The Chi Square Distribution

Statistics • Chi Square Tests

Written by STEM Calculators Team Published December 25, 2025 Updated February 24, 2026

Task

The chi-square distribution (χ²) depends on the degrees of freedom (df) and takes only nonnegative values. Use this calculator to move between areas (probabilities) and χ² values.

Degrees of freedom (df)

df must be an integer ≥ 1.

Area type

You can switch tails; the visualization updates to match.

Tail area

Enter a probability between 0 and 1 (exclusive). Example: 0.10.

Numeric precision

Higher precision uses more iterations.

χ² value (x)

x must be ≥ 0.

Show

Useful when you want the p-value in the right tail but also want the left area.

Lower value (a)

a ≥ 0.

Upper value (b)

b ≥ a.

Ready

χ² model visualization

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The shaded area updates after calculation.

Quick notes

χ² values are always ≥ 0.
The curve is right-skewed for small df and becomes more symmetric as df increases.
Right-tail areas are common when finding critical values for hypothesis tests.

Key relationships

\[ \begin{aligned} \text{Left area} &= P(\chi^2 \le x) \\ \text{Right area} &= P(\chi^2 \ge x) = 1 - P(\chi^2 \le x) \end{aligned} \]

Enter values and click Calculate.

Batch mode: paste CSV data (compute many rows at once)

Paste rows as CSV (comma-separated). Header is optional. Supported columns: task (crit/area/between), df, tail (right/left), and depending on task: area (for crit), x (for area), a, b (for between).

CSV input

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How to use this calculator

This tool works with the chi-square distribution (χ²) and a chosen degrees of freedom (df). It supports two common tasks:

Quantile (critical value): find the χ² value x that corresponds to a specified tail area.
Tail area: compute the probability in a tail when a χ² value x is given.

Mode 1: Quantile (critical value)

Use this when you know df and a tail area (often called α) and want the cutoff value x.

Most printed tables are organized by right-tail area. If your given area is in the left tail, convert it first using:

\[ \alpha_{\text{right}} = 1 - \alpha_{\text{left}} \]

Enter Degrees of freedom (df).
Select whether your given area is in the right tail \(P(X \ge x)\) or left tail \(P(X \le x)\).
Enter the tail area (between 0 and 1).
Click Calculate to get the critical value x.

Mode 2: Tail area

Use this when you know df and a chi-square value x and want a probability. The calculator can report either tail:

\[ \begin{aligned} \text{Left tail: } &P(X \le x) \\ \text{Right tail: } &P(X \ge x) = 1 - P(X \le x) \end{aligned} \]

Enter Degrees of freedom (df).
Enter the chi-square value x (must be ≥ 0).
Select which tail area you want to report.
Click Calculate to see the tail probability.

Batch (CSV) input

Use Batch (CSV) when you want multiple critical values at once. Paste rows with: df, tail (right/left), area. Header row is optional.

Example format (you can copy/paste into the calculator):

df,tail,area
7,right,0.10
12,left,0.05

For a left-tail area, the calculator converts it internally using \( \alpha_{\text{right}} = 1 - \alpha_{\text{left}} \).

Understanding the graph

The curve shows the χ² distribution for your chosen df.
The dashed vertical line marks the current x value (critical value or given value).
The shaded region is the selected tail area being used or reported.

Frequently Asked Questions

What is the chi-square distribution used for in statistics?

The chi-square distribution models sums of squared standard normal variables and depends on degrees of freedom. It is commonly used in chi-square tests (goodness of fit, independence, homogeneity) and for inference about a population variance.

What does degrees of freedom mean for a chi-square distribution?

Degrees of freedom (df) controls the shape of the chi-square curve. Small df produces a strongly right-skewed distribution, and the curve becomes more symmetric as df increases.

How do left-tail and right-tail areas relate for chi-square probabilities?

For a fixed df and value x, the left-tail area is P(chi-square <= x) and the right-tail area is P(chi-square >= x). They satisfy right tail = 1 - left tail.

What is a chi-square critical value?

A chi-square critical value is the cutoff x that leaves a specified tail area under the chi-square curve for a given df. It is often used as a decision threshold in hypothesis tests.

How is the probability between two chi-square values computed?

The probability between a and b is P(a <= chi-square <= b), which equals the difference between the left-tail areas at b and at a for the same df. The calculator reports this using your selected inputs.

Quick notes

Calculation steps

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Frequently Asked Questions

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