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 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

Dispersion of Data : Range, IQR, Variance, Standard Deviation

s
suvarna Last Updated : 29 Apr, 2021
6 min read
This article was published as a part of the Data Science Blogathon.

Introduction   

Dispersion of data used to understands the distribution of data. It helps to understand the variation of data and provides a piece of information about the distribution data. Range, IOR, Variance, and Standard Deviation are the methods used to understand the distribution data.

Dispersion of data helps to identify outliers in a given dataset. 

Overview 

In this article, I’ll cover the following questions related to the dispersion of data.

  1. Importance of a range.
  2. Why IQR preferred across the range?
  3. Why IQR accept data sorted in ascending order, not in descending order?
  4. Impact on outliers when data is sort in ascending order and descending order.
  5. Why does standard deviation accept over the mean absolute deviation?
  6. Why do we use n-1 in
    the denominator of standard deviation? Or the concept of degree of freedom.

Range

The range is the easiest dispersion of data or measure of variability. The range can measure by subtracting the lowest value from the massive Number. The wide range indicates high variability, and the small range specifies low variability in the distribution. To calculate a range, prepare all the values in ascending order, then subtract the lowest value from the highest value.

 .

         Range = Highest_value  – Lowest_value

 

Dispersion of Data range

The range of marks is 18.

The range can influence by outliers. If there is one extreme value that can change the value of a range.

Dispersion of Data range eg

 

Interquartile Range (IQR)

IQR is a range (the boundary between the first and second quartile) and Q3 (the boundary between the third and fourth quartile).IQR is preferred over a range as, like a range, IQR does not influence by outliers. IQR is used to measure variability by splitting a data set into four equal quartiles.

Dispersion of Data iqr
Dispersion of Data iqr example

IQR uses a box plot to find the outliers. ”To estimating IQR, all the values form (sort) in the ascending order else it will provide a negative value, and that influences to find the outliers.”

 

Formula to find outliers

        [Q1 – 1.5 * IQR, Q3 + 1.5 * IQR]

If the value does not fall in the above range it considers outliers.

Dispersion of Data outliers formula

 

Variance

Variance is a simple measure of dispersion. Variance measures how far each number in the dataset from the mean. To compute variance first, calculate the mean and squared deviations from a mean.

Population variance

variance formula

Sample variance 

sample variance Dispersion of Data

Observation near to mean value gets the lower result and far from means gets higher value.

Description of a variance formula with example.

Dispersion of Data example

Standard Deviation

Standard deviation is a squared root of the variance to get original values. Low standard deviation indicates data points close to mean.

The normal distribution is conventional bits of help to understand the standard deviation.

Standard Deviation

X indicates the mean value

68 % of values lie within 1 standard deviation.

95 % of values lies within 2 standard deviation.

99.7 % of values lie within 3 standard deviation.

 

SD

Specification of Standard deviation and Mean Absolute deviation (Why SD is more reliable than MAD) by an example.

Suppose we have 4 values are 5, 7, 9, and 3.

1. Calculate the Mean :

 

mean

2. To compute variance, first need to subtract the mean from all observations to find the distance of all values from the average (mean).

 

mean deviation

We noticed the result as zero because whenever we discussed the mean, the identity properties of a sample mean “deviation of a means is always sum to zero. “

Therefore we have to work on it. We don’t want negative values (as distance cannot be negative). We need something that secures negative values into positive numbers. We can practice either Mean absolute deviation or squared deviation (Standard deviation).

Mean absolute deviation (MAD):
MAD is the measure of spread (variability).

 

MAD

Here | gives the absolute value that means all negative deviation (distance) made positive.

The problem with MAD is that it is not well behaved as squared deviation dose in a normal distribution, and it is not differential. It is most salutary to use squared deviation than MAD as squared deviation has well mathematical properties for the normal distribution.

Squared Difference

 

Squared diffference

Squaring amplifies the effect of massive differences. Variance gives added weight to the values that impact outliers (the numbers that are far from
the mean and squaring of these numbers can skew the data like 10 square is 100, and 100 square is 10,000 ) to overcome the drawback of variance standard deviation came into the picture.

Standard deviation uses the square root of the variance to get original values.

 

SD 2

Now, you may have one question why do we use n-1 in the denominator of sample variance.

A population gives a true mean, and a sample statistic is an approximation population parameter which means a population mean is already known. The sample mean estimates the population means, and sample variance s2 estimates population variance. “ 10 different samples gives 10, different variances”. 

n-1 means the degree of from where 1, is we lost 1 degree of freedom.

Degree of Freedom (DOF): DOF means how many values are free to vary.

Example:

Suppose we draw four independent observations from a population where is the population mean is known μ = 80.

DOF

For the 4th value, we can place any number that will not impact the result as the population provides the true mean.

Therefore in the population, four numbers are free to vary.

When we talk about the sample mean, one property of sample mean is “Deviation from the sample means always summing to zero.”

 

sample mean

Suppose we have the same situation but, μ is unknown (as in sample means, we estimate the population means), and find the sample mean = 80.

sammple ,ean

Here for observation 4th, we cannot place either value as we have to satisfy sample mean property so that we have to add the value for the observation 4th that will sum to zero. Therefore we can state that for observation 4th, we lost one degree of freedom.

To satisfy the sample mean deviation property, we can place 84 so that 84-80 will return 4, and following that, when we combine all the values that sum to zero.

sample mean deviation

Summary of the article, the range is a difference between a large number and a small number. The range can influence by an outlier. In IQR, all the numbers should arrange in an ascending order else it will impact outliers. Variance uses squaring that can create outliers, and to overcome this drawback, we use standard deviation. In the denominator, n-1 indicates the degree of freedom (how many values are free to vary).

It is good practice to use standard deviation over MAD as standard deviation has a well mathematical property in the normal distribution.

The media shown in this article on Dispersion of Data are not owned by Analytics Vidhya and is used at the Author’s discretion. 

blogathondispersion of datastatistics
s
suvarna

Passionate about machine learning

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

NAGARAJU KAVYASRI
NAGARAJU KAVYASRI

Useful material thank you

123

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
Congratulations, You Did It!
Well Done on Completing Your Learning Journey. Stay curious and keep exploring!
Outlier Detection & Removal | How to Detect & Remove Outliers (Updated 2024)
Outlier Detection & Removal | How to Detect & Remove Outliers (Updated 2024)
Get Started with Statistics for Data Science
Get Started with Statistics for Data Science

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