Chi-Square Test (χ²)

What It Is

The Core Formula

χ² = Σ [ (O - E)² / E ]

1. Subtract expected from observed for each category: (O - E)
2. Square the difference: (O - E)²
3. Divide by expected count to normalize: (O - E)² / E
4. Sum all values to get χ²

Two Main Types

1. Chi-Square Goodness of Fit Test

• Purpose: Tests whether a single categorical variable follows an expected distribution.
• Question: "Does my observed data match a theoretical or expected distribution?"
• Example: You roll a die 60 times. Do all 6 faces come up equally (~10 times each)? You compare observed rolls to the expected 10 per face.
• Degrees of freedom: k - 1 (where k = number of categories)

2. Chi-Square Test of Independence

• Purpose: Tests whether two categorical variables are associated or independent.
• Question: "Is variable A related to variable B?"
• Example: Is smoking status (smoker/non-smoker) related to lung disease (yes/no)? Uses a contingency table.
• Degrees of freedom: (rows - 1) × (columns - 1)

Step-by-Step Procedure

1. State hypotheses:
   • H₀: No difference from expected / Variables are independent
   • H₁: Significant difference / Variables are associated
2. Choose significance level (α), typically 0.05
3. Calculate expected frequencies:
   • Goodness of fit: based on theoretical proportions
   • Independence: E = (Row total × Column total) / Grand total
4. Compute χ² statistic using the formula above
5. Find degrees of freedom (df)
6. Get the p-value from a chi-square distribution table or software
7. Decision:
   • If p < α → Reject H₀ (significant result)
   • If p ≥ α → Fail to reject H₀

Assumptions & Conditions

If expected cell counts are < 5, use Fisher's Exact Test instead (especially for 2×2 tables with small samples).

Worked Example: Test of Independence

Effect Size: Cramer's V

V = √(χ² / (n × min(r-1, c-1)))

When to Use Chi-Square vs. Alternatives

Common Applications

• Medicine/Epidemiology: Comparing disease rates between treatment and control groups
• Market research: Preference surveys across demographic groups
• Genetics: Testing Mendelian inheritance ratios
• Social sciences: Survey response analysis by group
• Quality control: Checking whether defects follow an expected pattern

Tests of Significance

Definition

Key Concepts

1. Null Hypothesis (H₀)

• The default assumption - states there is no effect, no difference, or no relationship.
• Example: "There is no difference in mean blood pressure between the drug group and placebo group."

2. Alternative Hypothesis (H₁ or Hₐ)

• What you want to prove - states there is an effect or difference.
• Can be one-tailed (directional) or two-tailed (non-directional).

3. Level of Significance (α)

• The threshold probability for rejecting H₀.
• Common values:

4. p-value

• The probability of obtaining results as extreme as or more extreme than observed, assuming H₀ is true.
• If p ≤ α → Result is statistically significant → Reject H₀
• If p > α → Result is not significant → Fail to reject H₀

5. Test Statistic

• A calculated value from sample data (e.g., z, t, F, χ²) used to determine the p-value.
• Each test has its own formula and sampling distribution.

6. Critical Value

• The boundary value from the relevant statistical table beyond which H₀ is rejected.
• If test statistic > critical value → Reject H₀

General Steps in a Test of Significance

Step 1: State H₀ and H₁
Step 2: Choose significance level (α)
Step 3: Select the appropriate test
Step 4: Check assumptions
Step 5: Compute the test statistic
Step 6: Find p-value or compare to critical value
Step 7: Make a decision (Reject or Fail to Reject H₀)
Step 8: State conclusion in context

Types of Tests of Significance

A. Parametric Tests

B. Non-Parametric Tests

Types of Errors

• Type I Error (False Positive): Rejecting a true H₀. Controlled by α.
• Type II Error (False Negative): Failing to reject a false H₀. Controlled by β.
• Power (1-β): Probability of correctly detecting a true effect. Ideally ≥ 0.80.

One-Tailed vs. Two-Tailed Tests

Choosing the Right Test

Is your outcome variable continuous or categorical?
│
├── Continuous
│   ├── Is data normally distributed?
│   │   ├── Yes (Parametric)
│   │   │   ├── 1 group vs. known value → One-sample t-test / z-test
│   │   │   ├── 2 independent groups → Independent t-test
│   │   │   ├── 2 paired groups → Paired t-test
│   │   │   └── 3+ groups → ANOVA
│   │   └── No (Non-parametric)
│   │       ├── 2 independent groups → Mann-Whitney U
│   │       ├── 2 paired groups → Wilcoxon signed-rank
│   │       └── 3+ groups → Kruskal-Wallis
│
└── Categorical
    ├── Large sample → Chi-square test
    └── Small sample (2×2) → Fisher's Exact Test

Worked Example: z-test

z = (x̄ - μ) / (σ/√n) = (78 - 75) / (10/√50) = 3 / 1.414 = 2.12

Statistical Significance vs. Practical Significance

• Effect size (Cohen's d, r, η², Cramer's V)
• Confidence intervals
• Clinical/practical context

Example: A drug lowers blood pressure by 1 mmHg with p = 0.001. Statistically significant - but clinically irrelevant.

Summary Table