Lesson 13

Adjusted R-Squared and Model Size

Big question

Why can R-squared rise when a variable adds little value?

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain adjusted r-squared and model size in plain language.
  • Use precision control correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

Ordinary R-squared never falls when a regressor is added. Adjusted R-squared adds a penalty for model size, so it can fall when a new variable is not useful enough.

Key terms

Precision control
A safe control that can reduce residual variation and improve precision.
Standardized coefficient
A slope measured in standard deviation units for both the explanatory variable and the dependent variable.
Turning point
The value of x where the fitted quadratic slope is zero.

Core formula

adjustedR2=1−(SSR/(n−k−1))/(SST/(n−1))adjusted R^2 = 1 - (SSR/(n-k-1))/(SST/(n-1))

Use plain-language interpretation before algebra.

Example

M6_POLICY_CONTROLS is a synthetic teaching dataset for adjusted r-squared and model size. It is designed to practice model fit without presenting fabricated real-world empirical findings.

Interactive visual

Add candidate controls and watch R-squared, adjusted R-squared, and interpretability move in different directions.

Original Module 6 visual for Adjusted R-Squared and Model Size.

wage_sample.csv

y variable

wage

The dependent variable. It is the outcome students want to explain.

x variable

education

The explanatory variable. It is used to describe changes in wage.

Live Python

Adjusted R-Squared and Model Size Python example

Adjusted R-Squared and Model Size Python example

Stdout

Run Python to see results here.

Status / stderr

Ready to run Python in your browser.

Line-by-line guide

  1. Line 1Load a Python library needed for data work or regression.
  2. Line 2Load a Python library needed for data work or regression.
  3. Line 4Load the dataset into a pandas DataFrame.
  4. Line 5Create or update a Python object used in the analysis.
  5. Line 6Run this Python instruction as part of the lesson workflow.
  6. Line 7Run this Python instruction as part of the lesson workflow.
  7. Line 8Run this Python instruction as part of the lesson workflow.
  8. Line 9Run this Python instruction as part of the lesson workflow.
  9. Line 10Run this Python instruction as part of the lesson workflow.
  10. Line 11Add an intercept column to the regression design matrix.
  11. Line 12Display a result so students can inspect the output.

Python walkthrough

  1. 1Load the synthetic teaching dataset from the Module 6 public data folder.
  2. 2Create transformed variables only after checking their meaning and valid support.
  3. 3Fit a regression that matches the lesson's interpretation target.
  4. 4Print coefficient or prediction summaries that students can connect to the formula.
  5. 5Use comments and output labels so no empirical result is presented without context.

Live notebook

Run this lesson as a notebook

Open an editable notebook cell-by-cell, run Python in the browser, and download the `.ipynb` file for later.

Open live notebookDownload .ipynbDownload .py

Related dataset

M6_POLICY_CONTROLS

Estimated time

25 to 40 min

Packages

pandas, numpy, statsmodels, patsy

Expected output

A regression or inference table with coefficients, uncertainty, and short interpretation notes.

Learning goals

  • Load and inspect M6_POLICY_CONTROLS.
  • Run the Python cells connected to Adjusted R-Squared and Model Size.
  • Interpret the output using adjusted R-squared and model comparison.

Common errors

  • File not found: check that M6_POLICY_CONTROLS.csv is installed or use the course data folder.
  • Package import error: use the browser notebook first, then download for local Jupyter if your local packages differ.
  • Column name error: compare your variable names with the dataset variables listed for this notebook.

Dataset path helper

import pandas as pd

df = pd.read_csv("/data/module-6/M6_POLICY_CONTROLS.csv")
df.head()

Interactive activity

AdjustedRSquaredPenaltyLab

Watch adjusted R-squared respond

Add variables and compare R-squared with adjusted R-squared.

M6_POLICY_CONTROLS

Inputs

Try it yourself

Write one plain-English sentence explaining the main idea from this lesson.

Common mistakes

Check these before you move on.

A regression coefficient describes a pattern unless the assumptions or research design support a causal interpretation.

Quick quiz

Which interpretation is most careful for Adjusted R-Squared and Model Size?

Quick quiz

What is the main mistake to avoid in Adjusted R-Squared and Model Size?

Quick quiz

Why is M6_WAGE_SCALING a reasonable practice dataset here?

Key takeaway

Adjusted R-Squared and Model Size helps students make multiple regression more flexible while keeping interpretation precise and honest.

Download dataDownload resultsDownload interpretation