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Chi-Square Distribution Shapes

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About This MicroSim

Explore how the chi-square distribution's shape changes with different degrees of freedom (df). This visualization helps you understand why larger chi-square values become more extreme as df changes.

How to Use

  • Adjust the Degrees of Freedom slider to see how the distribution shape changes
  • Click Multiple DFs to compare several distributions (df = 2, 5, 10, 15, 20) side-by-side
  • Toggle Critical: ON/OFF to show or hide the right-tail critical region
  • Select different significance levels (0.01, 0.05, 0.10) to see how critical values change

Key Insights

  • Chi-square distributions are always right-skewed (no negative values possible)
  • As df increases, the distribution becomes more symmetric and shifts right
  • The mean equals the degrees of freedom (df)
  • The variance equals 2 times the degrees of freedom (2df)
  • Critical values increase as df increases (for the same alpha level)

Lesson Plan

Learning Objective

Students will compare how the chi-square distribution's shape changes with different degrees of freedom, helping them understand why larger chi-square values are more extreme (Bloom's Taxonomy: Understanding).

Warmup Activity (3 minutes)

Have students predict: "If we increase the degrees of freedom from 5 to 20, will the distribution become more or less symmetric?" Then use the simulation to verify.

Main Activity (10 minutes)

  1. Start with Single DF mode and df = 2. Note the extreme right skew.
  2. Slowly increase df using the slider, observing changes at df = 5, 10, 15, 20.
  3. Switch to Multiple DFs mode to see all curves simultaneously.
  4. Enable Critical shading and observe how critical values change.

Discussion Questions

  • Why does the chi-square distribution only have positive values?
  • Why does the distribution become more symmetric as df increases?
  • How does understanding the shape help interpret chi-square test results?

Connection to Chi-Square Tests

  • Goodness-of-fit tests: df = (number of categories) - 1
  • Independence/Homogeneity tests: df = (rows - 1)(columns - 1)
  • Larger chi-square statistics indicate greater departure from expected values