Chi-Square Calculator
Print PageA Chi-Square Calculator (also known as a Chi-Square Test Calculator, Goodness of Fit Utility, Independence Test Calculator, Contingency Table Chi-Square Analyzer, or Cramér’s V Effect Size Calculator) computes the Chi-Square test statistic (χ2 = ∑ [ (O - E)2 ÷ E ]), expected cell counts (Eij = [Row Total · Col Total] ÷ N), degrees of freedom (df = [r - 1] · [c - 1]), exact p-values, Yates’ continuity correction, and Cramér’s V association strength for categorical data tables.
In digital marketing A/B testing, clinical medical trials, and genetic cross experiments, the Chi-Square test evaluates whether observed categorical frequencies deviate significantly from expected baseline frequencies or whether two categorical variables are statistically independent.
Our free online Chi-Square Calculator provides instant calculations across all contingency table matrices:
- Chi-Square Test Statistic (χ2):
χ2 = ∑ [ (Oi - Ei)2 ÷ Ei ]. - Expected Cell Frequency (Eij):
Eij = ( Rowi Sum · Colj Sum ) ÷ Grand Total N. - Degrees of Freedom (df for Independence):
df = (r - 1) · (c - 1). - Degrees of Freedom (df for Goodness-of-Fit):
df = k - 1(wherekis the number of outcome categories). - Yates’ Continuity Correction (2×2 Tables):
χ2Yates = ∑ [ ( | O - E | - 0.5 )2 ÷ E ]. - Cramér’s V Effect Size:
V = √[ χ2 ÷ ( N · min[r - 1, c - 1] ) ].
Master 2×2 A/B Testing Chi-Square Reference Table (N = 1,000 Users)
The table below displays observed counts, expected counts, individual cell contributions, and total Chi-Square test metrics for an e-commerce website redesign A/B test (Variant A vs. Variant B):
| A/B Test Variant Group | Converted (Observed / Expected) | Not Converted (Observed / Expected) | Total Users N | Cell Contribution (χ2 term) | Observed Conversion Rate |
|---|---|---|---|---|---|
| Control Variant A | 100 Obs / 120.0 Exp | 400 Obs / 380.0 Exp | 500 Users | 3.333 + 1.053 = 4.386 | 20.00% |
| New Variant B (Winner) | 140 Obs / 120.0 Exp | 360 Obs / 380.0 Exp | 500 Users | 3.333 + 1.053 = 4.386 | 28.00% (+8.0% Lift!) |
| Chi-Square Summary Result | Total χ2 = 8.772 | Degrees of Freedom df = 1 | P-Value = 0.00306 | Cramér’s V = 0.0937 | STATISTICALLY SIGNIFICANT! |
Step-by-Step 6-Sided Die Fairness Goodness-of-Fit Calculation
To evaluate whether a 6-sided die is fair after N = 120 total rolls (where an unbiased die expects E = 120 ÷ 6 = 20 rolls per face), observing face counts [15, 25, 18, 22, 16, 24]:
Step 1 (Calculate Cell 1 Term - Face 1): (15 - 20)^2 ÷ 20 = (-5)^2 ÷ 20 = 25 ÷ 20 = 1.250
Step 2 (Calculate Cell 2 Term - Face 2): (25 - 20)^2 ÷ 20 = (+5)^2 ÷ 20 = 25 ÷ 20 = 1.250
Step 3 (Calculate Cell 3 Term - Face 3): (18 - 20)^2 ÷ 20 = (-2)^2 ÷ 20 = 4 ÷ 20 = 0.200
Step 4 (Calculate Cell 4 Term - Face 4): (22 - 20)^2 ÷ 20 = (+2)^2 ÷ 20 = 4 ÷ 20 = 0.200
Step 5 (Calculate Cell 5 & 6 Terms): Face 5 &implies; (-4)^2 ÷ 20 = 0.800; Face 6 &implies; (+4)^2 ÷ 20 = 0.800
Step 6 (Sum Total Chi-Square Statistic χ^2): χ^2 = 1.250 + 1.250 + 0.200 + 0.200 + 0.800 + 0.800 = 4.500
Step 7 (Determine df & P-Value): df = 6 - 1 = 5 categories. Critical value at α=0.05 is 11.07. P-value = 0.4799
Because χ2 = 4.500 < 11.07 (and p = 0.4799 > 0.05), we fail to reject the null hypothesis, establishing that the die exhibits no statistically significant bias (fair die).
Chi-Square Test Types: Goodness-of-Fit vs. Test of Independence
Below is a comparative reference chart detailing the two major forms of the Chi-Square test:
| Chi-Square Test Type | Primary Research Question | Degrees of Freedom Formula | Common Real-World Example |
|---|---|---|---|
| Goodness-of-Fit Test | Does 1 sample distribution match a theoretical model? | df = k – 1 | Benford’s Law audit, Mendelian genetics 9:3:3:1 ratio. |
| Test of Independence | Are 2 categorical variables related or independent? | df = (r – 1) · (c – 1) | Website A/B testing conversion rates, medical drug side effects. |
History & Mathematics: 1900 Karl Pearson to 1934 Frank Yates
1900 Karl Pearson & Modern Hypothesis Testing
In 1900, English mathematician and biostatistician Karl Pearson published his seminal paper in Philosophical Magazine, introducing the Chi-Square test statistic (χ2) and laying the foundation for modern statistical hypothesis testing.
1934 Frank Yates & Continuity Correction
In 1934, English statistician Frank Yates introduced Yates’ continuity correction (± 0.5 adjustment) to adjust Chi-Square calculations for small 2 × 2 contingency tables.
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Frequently Asked Questions (FAQ)
What is the Chi-Square test formula?
The Chi-Square statistic formula is χ2 = ∑ [ (O - E)2 ÷ E ].
How do you calculate degrees of freedom for a Chi-Square test?
For a contingency table, df = (rows - 1) · (columns - 1). For a goodness-of-fit test, df = categories - 1.
What is the minimum expected cell count for a valid Chi-Square test?
A standard rule of thumb requires all expected cell counts to be at least 5 (E ≥ 5). For smaller expected counts, Fisher’s Exact Test should be used.