Chi-Square Calculator - Calculate Chi-Square Test Statistic
Calculate chi-square statistic, degrees of freedom, and statistical significance for categorical data analysis and hypothesis testing
Enter Observed & Expected Frequencies
Test Results
Detailed Breakdown
| Category | Observed (O) | Expected (E) | (O - E) | (O - E)² | (O - E)² / E |
|---|
What is a Chi-Square Calculator?
A Chi-Square Calculator performs chi-square tests to determine if observed frequencies differ significantly from expected frequencies, testing hypotheses about categorical data.
This calculator is used for:
- Goodness-of-Fit - Test if data fits expected distribution
- Independence Tests - Determine if variables are related
- Categorical Analysis - Compare frequency distributions
- Hypothesis Testing - Statistical significance testing
To analyze first-digit distributions using chi-square, explore our Benford's Law Calculator to detect fraud and validate data authenticity with statistical testing.
To calculate statistical confidence in your estimates, check out our Confidence Interval Calculator to determine ranges for population means and proportions.
To organize categorical data for chi-square testing, visit our Frequency Distribution Calculator to create frequency tables and analyze data patterns.
To understand sampling distributions that underlie hypothesis testing, use our Central Limit Theorem Calculator to see how sample means approach normal distribution.
How Chi-Square Test Works
The chi-square test uses this formula:
Where O is observed frequency, E is expected frequency, df is degrees of freedom, and n is the number of categories.
Key Concepts Explained
Degrees of Freedom
df = n - 1 for goodness-of-fit tests. It represents the number of independent categories that can vary freely.
Critical Value
The threshold value from chi-square distribution tables. If calculated χ² exceeds this, reject the null hypothesis.
Null Hypothesis
Assumes no significant difference between observed and expected frequencies or no association between variables.
How to Use This Calculator
Enter Observed Frequencies
Input actual counts from your data
Enter Expected Frequencies
Input theoretical or expected counts
Select Significance Level
Choose α (0.10, 0.05, or 0.01) and calculate
Benefits of This Calculator
- Instant Calculation - Get chi-square results immediately
- Detailed Breakdown - See contribution of each category
- Critical Values - Automatic critical value lookup
- Statistical Decision - Clear reject/fail to reject result
- Educational Tool - Learn chi-square methodology
- Research Ready - Professional statistical analysis
Factors Affecting Results
- Sample Size - Larger samples provide more reliable results
- Expected Frequencies - Should be at least 5 for validity
- Number of Categories - Affects degrees of freedom
- Significance Level - Lower α requires stronger evidence
- Data Type - Must be frequency counts, not percentages
- Independence - Categories must be mutually exclusive
Frequently Asked Questions
What is a chi-square test?
A chi-square test is a statistical test used to determine if there's a significant association between categorical variables or if observed frequencies differ significantly from expected frequencies. It measures how well observed data fits expected values.
What is the chi-square formula?
The chi-square formula is: χ² = Σ[(O - E)² / E], where O is the observed frequency, E is the expected frequency, and the sum is taken over all categories.
What are degrees of freedom in chi-square test?
Degrees of freedom (df) for a chi-square test equals (number of categories - 1) for goodness-of-fit tests, or (rows - 1) × (columns - 1) for independence tests. It's used to determine the critical value.
When do you use a chi-square test?
Use chi-square tests when analyzing categorical data, testing independence between variables, comparing observed vs. expected frequencies, or testing goodness-of-fit. Data must be in frequency counts, not percentages.
What is a good chi-square value?
A "good" chi-square value depends on degrees of freedom and significance level. If χ² is less than the critical value at your chosen significance level (e.g., 0.05), you fail to reject the null hypothesis.
What is the difference between chi-square and t-test?
Chi-square tests categorical data and frequencies, while t-tests compare means of continuous numerical data. Chi-square examines associations or goodness-of-fit; t-tests compare group means.