Non-Parametric Test
Chi-Square Goodness-of-Fit Test Calculator
Test whether your observed frequencies match expected frequencies — free online tool with step-by-step results, effect size, APA write-up, and 4 interactive charts.
χ² Statistic
p-Value
Cohen's w
Cramér's V
APA 7th Write-Up
Non-Parametric
4 Charts
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Step 1 — Enter Your Data
Enter Observed (O) and Expected (E) counts for each category. Click ＋ Add Category to add more rows. Category names are editable.
💡 Tip: For comma-separated quick entry, paste values in the Observed field. Or click a sample dataset above.
Supports .csv, .txt, .xlsx, .xls — headers detected automatically.
📌 Click the column buttons below to select Observed counts and (optionally) Expected counts columns. Selected columns load into the categories above.
Enter values directly into the table cells below. Click ＋ Add Row to add more categories.
| # | Category Name | Observed (O) | Expected (E) |
|---|
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Step 2 — Configuration
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Results Summary
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Visualizations
Chart 1 — Observed vs Expected Frequencies
Chart 2 — Chi-Square Contribution per Category
Chart 3 — χ² Distribution with Critical Region
Chart 4 — Deviation from Expected (O − E)
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Assumption Checks
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Detailed Interpretation of Results
Run the analysis to see a detailed interpretation.
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How to Write Your Results in Research
Run the analysis to see auto-filled write-up templates.
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Technical Notes — Formulas Used
View Formulas
①
Chi-Square Test Statistic
χ² = Σ [(O⊂i − E⊂i)² / E⊂i]
χ²Chi-square statistic — larger values mean greater discrepancy between observed and expected
O⊂iObserved frequency in category i — the actual count from your data
E⊂iExpected frequency in category i — the count you expect under H₀
ΣSummation across all k categories
②
Degrees of Freedom
df = k − 1
dfDegrees of freedom — determines the shape of the chi-square distribution used for the p-value
kNumber of categories in your data
NoteOne degree of freedom is lost because the total of expected counts must equal the total of observed counts
③
Expected Frequency (Equal Distribution)
E⊂i = N / k
E⊂iExpected count for each category when distribution is assumed uniform
NTotal sample size — sum of all observed counts
kNumber of categories
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Expected Frequency (Proportional / Custom)
E⊂i = N × p⊂i
E⊂iExpected count for category i under the specified probability
NTotal sample size
p⊂iTheoretical probability for category i (must sum to 1 across all categories)
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Cohen's w — Effect Size
w = √(χ² / N)
wCohen's w — standardised effect size for chi-square tests
χ²Chi-square test statistic from the analysis
NTotal sample size
Benchmarks0.10 small · 0.30 medium · 0.50 large (Cohen, 1988)
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Cramér's V — Effect Size
V = √[χ² / (N × (k − 1))]
VCramér's V — bounded 0–1; for goodness-of-fit (1 row), V equals Cohen's w when min(r-1,c-1)=1
k−1Degrees of freedom (number of categories minus one)
Benchmarks≈ 0.10 small · 0.30 medium · 0.50 large
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Chi-Square Contribution per Category
c⊂i = (O⊂i − E⊂i)² / E⊂i
c⊂iIndividual contribution of category i to the total chi-square statistic
O⊂i − E⊂iResidual — how far the observed count deviates from the expected count
UseCategories with high c_i are the main drivers of a significant result; inspect them first
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When to Use the Chi-Square Goodness-of-Fit Test
Use the chi-square goodness-of-fit test when you want to check whether the distribution of a single categorical variable matches a theoretically expected distribution.
✅ Good fit for this test
- One categorical variable with 2+ categories
- Data are counts (not proportions or means)
- Each observation falls in exactly one category
- All expected frequencies ≥ 5
- You have a theoretical expected distribution
❌ Consider alternatives if…
- Expected frequencies < 5 → use Fisher's Exact Test
- Testing association between two variables → use Chi-Square Test of Independence
- Data are continuous → use KS test or Anderson-Darling
- Paired or repeated measurements → use McNemar's test
🔬 Common applications
- Testing if a die is fair (equal frequencies)
- Hardy-Weinberg equilibrium in genetics
- Species habitat use vs. habitat availability
- Market share vs. expected proportions
- Survey response distribution checks
📐 Decision tree
- Categorical? → Yes
- One variable? → Yes → Goodness-of-Fit
- Two variables? → Yes → Test of Independence
- Expected counts all ≥ 5? → Yes → χ² GoF
- Small counts → Fisher's / Exact Test
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How to Use This Calculator — 10-Step Guide
1
Choose or enter your data
Enter observed counts for each category in the Type/Paste tab, upload a CSV/Excel file, or fill the manual table. Each row is one category (e.g., colour, day, habitat type).
2
Enter expected frequencies
For each category, enter the expected count. If you expect equal distribution, click Set Equal Expected to auto-fill proportional counts. Or enter custom values based on theory.
3
Name your categories
Click the category name field (e.g., "Category 1") and type a meaningful label such as "Red", "Monday", or "Dense Forest". This makes results easier to read.
4
Add or remove categories
Click ＋ Add Category to add more rows. Click the ✕ button on any row to remove it. You need at least 2 categories.
5
Set significance level
Choose α = 0.05 (most common), 0.01 (strict), or 0.10 (exploratory). You can also enter a custom value. Your choice affects the critical value and the significance decision.
6
Run the test
Click Run Chi-Square Goodness-of-Fit Test. The tool checks assumptions, calculates χ², df, p-value, Cohen's w, and Cramér's V instantly.
7
Read the summary cards
Six summary cards show the most important values at a glance: χ², df, p-value, Cohen's w, sample size N, and the significance decision. Green = significant, amber = borderline.
8
Interpret the charts
Chart 1 compares observed vs expected bars. Chart 2 shows which category contributes most to χ². Chart 3 shows where your statistic falls on the χ² distribution. Chart 4 shows O−E deviations.
9
Read the interpretation and write-up
Scroll to the Detailed Interpretation section. It explains the result in plain language. Then choose from five auto-filled write-up templates (APA 7th, thesis, plain language, abstract, pre-registration).
10
Download your report
Click Download Report (.txt) for a plain-text results file, or Download PDF to print or save as PDF directly from your browser.
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Frequently Asked Questions
What is the chi-square goodness-of-fit test?
The chi-square goodness-of-fit test checks whether the observed frequencies in a categorical variable match a set of expected frequencies. It answers the question: does my data fit the distribution I expected? For example, does a six-sided die produce equal counts for each number after 120 rolls?
What is the difference between goodness-of-fit and test of independence?
The goodness-of-fit test compares observed frequencies in one categorical variable to expected frequencies. The test of independence compares two categorical variables in a contingency table to check if they are related. Both use the χ² statistic, but they have different degrees of freedom and research questions.
What are the assumptions of the chi-square goodness-of-fit test?
There are four main assumptions: (1) data must be categorical or count-based — not means or proportions; (2) observations must be independent of each other; (3) each expected frequency must be at least 5 — the tool flags violations; and (4) categories must be mutually exclusive and exhaustive. Violating these can make the test inaccurate.
How do I interpret a chi-square goodness-of-fit p-value?
If p < α (your significance level, usually 0.05), you reject the null hypothesis and conclude that the observed distribution significantly differs from the expected distribution. If p ≥ α, you fail to reject H₀ — the data are consistent with the expected distribution. A small p-value means the observed data are unlikely under H₀, not that the effect is large.
What effect size should I report with this test?
Report Cohen's w — the standard effect size for chi-square goodness-of-fit. It is calculated as w = √(χ²/N). Benchmarks: small = 0.10, medium = 0.30, large = 0.50 (Cohen, 1988). Cramér's V is also commonly reported and equals Cohen's w for single-row goodness-of-fit tests. This calculator reports both.
What happens if my expected frequencies are less than 5?
The chi-square approximation becomes inaccurate when expected frequencies are small. The tool warns you if this happens. Solutions include: combining adjacent categories to increase expected counts, collecting more data, or using an exact multinomial test. The threshold is usually E ≥ 5 per cell, but some authorities accept E ≥ 1 if no more than 20% of cells are below 5.
How do I write chi-square goodness-of-fit results in APA format?
The standard APA 7th edition format is: χ²(df, N = n) = value, p = value, w = value. For example: χ²(3, N = 200) = 8.24, p = .041, w = 0.20. Report observed and expected frequencies in a table or in the text. Always state the significance level and whether you reject or retain H₀.
Is the chi-square goodness-of-fit test one-tailed or two-tailed?
The standard chi-square goodness-of-fit test is always an upper-tailed (one-tailed) test. A large χ² statistic — regardless of direction — indicates a poor fit. The p-value is always computed from the upper tail of the chi-square distribution. Two-tailed interpretations are not typical for this test.
Can I use this test for continuous data?
Not directly. Continuous data must first be grouped into discrete categories (bins). However, this introduces arbitrary choices about bin width and boundaries. For testing whether continuous data follow a specific distribution (e.g., normal), better alternatives include the Kolmogorov-Smirnov test or the Anderson-Darling test.
How do I use this test in ecology or biology?
The chi-square goodness-of-fit test is widely used in ecology. Common applications include: testing whether an animal species uses different habitat types in proportion to their availability (habitat selectivity), checking Hardy-Weinberg equilibrium in population genetics, testing whether seed dispersal events are uniformly distributed across directions, and checking whether species occurrences match a Poisson distribution.
What is the null hypothesis for this test?
The null hypothesis (H₀) states that the observed frequency distribution follows the expected distribution — meaning the proportions in the population match the specified theoretical proportions. The alternative hypothesis (H₁) states that at least one category has a frequency that differs from the expected value.
How many categories do I need?
You need at least 2 categories. There is no strict upper limit, but very many categories can reduce power if expected counts become small. Practically, most goodness-of-fit analyses involve 2 to 10 categories. Each category should have an expected frequency of at least 5.
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References
The chi-square goodness-of-fit test calculator on StatsUnlock is built on peer-reviewed methodology from statistics, ecology, and quantitative research. The following references support the goodness-of-fit test formulas, effect size calculations, and interpretation standards used in this tool.
- Pearson, K. (1900). 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, 50(302), 157–175. https://doi.org/10.1080/14786440009463897
- Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
- Cochran, W. G. (1952). The χ² test of goodness of fit. The Annals of Mathematical Statistics, 23(3), 315–345. https://doi.org/10.1214/aoms/1177729380
- Agresti, A. (2013). Categorical data analysis (3rd ed.). Wiley.
- Conover, W. J. (1999). Practical nonparametric statistics (3rd ed.). Wiley.
- Cramér, H. (1946). Mathematical methods of statistics. Princeton University Press.
- Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). Sage Publications.
- Tabachnick, B. G., & Fidell, L. S. (2019). Using multivariate statistics (7th ed.). Pearson.
- Zar, J. H. (2010). Biostatistical analysis (5th ed.). Prentice Hall. [Standard ecology statistics reference including chi-square goodness-of-fit for habitat use]
- Siegel, S., & Castellan, N. J. (1988). Nonparametric statistics for the behavioral sciences (2nd ed.). McGraw-Hill.
- Goodman, L. A. (1954). Kolmogorov-Smirnov tests for psychological research. Psychological Bulletin, 51(2), 160–168. https://doi.org/10.1037/h0054447
- Sokal, R. R., & Rohlf, F. J. (2012). Biometry: The principles and practice of statistics in biological research (4th ed.). W.H. Freeman.
- McDonald, J. H. (2014). Handbook of biological statistics (3rd ed.). Sparky House Publishing. http://www.biostathandbook.com/chigof.html
- Mangiafico, S. S. (2016). Summary and analysis of extension program evaluation in R, version 1.18.1. Rutgers Cooperative Extension. https://rcompanion.org/handbook/
- American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). APA. https://doi.org/10.1037/0000165-000
