Chi-Square Critical Values Table

Enter your degrees of freedom and a df range to look up chi-square critical values across all common significance levels. Exact values for df 1–30; Wilson-Hilferty approximation beyond.

Created by Luis GonzalezLast updated: May 9, 2026

How to Use This Calculator

1
Enter Degrees of Freedom
Input the degrees of freedom (df) for your chi-square test. This value is typically related to the number of categories in your data.
2
Specify Table Range — Min df
Set the starting degrees of freedom for the detailed table display. This helps narrow down the relevant section of the table.
3
Specify Table Range — Max df
Set the ending degrees of freedom for the detailed table display. For values above 30, the calculator uses an approximation.
4
Review Your Results
Observe the critical chi-square values for various common significance levels (α) at your specified degrees of freedom, and browse the full table range.

Example Calculation

A researcher conducting a chi-square test with 5 degrees of freedom needs to find the critical value for a significance level of α = 0.05 to determine statistical significance.

Degrees of Freedom

Table Range — Min df

Table Range — Max df

Results

11.070

Tips

Understand Degrees of Freedom

For a goodness-of-fit test, degrees of freedom are `k - 1`, where `k` is the number of categories. For a test of independence, it's `(rows - 1) × (cols - 1)`.

Interpret Significance Levels

A common significance level (α) is 0.05, meaning there's a 5% chance of rejecting a true null hypothesis. For stricter tests, use α = 0.01 or 0.001.

Use Approximation for Large df

For degrees of freedom (df) greater than 30, the chi-square distribution approximates a normal distribution. While this calculator uses the Wilson-Hilferty approximation, statistical software provides more precise values for very large df.

Understanding Chi-Square Critical Values

The Chi-Square Critical Values Table provides essential reference points for statistical hypothesis testing, allowing researchers and analysts to determine the significance of observed data. This tool helps you quickly find the critical chi-square value for a given number of degrees of freedom at various common significance levels (α), such as 0.10, 0.05, 0.025, 0.01, and 0.001. For instance, with 5 degrees of freedom, a critical value of 11.070 at α = 0.05 indicates the threshold for rejecting a null hypothesis in 2025.

Navigating Statistical Tables

Statistical tables, like the Chi-Square Critical Values Table, are fundamental tools in inferential statistics, providing pre-calculated values that simplify hypothesis testing. Instead of computing complex probabilities for every test, researchers can quickly look up the critical value associated with their specific degrees of freedom and chosen significance level. This allows for a swift comparison with their calculated test statistic. The primary purpose is to help researchers make informed decisions about whether to reject or fail to reject a null hypothesis, guiding conclusions about the relationships or distributions within their data. Mastering the use of these tables is a cornerstone of statistical literacy.

The Mathematics of Chi-Square Critical Values

The chi-square critical values are not generated by a simple formula but are derived from the inverse of the cumulative distribution function (CDF) of the chi-square distribution. For a given degrees of freedom (df) and a significance level (α), the critical value is the point x such that the area under the chi-square probability density function to the right of x equals α.

For example, to find the critical value at α = 0.05 for df:

P(X > critical value) = α

where X follows a chi-square distribution with df degrees of freedom. For df up to 30, these values are typically found in pre-computed tables. Beyond df = 30, approximations like the Wilson-Hilferty transformation are often used, which relates the chi-square distribution to a normal distribution for easier calculation.

💡 To understand the full properties of the chi-square distribution, including its mean, variance, and skewness, our Chi-Squared Distribution Calculator can provide a deeper dive into its characteristics.

Looking Up a Critical Value for a Five-Category Test

Consider a scenario where a social scientist is conducting a chi-square goodness-of-fit test to see if observed frequencies in five categories differ significantly from expected frequencies. For this test, the degrees of freedom (df) would be k - 1, where k is the number of categories. With five categories, df = 5 - 1 = 4. The researcher chooses a common significance level of α = 0.05.

To find the critical value:

Identify Degrees of Freedom: The degrees of freedom are 4.
Select Significance Level: The alpha level is 0.05.
Consult the Table: Using a chi-square critical values table, locate the row for 4 degrees of freedom and the column for α = 0.05.

The critical value found would be 9.488. If the calculated chi-square test statistic from the social scientist's data exceeds 9.488, they would reject the null hypothesis, indicating a statistically significant difference between observed and expected frequencies at the 5% level.

💡 If your data involves comparing observed vs. expected frequencies across multiple categories, our Chi-Squared Goodness of Fit Calculator can directly compute your test statistic and p-value.

The Origins of the Chi-Square Distribution

The chi-square (χ²) distribution and its critical values are cornerstones of modern statistics, largely attributed to the pioneering work of Karl Pearson. In 1900, Pearson introduced the chi-square test as a method to assess the "goodness of fit" between observed frequencies and those expected under a theoretical distribution. His seminal paper, "On the Criterion that a System of Deviations from the Probable in the Case of a Correlated System of Variables is such that it can be Reasonably Supposed to have Arisen from Random Sampling," laid the mathematical foundation. Pearson's work provided a robust framework for hypothesis testing, allowing scientists to quantify the discrepancy between empirical observations and theoretical predictions, which quickly became indispensable across various scientific disciplines.

Navigating Statistical Tables

Frequently Asked Questions

What is a chi-square critical value?

A chi-square critical value is a threshold used in hypothesis testing to determine whether an observed chi-square test statistic is statistically significant. If your calculated chi-square statistic exceeds this critical value for a given degrees of freedom and significance level (alpha), you reject the null hypothesis. These values are derived from the chi-square distribution and are essential for making inferential conclusions in statistical analysis.

How do degrees of freedom impact the critical value?

Degrees of freedom (df) directly impact the chi-square critical value by changing the shape of the chi-square distribution. As the degrees of freedom increase, the chi-square distribution becomes less skewed and shifts to the right, leading to larger critical values for the same significance level. This reflects that more variability is expected when more categories or independent pieces of information are involved in the calculation.

When should I use different alpha levels (significance levels)?

Different alpha levels are chosen based on the desired level of confidence and the consequences of making a Type I error (falsely rejecting the null hypothesis). A common alpha is 0.05, suitable for many social science or biological studies. For studies where false positives are more costly or undesirable, such as in medical trials or quality control, stricter alpha levels like 0.01 or 0.001 are used to demand stronger evidence for significance.