Answer First
The Chi‑Square Goodness‑of‑Fit Test checks whether one categorical variable follows a specific distribution. The Chi‑Square Test of Independence checks whether two categorical variables are related. Both use the same chi‑square formula, but they answer different questions and use different expected counts.
Problem Setup
1. Goodness‑of‑Fit (GOF)
Used for one categorical variable with k categories.
\[ H_0: \text{The data follow a specified distribution.} \]
Expected counts come from the hypothesized probabilities:
\[ E_i = n p_i \]
2. Test of Independence
Used for two categorical variables arranged in an r×c contingency table.
\[ H_0: \text{The two variables are independent.} \]
Expected counts come from row and column totals:
\[ E_{ij} = \frac{(\text{row total})(\text{column total})}{n} \]
Step-by-Step Solution
1. They answer different questions
- GOF: Does one variable follow a known distribution?
- Independence: Are two variables associated?
2. They use different data structures
- GOF: One list of counts.
- Independence: A full contingency table.
3. They compute expected counts differently
- GOF: Expected = n × hypothesized probability.
- Independence: Expected = (row total × column total) / n.
4. They have different degrees of freedom
- GOF: df = k − 1 − (# estimated parameters).
- Independence: df = (r − 1)(c − 1).
5. They use the same chi‑square statistic
\[ \chi^2 = \sum \frac{(O – E)^2}{E} \]
But the meaning of O and E differs between tests.
Intuition
The Goodness‑of‑Fit test asks whether one variable matches a known pattern. The Test of Independence asks whether two variables interact. Same math, different purpose.
Common Exam Mistakes
- Using GOF when the problem has two variables.
- Using Independence when the problem gives a hypothesized distribution.
- Computing expected counts incorrectly.
- Forgetting to adjust df when parameters are estimated.
Why This Matters
Chi‑square tests are used in marketing, HR analytics, operations, A/B testing, and survey analysis. Choosing the wrong test leads to incorrect conclusions about customer behavior, employee patterns, or market segments.
Final Summary
The Chi‑Square Goodness‑of‑Fit Test checks whether one categorical variable follows a specified distribution. The Chi‑Square Test of Independence checks whether two categorical variables are related. They use the same chi‑square formula but answer different questions and compute expected counts differently.
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