Lesson 2

Coefficients under Rescaling

Big question

What happens to a slope when x or y is divided by 100 or 1,000?

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain coefficients under rescaling in plain language.
  • Use standardized coefficient correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

If the dependent variable is divided by 1,000, the coefficient is also divided by 1,000. If an explanatory variable is divided by 1,000, the coefficient is multiplied by 1,000.

Key terms

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.
Bad control
A control variable that changes the target estimand or can introduce bias.

Core formula

pricethousands=alpha0+alpha1sqrft+alpha2bdrms+eprice_thousands = alpha_0 + alpha_1 sqrft + alpha_2 bdrms + e

Use plain-language interpretation before algebra.

Example

M6_HOUSING_LOGS is a synthetic teaching dataset for coefficients under rescaling. It is designed to practice coefficient rescaling without presenting fabricated real-world empirical findings.

Interactive visual

Use a before-and-after coefficient table to identify whether the dependent variable, the explanatory variable, or both were rescaled.

Original Module 6 visual for Coefficients under Rescaling.

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

Coefficients under Rescaling Python example

Coefficients under Rescaling 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 6Add an intercept column to the regression design matrix.
  6. Line 7Display a result so students can inspect the output.
  7. Line 8Display 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_WAGE_SCALING

Estimated time

25 to 40 min

Packages

pandas, numpy

Expected output

Printed Python results that can be compared with the lesson explanation.

Learning goals

  • Load and inspect M6_WAGE_SCALING.
  • Run the Python cells connected to Why Units Matter in Multiple Regression.
  • Interpret the output using scaling and unit conversion.

Common errors

  • File not found: check that M6_WAGE_SCALING.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_WAGE_SCALING.csv")
df.head()

Interactive activity

UnitChangePracticeTable

Read a rescaled coefficient table

Identify which variable was rescaled and rewrite the slope in original units.

M6_HOUSING_LOGS

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

What is the main mistake to avoid in Coefficients under Rescaling?

Quick quiz

What is the main mistake to avoid in Coefficients under Rescaling?

Quick quiz

Why is M6_HOUSING_LOGS a reasonable practice dataset here?

Key takeaway

Coefficients under Rescaling helps students make multiple regression more flexible while keeping interpretation precise and honest.

Download dataDownload resultsDownload interpretation