Understanding your chi-square result
The table below lists the standard chi-square critical values at the conventional α = 0.05 significance level for common degrees of freedom. An observed χ² statistic exceeding the listed value indicates a statistically significant departure from the expected distribution.
| Degrees of freedom (df) | Critical χ² at α = 0.05 |
|---|---|
| 1 | 3.841 |
| 2 | 5.991 |
| 3 | 7.815 |
| 4 | 9.488 |
| 5 | 11.070 |
| 6 | 12.592 |
| 8 | 15.507 |
| 10 | 18.307 |
- A widely used rule of thumb (found in standard statistics texts) is that the chi-square approximation is reliable when every expected count is at least 5; with smaller expected counts, an exact test is more appropriate.
- This calculator performs goodness-of-fit only — a single categorical variable against a hypothesized distribution — not the chi-square test of independence between two categorical variables, which uses a different (row-by-column) expected-count calculation.
- The observed and expected lists must have the same length, with at least 2 categories and every expected count strictly positive.
- A statistically significant result indicates the categories differ from the hypothesized proportions; it does not by itself indicate which category, or how large the practical difference is.
What is a chi-square goodness-of-fit test?
The chi-square goodness-of-fit test compares the observed counts in a set of categories against the counts that would be expected under a hypothesized distribution, and measures how much they diverge. This calculator performs the goodness-of-fit version only — a single categorical variable checked against expected proportions — and not the related chi-square test of independence, which examines the association between two categorical variables in a contingency table.
The test statistic sums the squared, scaled difference between each observed and expected count across all categories: larger differences relative to the expected counts produce a larger χ² value and stronger evidence against the hypothesized distribution.
Because χ² is always zero or positive, the test is inherently one-directional: only unusually large χ² values (relative to the chi-square distribution with the appropriate degrees of freedom) are treated as evidence of a poor fit — there is no equivalent of a 'negative' direction to test against.
How to use this chi-square test calculator
- Enter the observed counts for each category as a comma-separated list, for example 30, 25, 22, 23.
- Enter the expected counts for the same categories, in the same order, as a comma-separated list, for example 25, 25, 25, 25.
- Ensure both lists have the same length (at least 2 categories) and that every expected count is greater than zero.
- Read the χ² statistic, degrees of freedom (number of categories minus 1), and p-value, and compare the p-value to your chosen significance level (commonly α = 0.05).
The formula behind the chi-square goodness-of-fit test
The chi-square statistic is χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ, summed over all k categories, where Oᵢ is the observed count and Eᵢ is the expected count in category i. The degrees of freedom equal k − 1, one less than the number of categories.
The p-value is the probability of observing a χ² statistic at least as large as the one calculated, under the chi-square distribution with the given degrees of freedom, if the hypothesized (expected) distribution were true.
Worked example: observed counts 30, 25, 22, 23 against expected counts of 25 each. χ² = (30−25)²/25 + (25−25)²/25 + (22−25)²/25 + (23−25)²/25 = 1 + 0 + 0.36 + 0.16 = 1.52, with df = 4 − 1 = 3. This is well below the critical value of 7.815 for α = 0.05 at 3 degrees of freedom, so the result is not statistically significant.
Common mistakes
- Using this goodness-of-fit test when the real question involves two categorical variables and their association — that requires a chi-square test of independence with a contingency table of expected counts, not a single observed-versus-expected list.
- Violating the expected-count guideline: when expected counts are small (commonly below 5 in one or more categories), the chi-square approximation becomes unreliable and an exact test is more appropriate.
- Entering observed and expected lists of different lengths, or listing categories in a different order in each list, which misaligns the comparison.
- Entering an expected count of zero, which makes the (O − E)² / E term for that category undefined.
- Treating a statistically significant χ² as evidence of a large or important difference — significance indicates the deviation is unlikely to be due to chance, not that it is practically large.
Pertanyaan yang sering diajukan
What is a chi-square goodness-of-fit test used for?
It tests whether the observed counts in a set of categories match a hypothesized distribution's expected counts — for example, checking whether a die's observed roll frequencies match the expected uniform distribution, or whether survey responses match an assumed proportion.
How is the chi-square statistic calculated?
χ² = Σ (observed − expected)² / expected, summed across every category. Larger discrepancies between observed and expected counts, especially relative to the expected count, produce a larger χ² value.
What are the degrees of freedom in a goodness-of-fit test?
Degrees of freedom equal the number of categories minus 1 (df = k − 1). For 4 categories, as in the worked example on this page, df = 3.
Why do expected counts need to be at least 5?
The chi-square test statistic's distribution is only approximated well by the chi-square distribution when expected counts are not too small. A commonly cited rule of thumb requires every expected count to be at least 5; below that, the p-value from the chi-square approximation can be inaccurate and an exact test is preferred.
Is this the same as a chi-square test of independence?
No. This calculator performs the goodness-of-fit test, which compares one categorical variable's observed counts to a hypothesized distribution. The test of independence examines whether two categorical variables are associated, using a contingency table where expected counts are computed from row and column totals — a different calculation.
Referensi
- Pearson K. On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine Series 5, 1900; 50(302): 157-175.
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods — chi-square goodness-of-fit test. nist.gov.
- Moore DS, McCabe GP, Craig BA. Introduction to the Practice of Statistics. W. H. Freeman (chi-square goodness-of-fit test and expected-count guidelines).
- Snedecor GW, Cochran WG. Statistical Methods. Iowa State University Press (chi-square goodness-of-fit test).
- Agresti A. Categorical Data Analysis. Wiley (chi-square goodness-of-fit and independence tests).