution or Gaussian distribution or bell curve is one of the most important continuous probability distributions. The normal distribution is defined as the probability density function f(x) for the continuous random variable, say x, in the system. A normal distribution is a very important statistical data distribution pattern occurring in many natural phenomena, such as height, blood pressure, lengths of objects produced by machines, etc. Here, we are going to discuss the normal distribution formula and examples in detail.

Normal Distribution Formula

For a random variable x, with mean “μ” and standard deviation “σ”, the probability density function for the normal distribution is given by:

Normal Distribution Formula:

\(\begin{array}{l}f(x)= \frac{1}{\sqrt{2\pi \sigma ^{2}}}e^{\frac{-(x-\mu )^{2}}{2\sigma ^{2}}}\end{array} \)

Where 

μ = Mean

σ = Standard deviation

x = Normal random variable

Solved Example on Normal Distribution Formula

Example: 

Find the probability density function for the normal distribution where mean = 4 and standard deviation = 2 and x = 3.

Solution:

Given:

Mean,μ = 4

Standard deviation, σ = 2

Random variable, x = 3.

We know that the normal distribution formula is:

\(\begin{array}{l}f(x)= \frac{1}{\sqrt{2\pi \sigma ^{2}}}e^{\frac{-(x-\mu )^{2}}{2\sigma ^{2}}}\end{array} \)

Now, substitute the values in the formula, we get

\(\begin{array}{l}f(x)= \frac{1}{\sqrt{2\pi (2)^{2}}}e^{\frac{-(3-4 )^{2}}{2(2)^{2}}}\end{array} \)

\(\begin{array}{l}f(x)= \frac{1}{2\sqrt{2\pi }}e^{\frac{-1}{8}}\end{array} \)

\(\begin{array}{l}f(x)= 0.19947 \times e^{-0.125}\end{array} \)

\(\begin{array}{l}f(x)= 0.19947 \times 0.882496\end{array} \)

\(\begin{array}{l}f(x)= 0.17603\end{array} \)

Therefore, the probability density function for the normal distribution is 0.17603.

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Frequently Asked Questions on Normal Distribution Formula

What is the standard normal distribution?

The distribution is said to be a standard normal distribution if the mean is equal to zero and the standard deviation is equal to 1.

What is the normal distribution formula?

For a random variable x, with mean “μ” and standard deviation “σ”, the normal distribution formula is given by: f(x) = (1/√(2πσ2)) (e[-(x-μ)^2]/2σ^2).

Mention two characteristics of normal distribution.

The two characteristics of the normal distribution are:
The mean, median, and mode are equal.
The normal distribution is unimodal and symmetric.

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