Chi-Square Test
Test goodness of fit or independence.
Key idea
Test goodness of fit or independence.
In depth
Compares observed counts with expected counts: χ² = Σ(O − E)²/E. Used for goodness of fit and independence in contingency tables. Large values indicate a poor fit. Expected counts should be at least 5.
Worked example
Die rolled 60 times, expected 10 per face; observed 8, 12, 9, 11, 10, 10 gives χ² = 1.0, no evidence of bias.
Formula
Real-world uses
- Checking whether a die is fair or whether preference for a product depends on age group
Common mistake
Applying it to percentages instead of counts.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Chi-Square Test?
Test goodness of fit or independence. It is a topic at the expert level in Probability & Statistics.
What is the formula for Chi-Square Test?
The key formula is χ² = Σ(O−E)²/E.
Where is Chi-Square Test used in real life?
Real-world examples: Checking whether a die is fair or whether preference for a product depends on age group.
What should I learn before Chi-Square Test?
Make sure you are comfortable with Paired Tests, the previous topic in Probability & Statistics.
What is a common mistake with Chi-Square Test?
Applying it to percentages instead of counts.
How can I practise Chi-Square Test?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Chi-Square Test?
Move on to ANOVA next.
Related expert topics in Probability & Statistics
- Skewness and Kurtosis – Measures of asymmetry and tail heaviness.
- Residuals and R² – Error of a fit and proportion of variance explained.
- Simpson’s Paradox – A trend reverses when groups are combined.
- Law of Total Probability – Split an event over a partition.
- Bayes’ Theorem – Update beliefs with evidence.
- Medical Tests and Base Rates – Why positive results can still be probably false.