Perform Chi-Square tests for goodness-of-fit or independence. Enter data to compute the statistic, degrees of freedom, p-value, and interpretations, with optional Yates' correction for 2x2 tables.
Calculation Results
How the Chi-Square Calculator Works
This calculator supports two primary Chi-Square tests: goodness-of-fit for comparing observed data to an expected distribution and independence for analyzing relationships in contingency tables. Select the test type, input frequencies, set the significance level, and click Calculate to obtain results including visualizations.
For goodness-of-fit, provide observed and expected values as comma-separated lists. For independence, enter observed data as CSV rows. The tool computes expected values automatically for independence and applies Yates' correction if selected for 2x2 tables.
Understanding Chi-Square Tests
Chi-Square tests are essential in statistics for evaluating categorical data. The goodness-of-fit test assesses if observed frequencies align with a theoretical distribution, while the independence test determines if two variables are associated.
Key Formulas
The core statistic is:
\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \]
Where \(O_i\) is observed and \(E_i\) is expected. For Yates' correction in 2x2 tables:
Degrees of freedom: For goodness-of-fit, \(k-1\) (k categories); for independence, \((r-1)(c-1)\) (r rows, c columns).
Assumptions and Limitations
Assumptions include independent observations and expected frequencies ≥5 in at least 80% of cells. For small samples, use alternatives like Fisher's exact test.
Cumulative distribution function of the chi-squared distribution.
Step-by-Step Examples
For goodness-of-fit: Suppose observed frequencies are 60,40,40,60 with expected 50 each (total 200). The calculator yields χ²=8, df=3, p≈0.046, suggesting significance at α=0.05.
For independence: Using neighborhood-occupation data:
Input data to view detailed tables and charts highlighting deviations.
Applications in Statistical Analysis
These tests are widely used in research: goodness-of-fit in genetics for Mendelian ratios, independence in surveys for variable associations. For in-depth guidance, refer to resources like Wikipedia's Chi-Squared Test.
Visualizing Test Results
Line charts compare observed and expected frequencies, while polar area charts show per-category contributions to the statistic, aiding in identifying key deviations.
FAQ
How does sample size impact Chi-Square results? Larger samples enhance power but may detect minor differences; small samples risk violating assumptions.
Can this calculator handle unequal expected proportions? Yes, for goodness-of-fit, enter custom expected values that sum to the total observed.
What if expected frequencies are low? The tool alerts if >20% are below 5; consider pooling categories or alternative tests.
JMP: Chi-Square Goodness of Fit Test - Knowledge portal with real-world examples, like candy bag distributions, and integration tips for statistical software.
This page provides a versatile Chi-Square Calculator for goodness-of-fit and independence tests in categorical data analysis. It includes formulas, examples, visualizations, and interpretations for hypothesis testing in statistics. Suitable for students, researchers, and analysts in fields like sociology, biology, and marketing. Index under statistical tools, math calculators, and data analysis resources.
