*This article was written by Jim Frost from Minitab. He came to Minitab with a background in a wide variety of academic research. His role was the “data/stat guy” on research projects that ranged from osteoporosis prevention to quantitative studies of online user behavior. Essentially, his job was to design the appropriate research conditions, accurately generate a vast sea of measurements, and then pull out patterns and meanings from it. *

After you have fit a linear model using regression analysis, ANOVA, or design of experiments (DOE), you need to determine how well the model fits the data. To help you out, Minitab statistical software presents a variety of goodness-of-fit statistics. In this post, you will explore the R-squared (R2 ) statistic, some of its limitations, and uncover some surprises along the way. For instance, low R-squared values are not always bad and high R-squared values are not always good!

Linear regression calculates an equation that minimizes the distance between the fitted line and all of the data points. Technically, ordinary least squares (OLS) regression minimizes the sum of the squared residuals.

In general, a model fits the data well if the differences between the observed values and the model's predicted values are small and unbiased.

Before you look at the statistical measures for goodness-of-fit, you should check the residual plots. Residual plots can reveal unwanted residual patterns that indicate biased results more effectively than numbers. When your residual plots pass muster, you can trust your numerical results and check the goodness-of-fit statistics.

R-squared is a statistical measure of how close the data are to the fitted regression line. It is also known as the coefficient of determination, or the coefficient of multiple determination for multiple regression.

The definition of R-squared is fairly straight-forward; it is the percentage of the response variable variation that is explained by a linear model. Or:

R-squared = Explained variation / Total variation

R-squared is always between 0 and 100%:

- 0% indicates that the model explains none of the variability of the response data around its mean.
- 100% indicates that the model explains all the variability of the response data around its mean.

In general, the higher the R-squared, the better the model fits your data. However, there are important conditions for this guideline that I’ll talk about both in this post and my next post.

Plotting fitted values by observed values graphically illustrates different R-squared values for regression models.

The regression model on the left accounts for 38.0% of the variance while the one on the right accounts for 87.4%. The more variance that is accounted for by the regression model the closer the data points will fall to the fitted regression line. Theoretically, if a model could explain 100% of the variance, the fitted values would always equal the observed values and, therefore, all the data points would fall on the fitted regression line.

To learn more about regression analysis, click here.

**Top DSC Resources**

- Article: Difference between Machine Learning, Data Science, AI, Deep Learnin...
- Article: What is Data Science? 24 Fundamental Articles Answering This Question
- Article: Hitchhiker's Guide to Data Science, Machine Learning, R, Python
- Tutorial: Data Science Cheat Sheet
- Tutorial: How to Become a Data Scientist - On Your Own
- Tutorial: State-of-the-Art Machine Learning Automation with HDT
- Categories: Data Science - Machine Learning - AI - IoT - Deep Learning
- Tools: Hadoop - DataViZ - Python - R - SQL - Excel
- Techniques: Clustering - Regression - SVM - Neural Nets - Ensembles - Decision Trees
- Links: Cheat Sheets - Books - Events - Webinars - Tutorials - Training - News - Jobs
- Links: Announcements - Salary Surveys - Data Sets - Certification - RSS Feeds - About Us
- Newsletter: Sign-up - Past Editions - Members-Only Section - Content Search - For Bloggers
- DSC on: Ning - Twitter - LinkedIn - Facebook - GooglePlus

Follow us on Twitter: @DataScienceCtrl | @AnalyticBridge

© 2019 Data Science Central ® Powered by

Badges | Report an Issue | Privacy Policy | Terms of Service

**Most Popular Content on DSC**

To not miss this type of content in the future, subscribe to our newsletter.

**Technical**

- Free Books and Resources for DSC Members
- Learn Machine Learning Coding Basics in a weekend
- New Machine Learning Cheat Sheet | Old one
- Advanced Machine Learning with Basic Excel
- 12 Algorithms Every Data Scientist Should Know
- Hitchhiker's Guide to Data Science, Machine Learning, R, Python
- Visualizations: Comparing Tableau, SPSS, R, Excel, Matlab, JS, Pyth...
- How to Automatically Determine the Number of Clusters in your Data
- New Perspectives on Statistical Distributions and Deep Learning
- Fascinating New Results in the Theory of Randomness
- Long-range Correlations in Time Series: Modeling, Testing, Case Study
- Fast Combinatorial Feature Selection with New Definition of Predict...
- 10 types of regressions. Which one to use?
- 40 Techniques Used by Data Scientists
- 15 Deep Learning Tutorials
- R: a survival guide to data science with R

**Non Technical**

- Advanced Analytic Platforms - Incumbents Fall - Challengers Rise
- Difference between ML, Data Science, AI, Deep Learning, and Statistics
- How to Become a Data Scientist - On your own
- 16 analytic disciplines compared to data science
- Six categories of Data Scientists
- 21 data science systems used by Amazon to operate its business
- 24 Uses of Statistical Modeling
- 33 unusual problems that can be solved with data science
- 22 Differences Between Junior and Senior Data Scientists
- Why You Should be a Data Science Generalist - and How to Become One
- Becoming a Billionaire Data Scientist vs Struggling to Get a $100k Job
- Why do people with no experience want to become data scientists?

**Articles from top bloggers**

- Kirk Borne | Stephanie Glen | Vincent Granville
- Ajit Jaokar | Ronald van Loon | Bernard Marr
- Steve Miller | Bill Schmarzo | Bill Vorhies

**Other popular resources**

- Comprehensive Repository of Data Science and ML Resources
- Statistical Concepts Explained in Simple English
- Machine Learning Concepts Explained in One Picture
- 100 Data Science Interview Questions and Answers
- Cheat Sheets | Curated Articles | Search | Jobs | Courses
- Post a Blog | Forum Questions | Books | Salaries | News

**Archives**: 2008-2014 | 2015-2016 | 2017-2019 | Book 1 | Book 2 | More

**Most popular articles**

- Free Book and Resources for DSC Members
- New Perspectives on Statistical Distributions and Deep Learning
- Time series, Growth Modeling and Data Science Wizardy
- Statistical Concepts Explained in Simple English
- Machine Learning Concepts Explained in One Picture
- Comprehensive Repository of Data Science and ML Resources
- Advanced Machine Learning with Basic Excel
- Difference between ML, Data Science, AI, Deep Learning, and Statistics
- Selected Business Analytics, Data Science and ML articles
- How to Automatically Determine the Number of Clusters in your Data
- Fascinating New Results in the Theory of Randomness
- Hire a Data Scientist | Search DSC | Find a Job
- Post a Blog | Forum Questions

## You need to be a member of Data Science Central to add comments!

Join Data Science Central