🎓 Statistics & Data • Hypothesis Testing
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Chi-Square Test Calculator (χ² Goodness of Fit)
Computes Chi-Square statistic χ² = Σ [(O - E)² / E] to test whether observed frequencies fit expected distributions.
Significance Level (α):
Goodness-of-Fit Analysis
Chi-Square Test: χ² = Σ [ (O - E)² / E ]
Test whether observed sample categorical counts differ significantly from expected theoretical frequencies.
Chi-Square Test Statistic
χ² = 6.56
Fail to Reject H₀: Distribution conforms to expected frequencies (χ² = 6.56 ≤ crit = 7.7757)
Degrees of Freedom3
Critical Value (χ²_crit)7.7757
Alpha Level0.05
What is the Chi-Square Test?
The Chi-Square test assesses whether discrepancies between observed and expected frequencies in categorical data are due to chance or statistically significant effects.
Formula & Step-by-Step Calculation
χ² = Σ [ (O_i - E_i)² / E_i ], df = k - 1
Sum of squared normalized deviations across categories.
Worked Step-by-Step Examples
Example 1
Testing a die rolled 60 times with observed counts: 12, 7, 14, 15, 6, 6 (Expected: 10 each)
Solution: χ² = 7.80 (df = 5, Fail to reject H₀ at 0.05)
• Σ [(O - 10)² / 10] = (4 + 9 + 16 + 25 + 16 + 16)/10 = 86/10...
Common Real-World & Academic Use Cases
- ✓ Genetics Mendelian phenotypic ratio verification
- ✓ Customer demographic preference surveys
- ✓ Contingency table test of independence
How to Use the Chi-Square Test Calculator (χ² Goodness of Fit)
1
Enter Observed Counts
Input observed frequencies.
2
Enter Expected Counts
Input theoretical frequencies.
3
Read Chi-Square Value
Inspect χ² statistic and degrees of freedom.
Frequently Asked Questions
Q: Can expected frequencies be less than 5?
Generally, each expected frequency should be at least 5 for the Chi-Square approximation to remain valid.