Normal Distribution

What is it ?

It's one of the most popular forms of distribution in statistics also known as Gaussian, Gauss or Laplace-Gauss distribution), curved like a bell, hence the name of bell curve.

The Gaussian distribution is defined by 2 parameters σ (the standard deviation of the distribution) and μ (the mean of the distribution).

Characteristics of normal distribution :

• The majority of the data is centered around the mean.
• The mean determine the location of the distribution and the standard deviation determines how flat and wide the distribution is.
• Normal distribution is symmetric ; the area on the left is equal to the area on the right.
• The mean is equal to the mode and it's equal to the median.
• The area under the curve is equal to 1
• The law of 68.3,95,99.7 : 68% pf the events are in the middle(1 std from the mean),95% of the events are 2std away from the mean and finnaly 99.7% pf the events are away from the mean.

Laplace-Gauss distribution PDF :

• f(x) : probability density function
• σ : standard deviation
• μ = mean

This formula allows us to draw our normal distribution using the mean and the standard deviation. For example X～N(10,2) ; 10 is the mean and 2 is the standard deviation

Our normal distribution would be equal to :

Let's take an example to see where do we use it

Example :

Photo by Philip Myrtorp on Unsplash

You have a business appointment to schedule, however you don't know the time when you should set the appointment, taking into account that you must take a plane and then a taxi to attend the appointment.

You know that flying from point A to B provides the following distribution:

From the distribution, you know that it is most likely that the time needed to reach your destination would be between 3.5h to 4.5h with most likely the flight would be at 4 hours, basing on the flights flown. ,and that you have a low chance of arriving in 2.5 or 5 hours and more, so you decide to schedule your appointment at 4.5 hours after your shift so as not to arrive late!

You can see in the graph that the average is 4 hours with an approximate standard deviation (σ) of 0.5 hours.

In the next article we will explore the standard normal distribution as well as how to determine the probabilities, what would be the chances that the trip would only take 3 hours? ??

Let's build it on Python

from scipy.stats import norm
import numpy as np
import matplotlib.pyplot as plt

# PDF

# We need x , mu and sigma for normal distribution


def normal_dis_pdf(x, mu, sigma):
? ? return (1 / (sigma * np.sqrt(2 * np.pi))) * np.exp((-1 / 2)* ((x- mu / sigma) ** 2))


# Let's test it
# x will be a np.array, you can also do it with a single value
# μ would be 0 and σ is equal to 1


x = np.array([1,2,3])


# You can also just do , no need to hard code
norm.pdf(x,loc=0,scale=1)


norm.pdf(x)
# By default loc is equal to 0 and scale is equal to 1


# Let's do some visualization
# We will have values that start from -7 to 7 with 10000 values


x = np.linspace(-7,7,10000)


fig, ax = plt.subplots(figsize=(20,8))
ax.plot(x,normal_dis_pdf(x,0,1))
plt.show()

# CDF

norm.cdf(x,loc=0,scale=1)


fig, ax = plt.subplots(figsize=(20,8))
ax.plot(x,norm.cdf(x,0,1))
plt.show()