Chi Square Statistic

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Chi-Square (χ²) Statistic

Definition

The chi-square (χ²) statistic measures how well observed categorical data match expected values under a specified model. It is commonly used to test relationships between categorical variables (test of independence) and to assess how well a sample distribution matches a theoretical distribution (goodness-of-fit).

Key takeaways

  • Applies to categorical (especially nominal) data from a random sample.
  • Two main uses: test of independence and goodness-of-fit.
  • Test results depend on the χ² value, degrees of freedom, and sample size.
  • χ² does not establish causation and is sensitive to sample size and small expected counts.

Formula and terms

χ² = Σ (O_i − E_i)² / E_i

where:
* O_i = observed frequency for category i
* E_i = expected frequency for category i
* Σ sums over all categories

Degrees of freedom (df):
* Goodness-of-fit: df = k − 1 (k = number of categories)
* Test of independence: df = (r − 1)(c − 1) (r = rows, c = columns)

For tests of independence, expected cell frequency = (row total × column total) / grand total.

Types of chi-square tests

  1. Test of independence
  2. Examines whether two categorical variables are related (e.g., gender and course choice).

  3. Compare observed cross-tabulation frequencies to expected frequencies under independence.

  4. Goodness-of-fit test

  5. Compares observed frequencies across categories to a specified theoretical distribution (e.g., a fair coin).
  6. Determines whether a sample matches the expected population proportions.

Example (coin toss)

If a fair coin is expected to produce 50 heads and 50 tails in 100 tosses:
* Observed: maybe 60 heads, 40 tails
* Expected: 50 heads, 50 tails
Compute χ² = (60−50)²/50 + (40−50)²/50 = (100/50) + (100/50) = 4.0
Compare this χ² to the critical value for df = 1 to decide whether deviation from fairness is statistically significant.

When to use χ²

Use chi-square tests when:
* Data are categorical (nominal or ordinal treated as categories).
* Samples are random and observations are independent.
* Expected frequencies per cell are adequate (rule of thumb: all expected ≥ 1 and ≥ 5 for most cells).

How to perform a chi-square test

  1. State the null hypothesis (e.g., variables are independent or distribution matches expectations).
  2. Build a table of observed frequencies.
  3. Calculate expected frequencies for each cell.
  4. Compute χ² = Σ (O − E)² / E.
  5. Determine df and find the critical χ² value (chi-square table or software) for chosen significance level.
  6. Compare computed χ² to critical value:
  7. If χ² > critical value, reject the null hypothesis.
  8. Otherwise, fail to reject the null hypothesis.

Interpretation

  • A large χ² indicates a substantial difference between observed and expected frequencies.
  • Statistical significance indicates unlikely to be due to random chance, given assumptions.
  • χ² does not indicate direction of effect or causality; examine contingency tables or follow-up tests for more detail.

Limitations and cautions

  • Sensitive to sample size: very large samples can produce significant χ² for trivial differences.
  • Requires independent observations; not appropriate for paired or matched data without adjustment.
  • Small expected counts distort χ² distribution; consider combining categories or using exact tests (e.g., Fisher’s exact) when expected counts are low.
  • Only tests association or fit, not causal relationships.

Practical users

Researchers working with survey or demographic data, market researchers, political scientists, and others analyzing categorical variables commonly use chi-square tests.

Summary

The chi-square statistic is a straightforward, widely used method for testing whether categorical data conform to expected distributions or whether two categorical variables are associated. Proper use requires attention to sample randomness, independence, sufficient expected counts, and cautious interpretation with respect to practical significance and causality.