Question 1

What does the chi-square goodness of fit test test?

Accepted Answer

H₀: the observed frequencies match the expected frequencies (the data fits the distribution). H₁: the observed frequencies differ significantly from expected. Example: rolling a die 60 times, you expect 10 in each face. If you observe 8, 14, 9, 12, 7, 10, χ² measures how surprising this deviation is. A significant χ² means the die is likely unfair.

Question 2

How do I specify expected frequencies?

Accepted Answer

You can specify expected frequencies in three ways: (1) Equal distribution: all categories have n/k expected (e.g., a fair die has 1/6 probability each). (2) Known probabilities: multiply total n by each probability (e.g., genetic ratios 3:1 → 75% and 25%). (3) Custom values: enter any non-negative expected frequencies — they will be scaled to match the observed total. Every expected value must be ≥ 1, ideally ≥ 5.

Question 3

What are the assumptions of the chi-square goodness of fit test?

Accepted Answer

Three assumptions: (1) Independence — each observation must be independent (no repeated measures). (2) Minimum expected frequency — all expected values should be ≥ 5. If any are less than 5, combine adjacent categories or use an exact test. (3) Sufficient sample size — typically n ≥ 20. The test uses a continuous chi-square approximation to a discrete distribution, which requires adequate sample size.

Question 4

What is Cohen's w as an effect size?

Accepted Answer

Cohen's w = √(χ²/n) measures the magnitude of departure from the null distribution. Benchmarks: w = 0.1 (small), 0.3 (medium), 0.5 (large). Alternative: Cramér's V = √(χ²/(n×(k−1))) for multi-category distributions. For a 2-category test (k=2), w = |p₁−p₂|/√(p̂(1−p̂)) — equivalent to a standardized proportion difference.

Question 5

What is the difference between chi-square goodness of fit and chi-square independence test?

Accepted Answer

Goodness of fit: one categorical variable, compare observed vs theoretically expected frequencies. Tests if data matches a specific distribution. Independence test: two categorical variables in a contingency table, tests if the variables are associated. Both use the χ² = Σ(O−E)²/E formula but differ in how expected values are computed and the degrees of freedom formula (df = k−1 for goodness of fit; df = (r−1)(c−1) for independence).

Question 6

Can I test for a specific distribution (e.g., uniform or Poisson)?

Accepted Answer

Yes. For a uniform distribution, set all expected frequencies equal to n/k. For a Poisson distribution with known mean λ, compute expected frequencies as n × P(X=k) for each count category. For a normal distribution, compute expected frequencies from the normal CDF across bins. Note: when you estimate distribution parameters from the data itself, subtract additional degrees of freedom (df = k − 1 − number of estimated parameters).

Symbol	Meaning
Oᵢ	Observed count in category i (from your data)
Eᵢ	Expected count in category i under H₀ (can be theoretical or uniform)
k	Number of categories
df	Degrees of freedom = k − 1 (one constraint: observed total is fixed)
Cohen's w	√(χ²/n): small ≥ 0.1 · medium ≥ 0.3 · large ≥ 0.5

Day	O	E (uniform)	O − E	(O−E)²/E
Mon	18	20	-2	0.200
Tue	22	20	2	0.200
Wed	15	20	-5	1.250
Thu	25	20	5	1.250
Fri	20	20	0	0.000
Total	100	100		2.900

df (k−1)	χ²_0.05	χ²_0.01
1	3.841	6.635
2	5.991	9.210
3	7.815	11.345
4	9.488	13.277
5	11.070	15.086
6	12.592	16.812
9	16.919	21.666

Chi-Square Goodness of Fit Calculator

Formula

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Chi-Square Goodness of Fit: Formula

Worked Example: Are Support Tickets Uniform Across Weekdays?

Assumptions & Chi-Square Critical Values

Assumptions

χ² critical values (α = 0.05)

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