Chi Squared Tests

What is bivariate categorical data?

Data classifying each item by two categorical variables (e.g. gender × preferred subject), usually displayed in a contingency table.

What is a contingency table?

A two-way table showing observed frequencies for each combination of two categorical variables, with row and column totals.

What does a χ² (chi-squared) test on a contingency table test?

Whether the two categorical variables in a contingency table are associated (related) in the population, or independent.

H₀ and H₁ for a χ² contingency-table test

H₀: there is no association between the two factors (variables are independent). H₁: there is an association.

Formula for expected frequency in a contingency table

Expected = (row total × column total) / overall total.

Test statistic for χ² test

X² = Σ (O − E)² / E, summed over all cells of the table.

Degrees of freedom for an r × c contingency table

df = (r − 1)(c − 1), where r is the number of rows and c the number of columns.

How do you carry out a χ² test for a contingency table?

1. State H₀ and H₁. 2. Calculate expected values. 3. Calculate X² = Σ(O−E)²/E. 4. Find df. 5. Compare X² to the critical value at the chosen significance level. 6. Reject or fail to reject H₀; conclude in context.

Decision rule using critical value (χ² test)

If X² > critical value, reject H₀. Otherwise, do not reject H₀.

Decision rule using p-value (χ² test)

If p ≤ significance level, reject H₀. Otherwise, do not reject H₀.

How is a χ² test conclusion phrased?

Non-assertively in context, e.g. "There is sufficient evidence to suggest an association between favourite subject and gender" or "There is insufficient evidence at the 5% level to suggest these factors are associated."

What does it mean if one cell's (O−E)²/E contribution is very large?

That cell shows the biggest discrepancy between observed and expected — it's where the relationship "breaks" independence most strongly. Useful for interpreting WHICH categories are linked.

Why might you "combine cells" in a χ² test?

If an expected frequency is too small (often < 5), the χ² approximation is poor. Combining adjacent categories raises the expected count; OCR won't ask you to make this decision in exams but you should know it happens.

What is Yates' continuity correction?

A correction for 2×2 tables that subtracts 0.5 from |O − E| before squaring. Not required in OCR Y432, but its use won't be penalised.

What is a χ² test for goodness of fit?

A test of whether a given distribution (e.g. uniform, binomial, Poisson) is a suitable model for an observed data set.

H₀ for a goodness-of-fit test

H₀: the given distribution fits the data (or: the given model is suitable).

Test statistic for a goodness-of-fit test

X² = Σ (O − E)² / E, summed over all categories.

Degrees of freedom for goodness of fit

df = (number of categories used) − (number of parameters estimated from the data) − 1.

df for goodness of fit to uniform distribution on k categories

df = k − 1 (no parameters estimated).

df for goodness of fit to Poisson when λ is estimated from the data

df = (number of categories used) − 1 − 1 = k − 2 (since λ is estimated).

df for goodness of fit to binomial when p is estimated from the data

df = k − 1 − 1 = k − 2.

df for goodness of fit to binomial when p is known (given, not estimated)

df = k − 1.

What does "−1 for estimating each parameter" mean intuitively?

Estimating a parameter from the data uses up information, leaving less to test the fit. Each estimated parameter costs you one degree of freedom.

How do you find expected frequencies for goodness of fit?

Multiply n (sample size) by the model's probability for each category: E = n · P(X is in that category).

Decision and conclusion for a goodness-of-fit test

If X² > critical value (or p ≤ α), reject H₀; conclude the model does NOT fit. Otherwise, the model is consistent with the data — conclude there is no reason to doubt the model.

Example conclusion phrasing for goodness of fit

"There is sufficient evidence at the 5% level to suggest that the Poisson model is NOT suitable for this data." Or: "It is reasonable to believe that the binomial model fits."

Why is the χ² test always one-tailed?

Because (O − E)² is always non-negative, large X² values (in only one tail) indicate poor fit; there's no notion of "too small" a discrepancy.

What does "degrees of freedom" represent for χ²?

The number of independent quantities that can vary freely after constraints are applied (e.g. totals fixed, parameters estimated).

What software output might Y432 expect you to interpret for a χ² test?

An observed-vs-expected table, the X² test statistic, degrees of freedom, and a p-value. You may need to interpret the p-value to make a decision.

Can a χ² test prove H₀ is true?

No — failing to reject H₀ only means there's not enough evidence to reject it. The model is consistent with the data, but not proven correct.

What's the difference between a contingency-table χ² test and a goodness-of-fit χ² test?

Contingency table tests for ASSOCIATION between two categorical variables. Goodness of fit tests whether ONE variable follows a SPECIFIED distribution.

Both tests use the same test statistic; what differs?

The hypotheses, the way expected frequencies are computed, and the formula for degrees of freedom.

Why are categorical data tests called "non-parametric"?

They don't assume a specific population distribution for the variable; they only assess frequencies in categories.

What is meant by "this is uninformative because the test is too sensitive"?

With huge samples, even tiny meaningless departures from H₀ may be flagged as "significant"; the practical effect size matters as much as the p-value.