Chi-Square (χ²) Test

Introduction / Background

The Chi-Square (χ²) test is a non-parametric statistical test used to examine the association between categorical variables, often organized into a cross-tabulation (or contingency table). Introduced by Karl Pearson in 1900, it is one of the earliest formal tests in statistics.

Unlike parametric tests, the Chi-Square test does not assume a normal distribution. It is widely used in fields such as social sciences, agricultural research, psychology, and health studies.

This test allows researchers to evaluate hypotheses about relationships in categorical data, such as preferences, treatment outcomes, or survey responses.

Types of Chi-Square Tests

1. Chi-Square Test of Independence

• Determines if there is a relationship between two categorical variables.
• Example: Is crop preference related to region?

2. Chi-Square Goodness-of-Fit Test

• Tests whether observed frequencies fit a specified theoretical distribution.
• Example: Does the distribution of wheat varieties in a field follow equal proportions?

Formulas / Key Calculations

To quickly compute the final result and verify your manual calculations, use our dedicated free Chi-Square Calculator tool.

1. Chi-Square Statistic

χ² = Σ ((O - E)² / E)

• O = Observed frequency
• E = Expected frequency

2. Expected Frequency Calculation

E = (Row Total × Column Total) / Grand Total

3. Degrees of Freedom

• For independence test: df = (r - 1) × (c - 1)
• For goodness-of-fit test: df = k - 1

Conceptual Method of Calculation

While this conceptual method shows the underlying statistics, many researchers use software like SPSS or R for computation. Understanding these steps is crucial for accurately interpreting the output from those programs.

1. Create a table of observed frequencies (O).
2. Calculate expected frequencies (E) for each cell.
3. Compute the difference: O - E.
4. Square the differences: (O - E)².
5. Divide by expected frequency: (O - E)² / E.
6. Sum all values to get χ².
7. Determine degrees of freedom (df).
8. Compare χ² to critical value at chosen significance level (e.g., 5%).
9. Interpret result:
   • χ² > critical → significant association
   • χ² ≤ critical → not significant

Illustrative Example

Suppose 50 farmers in two regions prefer three types of fertilizers. Observed data:

Fertilizer	Region A	Region B	Total
F1	10	12	22
F2	8	10	18
F3	5	5	10
Total	23	27	50

Step 1: Calculate Expected Frequencies (E)

For F1, Region A: E = (22 × 23) / 50 = 10.12

Similarly, compute E for all cells.

Step 2: Compute χ²

χ² = Σ ((O-E)² / E) = 0.5 (example total)

Step 3: Degrees of Freedom

df = (3-1)(2-1) = 2

Step 4: Compare with Critical Value

At 5% significance, χ² critical = 5.991

Since 0.5 < 5.991 → Not significant; no association between region and fertilizer preference. For **academic reporting**, ensure your results are presented in APA format.

Fields / Disciplines of Use

• Agriculture: Crop choice, fertilizer preference, disease incidence
• Sociology / Psychology: Survey responses, behavior studies
• Health Sciences: Treatment outcomes, prevalence studies
• Marketing / Business: Consumer preference analysis

Comparison with Similar Tools

• Fisher’s Exact Test: Small sample sizes where expected frequency < 5
• ANOVA: For continuous variables instead of categorical

Common Mistakes / Misconceptions

• Expected frequency in any cell should not be less than 5
• χ² is sensitive to sample size; large samples may show significance even with small differences
• Only applicable to categorical data, not continuous

Summary / Key Points

• Non-parametric test for categorical data
• Two main types: Independence and Goodness-of-Fit
• χ² formula compares observed vs expected frequencies
• Widely used across agriculture, social sciences, health, and marketing

Related Statistical Articles

Deepen your understanding of statistical testing and categorical data analysis by exploring these related methods and tools:

• Non-Parametric Tests: A Complete Guide for Researchers and Students
• Guide to ANOVA: From Basics to Examples (Analysis of Variance)
• Kruskal-Wallis H Test: A Non-Parametric Alternative to ANOVA
• Frequency Distribution: Making Sense of Raw Data in Simple Steps
• Kruskal-Wallis H Test: Direct Tool Page