# Chi-Square Distribution Calculator Calculate chi-square distribution PDF, CDF, and right-tail probability from a statistic and degrees of freedom > Canonical page: https://elysiatools.com/en/tools/chi-square-distribution-calculator - **Category:** Math & Numbers - **Keywords:** chi-square distribution, chi-square test, p value, statistics ## Overview Calculate probabilities for chi-square tests, goodness-of-fit tests, and variance-related statistics. ## Inputs - **Chi-Square Value** (number) - **Degrees of Freedom** (number) - **Probability Mode** (select) - **Decimal Places** (number) ## When to use - Evaluating goodness-of-fit to determine if observed sample data matches an expected population distribution. - Testing the independence of two categorical variables using data from a contingency table. - Calculating exact p-values for variance-related statistical tests to determine statistical significance. ## How it works - Enter your calculated chi-square statistic into the primary input field. - Specify the degrees of freedom associated with your specific statistical test. - Select your preferred probability mode (left-tail or right-tail) and set the desired number of decimal places. - The calculator instantly computes the corresponding probabilities and returns the results in a structured JSON format. ## Use cases - Biology researchers analyzing genetic cross data to verify if observed traits fit expected Mendelian inheritance ratios. - Market analysts determining if customer purchasing preferences are independent of specific demographic segments. - Quality control engineers testing whether the variance of a manufacturing process meets specified operational standards. ## Frequently asked questions ### What is a chi-square distribution? It is a continuous probability distribution of the sum of the squares of independent standard normal variables, widely used in hypothesis testing and confidence interval estimation. ### How do I determine the degrees of freedom? For a goodness-of-fit test, it is typically the number of categories minus one. For a test of independence in a contingency table, it is calculated as (rows - 1) multiplied by (columns - 1). ### What does the right-tail probability represent? The right-tail probability represents the p-value. It indicates the likelihood of observing a chi-square statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. ### Can I adjust the precision of the calculated probabilities? Yes, you can use the decimal places setting to round the calculated probabilities to your required level of precision, supporting up to 10 decimal places. ### What is the difference between left-tail and right-tail modes? The left-tail mode calculates the cumulative probability up to your statistic (CDF), while the right-tail mode calculates the probability exceeding your statistic, which is commonly used for finding p-values. ## Related tools - [Standard Normal Calculator](https://elysiatools.com/en/tools/standard-normal-calculator): Calculate standard normal PDF, CDF, tail probability, two-tail probability, and central area from a z-score - [Exponential Distribution Calculator](https://elysiatools.com/en/tools/exponential-distribution-calculator): Calculate PDF, CDF, survival probability, and interval probability for an exponential distribution - [Fitness Workout PDF Generator](https://elysiatools.com/en/tools/fitness-workout-pdf-generator): Generate a printable workout-plan PDF from JSON or CSV data with exercise tables, progress boxes, and optional QR video links - [OCR PDF to Structured JSON Bridge](https://elysiatools.com/en/tools/ocr-pdf-to-structured-json-bridge): Extract the PDF text layer with geometry (lines by y-position, tables by column gaps, headings by font size, colon key-value pairs), then fill a user-supplied JSON Schema field by field — labels matched by normalized keys, values coerced to declared types and validated with ajv. - [PDF Extract Selected Pages](https://elysiatools.com/en/tools/pdf-extract-selected-pages): Extract selected pages from one PDF and generate a new PDF - [PDF Form Data Export](https://elysiatools.com/en/tools/pdf-form-data-export): Read AcroForm field names, types and current values from a PDF and export them to JSON - [PDF to JSON Structure Explorer](https://elysiatools.com/en/tools/pdf-to-json-structure-explorer): Extract structured OpenDataLoader JSON from a PDF and browse headings, paragraphs, tables, lists, pages, and bounding boxes in an explorer view - [T Distribution Calculator](https://elysiatools.com/en/tools/t-distribution-calculator): Calculate Student t distribution PDF, CDF, tail probability, two-tail probability, and central area ## Samples - [Web Image Processing Python Samples](https://elysiatools.com/en/samples/web-image-processing-python): Web Python image processing examples using PIL/Pillow including reading, saving, resizing, and format conversion - [PDF Samples](https://elysiatools.com/en/samples/pdf-samples): Generated PDF samples from tools dated 2026-02-01 to 2026-02-10 - [Android Image Processing Java Samples](https://elysiatools.com/en/samples/android-image-processing-java): Android Java image processing examples including reading/saving images, scaling, and format conversion - [Android Image Processing Kotlin Samples](https://elysiatools.com/en/samples/android-image-processing-kotlin): Android Kotlin image processing examples including reading/saving images, scaling, and format conversion ## Related content - [PDF Conversion and Document Export Tools](https://elysiatools.com/en/hubs/pdf-convert): Compare tools that convert documents, images, and structured extractions into or out of PDF in one hub for publishing, sharing, and downstream processing. - [PDF Assembly, Layout, and Protection Tools](https://elysiatools.com/en/hubs/pdf-utility): Compare PDF page assembly, layout control, watermarking, stationery overlays, anonymization, password protection, and redaction helper tools in one hub. - [Printable PDF Layout and Template Generators](https://elysiatools.com/en/hubs/pdf-generate): Curated tools for printable PDF layout generation and reusable document templates in one hub. - [Document OCR and Structured Extraction Tools](https://elysiatools.com/en/hubs/document-ocr-extraction): Extract text, Markdown, JSON, tables, captions, and RAG-ready chunks from scanned PDFs and document images with OCR and structure-aware workflows.