Ceteris LabInteractive Econometrics

Lesson 5

What does beta1 mean?

Big question

How do we turn the slope coefficient into an economic sentence?

Lesson progress

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Learning objectives

  • Explain what does beta1 mean? in plain language.
  • Use beta1 correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

The slope tells how much the predicted outcome changes when the explanatory variable increases by one unit. The units matter: if y is dollars per hour and x is years of education, the slope is dollars per hour per additional year.

Key terms

beta1
The population slope in the simple regression model.
Slope estimate
The sample estimate of beta1, usually written beta1-hat.
Unit change
A one-unit increase in the explanatory variable.
Coefficient
An estimated number multiplying a variable in a regression equation.

Estimated slope

β^1=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2\hat{\beta}_1 = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2}

The numerator measures how x and y move together. The denominator scales by variation in x.

Example

If the estimated slope is 2.33, a worker with one more year of education has a predicted wage about $2.33 higher in this sample.

Interactive visual

Slope as a one-unit change

Move the slope and watch the predicted wage change for one more year of education.

wage_sample.csv
12141618201824303612 years education, $18.50 wage16 years education, $24.20 wage18 years education, $31.80 wage14 years education, $21.10 wage16 years education, $28.40 wage13 years education, $19.70 wage20 years education, $35.60 wage15 years education, $26.90 wage14 years education, $22.30 wage17 years education, $30.10 wage12 years education, $17.90 wage18 years education, $33.40 wageEducationWage

Adjust the slope

Moving from 16 to 17 years of education changes predicted wage from $27.18 to $29.51.

Interactive activity

Slope and intercept diagram

The intercept anchors the line. The slope controls how steeply predicted y changes as x rises.

β0 interceptrise over run

Coefficient interpreter

For wage = 1.20 + 0.60 education, what does 0.60 mean?

Prediction calculator

Use wage-hat = 1.20 + 0.60 education.

Predicted wage = 1.20 + 0.60(12) = 8.40

Try it yourself

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

Common mistakes

Check these before you move on.

Always say one more unit of x is associated with how many more units of y.

Quick quiz

How should a slope of 2.33 be interpreted when y is hourly wage and x is years of education?

Quick quiz

What should always appear in a clear coefficient interpretation?

Key takeaway

The slope is the heart of simple regression interpretation: one more x, how much different is predicted y?

Download dataDownload resultsDownload interpretation