Gaussian distribution is a bell-shaped curve, it follows the normal distribution with the equal number of measurements right side and left side of the mean value. Mean is situated in the center of the curve, the right side values from the mean are greater than mean value and the left side values from mean are smaller than the mean. It is used for mean, median, and mode for continuous values. You all know the basic meaning of mean, median, and mod. The mean is an average of the values, the median is the center value of the distribution and the mode is the value of the distribution which is frequently occurred. In the normal distribution, the values of mean, median, and are all same. If the values generate skewness than it is not normally distributed. The normal distribution is very important in statistics because it fits for many occurrences such as heights, blood pressure, measurement error, and many numerical values.

A gaussian and normal distribution is the same in statistics theory. Gaussian distribution is also known as a normal distribution. The curve is made with the help of probability density function with the random values. F(x) is the PDF function and x is the value of gaussian & used to represent the real values of random variables having unknown distribution.

There is a property of Gaussian distribution which is known as Empirical formula which shows that in which confidence interval the value comes under. The normal distribution contains the mean value as 0 and standard deviation 1.

The empirical rule also referred to as the three-sigma rule or 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ). Broken down, the empirical rule shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).

Python code for plotting the gaussian graph:

import matplotlib.pyplot as plt
import numpy as np
import scipy.stats as stats
import math

mu = 0
variance = 1
sigma = math.sqrt(variance)
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)
plt.plot(x, stats.norm.pdf(x, mu, sigma)) 

The above code shows the Gaussian distribution with 99% of the confidence interval with a standard deviation of 3 with mean 0.

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