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Normal Distribution Calculator
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Normal Distribution Calculator is also known as Gaussian Distribution Calculator, is used to find the Normal Probability density function for the given standard deviation and mean. Normal Distribution is a continuous probability distribution also known as Gaussian function and the curve related with it is called as Bell Curve.

Normal Distribution Definition
The most important continuous probability function in statistics Normal Distribution or Gaussian Distribution, which was innovated by Gauss to find the errors in immense observations. By which the Bell curved shape is called as Gaussian function or Bell curve is used to map the statistical distribution of probability density function.
Normal Distribution Formula
f(x) = $\frac{1}{\sqrt{2 \pi \sigma^{2}}}$ $e^{\frac{(x-\mu)^2}{2\sigma^{2}}}$

where,
$\mu$ = Mean,
$\sigma$ = Standard Distribution.
If mean($\mu$) = 0 and standard deviation($\sigma$) = 1, then this distribution is known to be normal distribution.
x = Normal random variable.

You can see a default mean, standard deviation with its minmum and maximum values given below. Click on "Calculate", it will calculate the z-value using the formula $z=\frac{x-\mu}{\sigma}$ and find the area under the curve for z-value. Normal distribution is calculated by subtracting upper limit area from lower limit area.

How to Calculate Normal Distribution

Step 1 :

Observe the value of given mean, standard deviation and limits.

Step 2 :

Find the z-value for the respective random variable by using the formula

z = $\frac{x - \mu}{\sigma}$

Where,

x = Normal random variable

$\mu$ = mean

$\sigma$ = standard deviation.

Step 3 :

Now, find the area under the standard normal curve for the z-value obtained from step2 by using Z-table.

Step 4 :

Subtract area of upper limit from area of lower limit gives the normal probabilty density function.

Normal Distribution Problems

1. If mean($\mu$) = 40 and standard deviation($\sigma$) = 5, then find the P(40 $\leq$ x $\leq$ 45), where x is the normal random variable.

Step 1 :

Given mean($\mu$) = 40 and standard deviation($\sigma$) = 5

P(40 $\leq$ x $\leq$ 45) = ?

Step 2 :

For x = 40, z = $\frac{x - \mu}{\sigma}$

z = $\frac{40 - 40}{5}$ = 0

For x = 45, z = $\frac{x - \mu}{\sigma}$

z = $\frac{45 - 40}{5}$ = 1

Step 3 :

P(40 $\leq$ x $\leq$ 45) = P(0 $\leq$ x $\leq$ 1) = [area to the left of z = 1] - [area to the left of z = 0]

Step 4 :

P(0 $\leq$ x $\leq$ 1) = [area to the left of z = 1] - [area to the left of z = 0]
$\Rightarrow$ 0.8413 - 0.5
$\Rightarrow$ 0.3413

P(40 $\leq$ x $\leq$ 45) = 0.3413

2. If mean($\mu$) = 20 and standard deviation($\sigma$) = 4, then find the P(20 $\leq$ x $\leq$ 25), where x is the normal random variable.

Step 1 :

Given mean($\mu$) = 20 and standard deviation($\sigma$) = 4

P(20 $\leq$ x $\leq$ 25) = ?

Step 2 :

For x = 20, z = $\frac{x - \mu}{\sigma}$

z = $\frac{20 - 20}{4}$ = 0

For x = 25, z = $\frac{x - \mu}{\sigma}$

z = $\frac{25 - 20}{4}$ = 1.25

Step 3 :

P(20 $\leq$ x $\leq$ 25) = P(0 $\leq$ x $\leq$ 1.25) = [area to the left of z = 1.25] - [area to the left of z = 0]

Step 4 :

P(0 $\leq$ x $\leq$ 1.25) = [area to the left of z = 1.25] - [area to the left of z = 0]
$\Rightarrow$ 0.8944 - 0.5
$\Rightarrow$ 0.3944

P(20 $\leq$ x $\leq$ 25) = 0.3944