Lesson 3

Probability Limits and the Law of Large Numbers

Big question

Why do sample averages become population information?

Lesson progress

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain probability limits and the law of large numbers in plain language.
  • Use finite sample properties correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

A probability limit is the value a statistic approaches as the sample grows. The law of large numbers explains why averages stabilize.

Key terms

finite sample properties
Properties that hold at a fixed sample size under stated assumptions.
inconsistency
Failure to converge to the true parameter.
asymptotic t statistic
A standardized coefficient statistic with an approximate normal reference distribution.

Core formula

n−1sumxi−>E(x);n−1sumxi∗ui−>Cov(x,u)n^{-1} sum x_i -> E(x); n^{-1} sum x_i*u_i -> Cov(x,u)

Use plain-language interpretation before algebra.

Example

Probability Limits and the Law of Large Numbers uses simulated data so students can see the large-sample mechanism without inventing empirical results.

Interactive visual

ProbabilityLimitExplorer

Original Module 5 visual for Probability Limits and the Law of Large Numbers.

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

Probability Limits and the Law of Large Numbers Python example

Probability Limits and the Law of Large Numbers 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 3Load a Python library needed for data work or regression.
  4. Line 5Run this Python instruction as part of the lesson workflow.
  5. Line 6Create or update a Python object used in the analysis.
  6. Line 7Create or update a Python object used in the analysis.
  7. Line 8Run this Python instruction as part of the lesson workflow.
  8. Line 9Create or update a Python object used in the analysis.
  9. Line 10Run this Python instruction as part of the lesson workflow.
  10. Line 11Create or update a Python object used in the analysis.
  11. Line 12Display a result so students can inspect the output.
  12. Line 13Create or update a Python object used in the analysis.
  13. Line 14Create or update a Python object used in the analysis.
  14. Line 15Run this Python instruction as part of the lesson workflow.
  15. Line 16Run this Python instruction as part of the lesson workflow.
  16. Line 17Run this Python instruction as part of the lesson workflow.
  17. Line 18Run this Python instruction as part of the lesson workflow.
  18. Line 19Run this Python instruction as part of the lesson workflow.

Python walkthrough

  1. 1Load libraries and data or set a simulation seed.
  2. 2Build the model or simulation that matches the lesson question.
  3. 3Compute the statistic, graph, or summary table.
  4. 4Interpret the result as large-sample evidence, not automatic causality.

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

SIMULATION

Estimated time

25 to 40 min

Packages

pandas, numpy, scipy

Expected output

Simulation output showing how estimates or test statistics behave as sample size changes.

Learning goals

  • Load and inspect SIMULATION.
  • Run the Python cells connected to What Consistency Means.
  • Interpret the output using consistency and simulation.

Common errors

  • File not found: check that WAGE1.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-5/WAGE1.csv")
df.head()

Interactive activity

ProbabilityLimitExplorer

Explore probability limits

Move the sample size and compare a sample average with a population target.

Simulation

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

Why do probability limits appear in OLS consistency arguments?

Quick quiz

Which reporting habit is most important in Probability Limits and the Law of Large Numbers?

Quick quiz

Why is BWGHT a reasonable practice dataset here?

Key takeaway

Probability Limits and the Law of Large Numbers helps students separate large-sample approximation from valid research design.

Download dataDownload resultsDownload interpretation