Chi-Squared Goodness-of-Fit Test

chi-square

goodness-of-fit

categorical

kolmogorov-smirnov

Comparing observed frequencies to an expected distribution for a single categorical variable; with Kolmogorov-Smirnov note for continuous data

Published

April 17, 2026

Research question

The chi-squared goodness-of-fit test compares observed counts in each category to an expected distribution. Biomedical examples: (1) in a paediatric ICU audit, are the observed frequencies of admission causes (respiratory, cardiac, sepsis, trauma, other) different from the national reference distribution reported in a 2023 registry?; (2) does the distribution of ABO blood groups in a local donor pool match the national population (A 37 %, B 10 %, AB 3 %, O 50 %)?

For continuous variables tested against a theoretical distribution (e.g., is a biomarker normal?), the Kolmogorov-Smirnov test is the standard non-parametric choice; a brief note is included below.

Assumptions

Assumption	How to verify in R
Independent observations	design
Categorical outcome with \(k\) mutually exclusive categories	scale level
Expected frequency in each cell >= 5	`expected <- p0 * sum(observed); all(expected >= 5)`

When expected frequencies are too small, use Fisher’s exact test or combine sparse categories.

Hypotheses

\[H_0: \text{the observed distribution matches the expected}\qquad H_1: \text{it does not}\]

R code

library(tidyverse); library(rstatix)

## Scenario: ABO blood-group comparison
observed <- c(A = 154, B = 46, AB = 18, O = 182)
expected_p <- c(A = 0.37, B = 0.10, AB = 0.03, O = 0.50)

chi <- chisq.test(x = observed, p = expected_p)
chi

chi$expected
chi$residuals   # standardised residuals

## Tidy workflow with rstatix
tibble(group = names(observed), observed = as.integer(observed)) |>
  rstatix::chisq_test(observed ~ group, p = expected_p)

## Kolmogorov-Smirnov, one-sample: is a biomarker standard normal?
set.seed(42)
biomarker <- rnorm(80, mean = 0, sd = 1)
ks.test(biomarker, "pnorm", mean = 0, sd = 1)

Interpreting the output

For the ABO example, \(\chi^2(3) \approx 3.9\), \(p = .27\): the donor-pool distribution is consistent with the national reference. The standardised residuals are all within \(\pm 2\), indicating no single category drives the comparison.

For the KS example, \(D \approx 0.08\), \(p = .66\): no evidence against normality of the biomarker at \(n = 80\).

Effect size

Cohen’s \(w = \sqrt{\chi^2 / n}\). Thresholds: small 0.10, medium 0.30, large 0.50. For the ABO example, \(w \approx 0.09\) – below the small threshold, consistent with the non-significant result.

Reporting (APA 7)

The ABO blood-group distribution in the donor pool did not differ from the national reference (chi-squared(3) = 3.94, p = .27, Cohen’s w = .09).

Common pitfalls

Expected frequencies below 5 invalidate the chi-squared approximation; use fisher.test() instead.
Specifying the expected proportions with small rounding errors that do not sum to 1; R warns but still computes.
Interpreting a large \(\chi^2\) without examining the standardised residuals to identify which categories drive it.

Parametric vs. non-parametric alternative

The chi-squared goodness-of-fit is the default categorical-variable test. For continuous variables, the Kolmogorov-Smirnov and Anderson-Darling tests are the standard one-sample alternatives against a specified distribution.