Chi-Square Distribution Calculator

Calculator Inputs

Degrees of Freedom

X Value

Lower-Tail Probability p

Significance Level α

Range Start

Range End

Calculate Reset

Example Data Table

This sample table uses 8 degrees of freedom.

Formula Used

Probability density function:

f(x; k) = 1 / (2^k/2 Γ(k/2)) × x^(k/2)-1 × e^-x/2, for x ≥ 0

Cumulative distribution function:

F(x; k) = P(k/2, x/2)

Here, P is the regularized lower incomplete gamma function.

Upper-tail probability:

P(X > x) = 1 - F(x; k)

Inverse chi-square:

Find x such that F(x; k) = p

Key moments:

Mean = k, Variance = 2k, Standard Deviation = √(2k), Mode = max(k - 2, 0)

How to Use This Calculator

1. Enter the degrees of freedom. This controls the distribution shape.
2. Enter an x value to compute the density and cumulative probability.
3. Enter a lower-tail probability p to estimate the inverse chi-square value.
4. Enter a significance level α to get left, right, and two-tailed critical values.
5. Enter a start and end point to estimate probability across an interval.
6. Press Calculate. The result will appear below the header and above the form.
7. Use the export buttons to save the current result as CSV or PDF.

About This Chi-Square Distribution Calculator

What this tool does

This chi-square distribution calculator helps you study distribution behavior from a chosen degrees of freedom value. It estimates probability density, lower-tail probability, upper-tail probability, interval probability, and inverse chi-square values. It also reports critical values for left-tail, right-tail, and two-tailed tests. These outputs are useful in statistics classes. They also help in research, auditing, process control, and hypothesis testing.

Why degrees of freedom matter

Degrees of freedom control the shape of the chi-square curve. Small values create a strong right skew. Larger values make the distribution wider and more balanced. The mean equals the degrees of freedom. The variance equals twice that amount. Because of this, a change in degrees of freedom changes both the center and spread. That is why a reliable chi-square distribution calculator should show more than one summary value.

How the results help

The PDF tells you the curve height at one x value. The CDF tells you the probability of observing a value at or below x. The upper-tail result gives the area to the right. This is often used as a p-value component in statistical tests. The inverse function works in the opposite direction. It finds the x value linked to a chosen cumulative probability. Critical values help you set rejection regions for chi-square procedures.

Common use cases

You can use this page for chi-square goodness-of-fit tests, variance confidence intervals, and contingency table work. It is also helpful when checking thresholds in quality studies or analytics tasks. The example table gives a quick reference point. The export tools help you move results into reports or worksheets. This calculator keeps the workflow simple, but it still covers the core outputs needed for serious statistical work. The layout also supports quick review during lessons, meetings, and written reporting. It reduces manual lookup and helps verify textbook or spreadsheet answers with less friction during repeated review tasks.

FAQs

1. What does this calculator return?

It returns core chi-square distribution results. These include PDF, lower-tail probability, upper-tail probability, inverse values, interval probability, and critical values. It also shows summary measures such as mean, variance, standard deviation, mode, median approximation, and skewness.

2. What are degrees of freedom?

Degrees of freedom define the shape of the chi-square distribution. Lower values produce stronger skew. Higher values spread the curve and move the center rightward. Many statistical procedures choose degrees of freedom from sample size or table structure.

3. What is the lower-tail probability?

The lower-tail probability is P(X ≤ x). It measures the area under the chi-square curve from zero to the selected x value. This is the cumulative distribution function, also called the CDF.

4. What is the upper-tail probability?

The upper-tail probability is P(X > x). It equals one minus the lower-tail probability. This value is important when you evaluate right-tail chi-square tests or compare an observed statistic with a critical threshold.

5. What does the inverse chi-square result mean?

The inverse result gives the x value tied to a chosen lower-tail probability. For example, if p is 0.95, the calculator finds the chi-square point where 95 percent of the distribution lies to the left.

6. Why are critical values useful?

Critical values define cutoff points for statistical decisions. You can use them in variance tests, confidence interval construction, and chi-square hypothesis testing. Left-tail, right-tail, and two-tailed values support different decision rules.

7. Can I calculate interval probability?

Yes. Enter a range start and range end. The calculator estimates the probability that the chi-square variable falls between those two x values. It does this by subtracting one cumulative probability from another.

8. Can I export the result?

Yes. After you calculate, use the CSV button to download the result table. Use the PDF button to save a clean report of the same metrics for records, sharing, or documentation.