Normal Distribution: Definition and Examples

# Normal Distribution: Definition and Examples

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Vikram Singh
Assistant Manager - Content
Updated on Jun 17, 2022 10:50 IST

## Introduction:

A probability distribution is a function that gives the relationship between the outcome of a random variable in any random experiment and its probable values.

In this article, we will discuss one of the probability distributions which is commonly used in Data Science,

Normal Distribution or Gaussian Distribution.

To know more about Random Variable, read the article Introduction to Probability.

To know about other probability distributions, read the article Probability Distribution used in Data Science.

## Normal Distribution:

Normal Distribution or Gaussian Distribution (named after German mathematician Carl Friedrich Gauss) is a continuous Probability distribution,

which is symmetric about its mean value (i.e. data near the mean value are more frequently occurring).

Example:

• Height of Students in the school
• The score of the student in any exam.

Note: Normal distribution is often known as the bell-shaped curve.

Before going further let’s have an example.

Consider the experiment of Number of books read by students in a school

## Mathematical Definition:

To know more about the mean, and variance read the article on Measures of Central Tendency and Measures of Dispersion.

Example: Normal Distribution curve using Python

```# importing libraries

import numpy as np
import matplotlib.pyplot as plt
import statistics as st
from scipy.stats import norm  #norm : normal distribution function

# distribution parameters

sample_set = np.arange(-20, 20, 0.1)
#arange (start, stop, step-size): used to generate linear sequence with a constant step size
mean = st.mean(sample_set)
sd = st.stdev(sample_set)

# plot the normal distribution function with the defined mean and standard deviation

plt.plot(sample_set, norm.pdf(sample_set, mean, sd))```

## Effect of Mean and standard deviation on Normal Distribution:

### Same Mean – Different Standard Deviation

```# import libraries

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

# distribution parameters
sample_set = np.arange(-20, 20, 0.1)
#arange (start, stop, step-size): used to generate linear sequence with a constant step size

# define multiple normal distributions: same mean and different standard deviation

plt.plot(sample_set, norm.pdf(sample_set, 0, 1), label='μ: 0, σ: 1')
plt.plot(sample_set, norm.pdf(sample_set, 0, 1.5), label='μ:0, σ: 1.5')
plt.plot(sample_set, norm.pdf(sample_set, 0, 2), label='μ:0, σ: 2')

plt.legend()```

From the above, we get, if we change the standard deviation keeping the mean constant the larger standard deviation will give a flatter curve.

### Different Mean – Same Standard Deviation

```#import libraries

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

# distribution parameters

sample_set = np.arange(-20, 20, 0.1)
#arange (start, stop, step-size): used to generate linear sequence with a constant step size

#define multiple normal distributions: different mean and same standard deviation

plt.plot(sample_set, norm.pdf(sample_set, -2, 2), label='μ: -2, σ: 2')
plt.plot(sample_set, norm.pdf(sample_set, 0, 2), label='μ:0, σ: 2')
plt.plot(sample_set, norm.pdf(sample_set, 2, 2), label='μ:2, σ: 2')

plt.legend()```

From the above, we get if we change the mean, the curve will shift either on the right or the left side.

## Properties of Normal Distribution:

• Symmetric The shape of the normal distribution is perfectly symmetric about the mean i.e. the equal number of observations lies on both sides of the mean.
• Mean = Median = Mode At the center of Normal distribution, all the measures of central tendency lie.
• The total area under the curve is 1.
• Empirical Rule:

In a Normal Distribution, data is distributed in constant proportion, which is given by the Empirical rule.

The empirical rule or 68 – 95 – 97 rule or the Three sigma rule.

It states that in a Normal Distribution:

• 68% of the data will be within one Standard Deviation of the Mean
• 95% of the data will be within two Standard Deviations of the Mean
• 99.7 of the data will be within three Standard Deviations of the Mean

## Standard Normal Distribution:

It is a special case of Normal Distribution for which mean = 0 and standard deviation = 1.

For any random variable X, Standard Normal Distribution is given by:

## Conclusion:

In this article, we have discussed about one of the most important probability distribution Normal Distribution , with examples in python.

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Vikram Singh
Assistant Manager - Content

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio