Interview Prep
Career
GenAI
Prompt Engg
ChatGPT
LLM
Langchain
RAG
AI Agents
Machine Learning
Deep Learning
GenAI Tools
LLMOps
Python
NLP
SQL
AIML Projects

Reading list

Machine Learning Basics for a Newbie
6 Steps of Machine learning Lifecycle Introduction to Predictive Modeling
Introduction to Exploratory Data Analysis & Data Insights Descriptive Statistics Inferential Statistics How to Understand Population Distributions?
Reading Data Files into Python Different Variable Datatypes
Probability for Data Science Basic Concepts of Probability Axioms of Probability Conditional Probability
Central Tendencies for Continuous Variables Spread of Data KDE plots for Continuous Variable Overview of Distribution for Continuous variables Normal Distribution Skewed Distribution Skeweness and Kurtosis Distribution for Continuous Variable
Central Tendencies for Categorical Variables Understanding Discrete Distributions Performing EDA on Categorical Variables
Dealing with Missing Values Understanding Outliers Identifying Outliers in Data Outlier Detection in Python Outliers Detection Using IQR, Z-score, LOF and DBSCAN
Sample and Population Central Limit Theorem Confidence Interval and Margin of Error
Bivariate Analysis Introduction
Covariance Pearson Correlation Spearman's Correlation & Kendall's Tau Correlation versus Causation Tabular and Graphical methods for Bivariate Analysis Performing Bivariate Analysis on Continuous-Continuous Variables
Tabular and Graphical methods for Continuous-Categorical Variables Introduction to Hypothesis Testing P-value Two sample Z-test T-test T-test vs Z-test Performing Bivariate Analysis on Continuous-Catagorical variables
Chi-Squares Test Bivariate Analysis on Categorical Categorical Variables
Multivariate Analysis A Comprehensive Guide to Data Exploration The Data Science behind IPL
Supervised Learning vs Unsupervised Learning Reinforcement Learning Generative and Descriminative Models Parametric and Non Parametric model
Machine Learning Pipeline Preparing Dataset Build a Benchmark Model: Regression Build a Benchmark Model: Classification
Evaluation Metrics for Machine Learning Everyone should know Confusion Matrix Accuracy Precision and Recall AUC-ROC Log Loss R2 and Adjusted R2
Dealing with Missing Values Replacing Missing Values Imputing Missing Values in Data Working with Categorical Variables Working with Outliers Preprocessing Data for Model Building
Understanding Cost Function Understanding Gradient Descent Math Behind Gradient Descent Assumptions of Linear Regression Implement Linear Regression from Scratch Train Linear Regression in Python Implementing Linear Regression in R Diagnosing Residual Plots in Linear Regression Models Generalized Linear Models Introduction to Logistic Regression Odds Ratio Implementing Logistic Regression from Scratch Introduction to Scikit-learn in Python Train Logistic Regression in python Multiclass using Logistic Regression How to use Multinomial and Ordinal Logistic Regression in R ? Challenges with Linear Regression Introduction to Regularisation Implementing Regularisation Ridge Regression Lasso Regression
Introduction to K Nearest Neighbours Determining the Right Value of K in KNN Implement KNN from Scratch Implement KNN in Python
Bias Variance Tradeoff Introduction to Overfitting and Underfitting Visualizing Overfitting and Underfitting Selecting the Right Model What is Validation? Hold-Out Validation Understanding K Fold Cross Validation
Introduction to Feature Selection Feature Selection Algorithms Missing Value Ratio Low Variance Filter High Correlation Filter Backward Feature Elimination Forward Feature Selection Implement Feature Selection in Python Implement Feature Selection in R
Introduction to Decision Tree Purity in Decision Tree Terminologies Related to Decision Tree How to Select Best Split Point in Decision Tree? Chi-Squares Information Gain Reduction in Variance Optimizing Performance of Decision Tree Train Decision Tree using Scikit Learn Pruning of Decision Trees
Introduction to Feature Engineering Feature Transformation Feature Scaling Feature Engineering Frequency Encoding Automated Feature Engineering: Feature Tools
Introduction to Naive Bayes Conditional Probability and Bayes Theorem Introduction to Bayesian Adjustment Rating: The Incredible Concept Behind Online Ratings! Working of Naive Bayes Math behind Naive Bayes Types of Naive Bayes Implementation of Naive Bayes
Understanding how to solve Multiclass and Multilabled Classification Problem Evaluation Metrics: Multi Class Classification
Introduction to Ensemble Techniques Basic Ensemble Techniques Implementing Basic Ensemble Techniques Finding Optimal Weights of Ensemble Learner using Neural Network Why Ensemble Models Work well?
Introduction to Stacking Implementing Stacking Variants of Stacking Implementing Variants of Stacking Introduction to Blending Bootstrap Sampling Introduction to Random Sampling Hyper-parameters of Random Forest Implementing Random Forest Out-of-Bag (OOB) Score in the Random Forest IPL Team Win Prediction Project Using Machine Learning Introduction to Boosting Gradient Boosting Algorithm Math behind GBM Implementing GBM in python Regularized Greedy Forests Extreme Gradient Boosting Implementing XGBM in python Tuning Hyperparameters of XGBoost in Python Implement XGBM in R/H2O Adaptive Boosting Implementing Adaptive Boosing LightGBM Implementing LightGBM in Python Catboost Implementing Catboost in Python
Different Hyperparameter Tuning methods Implementing Different Hyperparameter Tuning methods GridsearchCV RandomizedsearchCV Bayesian Optimization for Hyperparameter Tuning Hyperopt
Understanding SVM Algorithm SVM Kernels In-depth Intuition and Practical Implementation SVM Kernel Tricks Kernels and Hyperparameters in SVM Implementing SVM from Scratch in Python and R
Introduction to Principal Component Analysis Steps to Perform Principal Compound Analysis Computation of Covariance Matrix Finding Eigenvectors and Eigenvalues Implementing PCA in python Visualizing PCA A Brief Introduction to Linear Discriminant Analysis Introduction to Factor Analysis
Introduction to Clustering Applications of Clustering Evaluation Metrics for Clustering Understanding K-Means Implementation of K-Means in Python Implementation of K-Means in R Choosing Right Value for K Profiling Market Segments using K-Means Clustering Hierarchical Clustering Implementation of Hierarchial Clustering DBSCAN Defining Similarity between clusters Build Better and Accurate Clusters with Gaussian Mixture Models
Understand Basics of Recommendation Engine with Case Study
8 Ways to Improve Accuracy of Machine Learning Models
Introduction to Dask Working with CuML
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to AutoML Implementation of MLBox Introduction to PyCaret TPOT Auto-Sklearn EvalML
Pickle and Joblib Introduction to Model Deployment
Deploying Machine Learning Model using Streamlit Deploying ML Models in Docker Deploy Using Streamlit Deploy on Heroku Deploy Using Netlify Introduction to Amazon Sagemaker Setting up Amazon SageMaker Using SageMaker Endpoint to Generate Inference Deploy on Microsoft Azure Cloud Introduction to Flask for Model Deploying ML model using Flask
Model Deployment in Android Model Deployment in Iphone

Empirical Rule in Gaussian distribution in Statistics

B
Bhanu Last Updated : 31 Jul, 2021
6 min read

This article was published as a part of the Data Science Blogathon

Introduction

This article is based on the very famous and important rule, the rule of 68-95-99.7% rule or Empirical rule which is a very beautiful and powerful application of Normal Distribution or Gaussian Distribution. With the help of this rule, you can answer the data distribution without knowing too many things about the data. But before we start learning about this amazing concept we should know about the basic concept of probability and statistics such as what is distribution, mean, variance, pdf,  CDF, and much more things. So without wasting our time let’s just start it.

What are Data Distributions in Probability and statistics?

One of the most important ideas in probability and statistics is the concept of Distributions. Distributions are shown us the nature of random data, it shows us how the data is spread in the given minimum and maximum value of a random variable called range. Data Distributions are nothing but the statical functions that give us the important intuitions about the data and give the various values that are very useful for making a crucial decision about the data. In other words, we can say that the probability distribution is a statical function that gives us all possible information and values that a random variable can consist of. The distribution of the data contains a range, the statical distribution functions produce the results between the range, the range is nothing but the minimum and maximum values of the random variable can have.
If we talk about the example of data distributions then we found that our nature is full of distributions any feature that exists in this nature is probably follows some type of distribution. Let’s talk about the heights of the human being if we just measure the heights of random 50 people then we found that the heights of random people follow some kind of distribution. and now if we want to gather information about the heights of people then we will use the probability distribution.

Why we use Distributions in Probability and Statistics:

So the question is why we are using these distributions in probability and statistics. The intuition of using these distributions is very straightforward, we simply want to know the answer to various questions about the data or random variable. And here comes the two amazing fields of mathematics called probability and statistics. Probability and statistics have the strength to give the answers to our questions using some simple calculations.

Statistics and probability provide us the power to know about the data without even looking at the data. So probability and statistics provide us the various types of distributions such as Gaussian (aka Normal), Bernoulli, Binominal, Log-Normal, Perito distribution, etc. If we just know that the data or random variable follows any of the distribution available in probability and statistics we can answer tons of questions related to the data, that’s why we use distributions in probability and statistics. 

Mean, Variance, Pdf, and CDF:

Before we start learning about the Empirical rule we need to understand the concept of Mean, variance, pdf, and CDF because without knowing these terms we cant imagine the distributions so let’s learn about these topics:

Mean: Mean is nothing but the average of all data points. We simply calculate the mean by dividing the sum of all data points by the number of all data points and it is denoted by μ. Mean it gives us the central tendency of data points.

                The formula of Mean is as follows:

Empirical Rule mean

created in paint

Where xi is each dada point in random variable X, and n is the number of data points.

Variance: Variance is nothing but the average squared distance between each data point to the mean value.  variance tells about the spread of our distribution. in other words the variance tells us how far the data point is spread from the mean value. the variance is denoted by the symbol σ2.

               The formula of Variance is as follows:

variance Empirical Rule

https://getcalc.com/formula/statistics/variance-population-data.png

        and if we take the square root of the variance σ2 Then it’s become a Standard Deviation which is denoted by σ .

Pdf ( Probability Density Function ): Pdf is the smooth form of histograms which, pdf is defined as the density of data points on the particular value of the random variable.  in other words, pdf is the measure of the density of data points in the given range of the data.

Empirical Rule pdf
http://reliawiki.org/images/thumb/4/46/3.3.png/400px-3.3.png

In the above figure, the density of points or the counts of the points between the range a and b. By looking at the figure we literally can say that there are more data points of random variable lies between the range of a and b.

CDF ( Cumulative Distribution Function ): The cumulative distribution function is the method to describe the random variable. This function shows us that how much % of data points lie in the range of 0 to a given point. Or in other words, it tells us what is the location of our given data point on the 100% distribution of data.

Empirical Rule cdf
.ipynb notebook screenshot

In the above figure, the blue line is called Pdf and the orange line is called CDF, the value of CDF is always lie between 0 to 1. if we draw the perpendicular on the x-axis then there is a point that exists on the graph somewhere cut the CDF function, and if we perpendicularly join the cross point of CDF to the y axis then you find the probability of data point of the x-axis on the y axis. for example, if your value of the x-axis is 1.5, then if you follow the above method of drawing perpendicular, then you will find that there are 60% of points exist in the distribution which value is less than or equal to 1.5.

 68-95-99.7 % Rule or Empirical Rule: 

We get to see this rule under the Normal or Gaussian distribution. whenever a data or random variable follows the normal distribution, then we can apply this rule to the data. So let’s get to know a little bit about the Gaussian distribution.

Gaussian distribution is symmetric distribution. This means if we draw the Probability Density Function ( Pdf ) of normal distribution then the Pdf of both sides of the mean value will be the mirror image of each other. The Pdf of the Gaussian distribution is a bell-shaped curve that is symmetric.

Let’s assume that X is a random variable and follows Gaussian distribution with some mean (μ) and Standard deviation (σ)

gaussian dist

created in paint

So in the gaussian distribution, inside the bell curve the central point is our mean value, and if we just add standard deviation to the mean (μ + σ) and subtract standard deviation to the mean (μ – σ), then the range we get is called the range of first standard deviation. similarly, if we add 2 * standard deviation to the mean ( μ + 2σ) and subtract 2 * standard deviation to the mean (μ – 2σ) then we got the range of second standard deviation.

gaussian
https://theeducationlife.com/wp-content/uploads/2020/07/image-638×400.png

So according to the Empirical rule, if a random variable follows Gaussian distribution then it has also three properties, and these properties are also called the Empirical formula or 68-95-99.8 % formula, and the three properties of the Empirical formula are as follows:

 1. P [ μ – σ <= X <= μ + σ ] ≈ 68 %

So the first formula basically says that the probability of a variable that falls within the range of μ – σ and μ + σ is 68 %. which means 68 % of the data points belonging to the random variable X fall within the range of the first standard deviation. 

 

2. P [ μ – 2σ <= X <= μ + 2σ ] ≈ 95 %

So the second formula basically says that the probability of a variable that falls within the range of μ – 2σ and μ + 2σ is 95 %. which means 95% of the data points belonging to the random variable X fall within the range of the second standard deviation.

 

3. P [ μ – 3σ <= X <= μ + 3σ ] ≈ 99.8 %

So the second formula basically says that the probability of a variable that falls within the range of μ – 3σ and μ + 3σ is 99.8 %. which means 99.8% of the data points belonging to the random variable X fall within the range of the third standard deviation.

99.8
https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/1920px-Standard_deviation_diagram.svg.png

 

So I hope you understand the idea of Empirical rule,

Thank you.

The media shown in this article on Empirical Rule are not owned by Analytics Vidhya and are used at the Author’s discretion.
B
Bhanu

Free Courses

Generative AI 4.7

Generative AI - A Way of Life

Explore Generative AI for beginners: create text and images, use top AI tools, learn practical skills, and ethics.
Large Language Models 4.5

Getting Started with Large Language Models

Master Large Language Models (LLMs) with this course, offering clear guidance in NLP and model training made simple.
Prompt Engineering 4.6

Building LLM Applications using Prompt Engineering

This free course guides you on building LLM apps, mastering prompt engineering, and developing chatbots with enterprise data.
LlamaIndex 4.8

Improving Real World RAG Systems: Key Challenges & Practical Solutions

Explore practical solutions, advanced retrieval strategies, and agentic RAG systems to improve context, relevance, and accuracy in AI-driven applications.
RAG systems using LlamaIndex 4.7

Microsoft Excel: Formulas & Functions

Master MS Excel for data analysis with key formulas, functions, and LookUp tools in this comprehensive course.

Recommended Articles

Responses From Readers

Clear

Write for us

Write, captivate, and earn accolades and rewards for your work
  • Reach a Global Audience
  • Get Expert Feedback
  • Build Your Brand & Audience
  • Cash In on Your Knowledge
  • Join a Thriving Community
  • Level Up Your Data Science Game
imag

We use cookies essential for this site to function well. Please click to help us improve its usefulness with additional cookies. Learn about our use of cookies in our Privacy Policy & Cookies Policy.

Show details