# Normal Distribution Examples and Solutions

Normal distribution can also be known as Gaussian distribution. In statistics, the normal distributions are used to represent real-valued random variables with unknown distributions. Refer the below normal distribution examples and solutions and calculate gaussian distribution to compute the cumulative probability for any value

## Gaussian Distribution Examples

###### Example 1:

Let us consider the Gaussian distribution example: The time taken to assemble a car in a certain plant is a random variable having a normal distribution of 30 hours and a standard deviation of 4 hours. What is the probability that a car can be assembled in a period of time greater than 21 hours?

###### Solution:

We can calculate the normal distribution using the given formula.

#### Formula :

Z = (X - m) / ρ Where, m = Mean σ = Standard Deviation X = Normal Random Variable

Substituting the values in the formula,

Gaussian Distribution Z = (21 - 30) / 4 = - 2.25 P(x > 21) = P(z > -2.25) Looking up the z-score in the z-table, we get 1 – 0.0122 = 0.9878

Therefore, the value of Normal Distribution is 0.9878.

###### Example 2:

Refer the below Gaussian distribution worked example. A large group of students took a test in Physics and the final grades have a mean of about 70 and a standard deviation of 10. If we can approximate the distribution of these grades by a normal distribution, what percent of the students should fail the test (i.e) less than 60?

###### Solution:

Substituting the values in the above given formula,

Normal Distribution Z = (60 - 70) / 10 z = -1 P(x < 60) = P(z < -1) Looking up the z-score in the z-table, we get 1 - 0.8413 = 0.1587

Therefore, Normal Distribution is 0.1587.