Solve chi-square density, probability, and inverse values easily. Built for engineering analysis and process validation. Use clean inputs, fast outputs, and practical reference tables.
| Case | Degrees of freedom | x value | Left-tail CDF | Right-tail | Engineering use | |
|---|---|---|---|---|---|---|
| 1 | 4 | 2.50 | 0.1791 | 0.3554 | 0.6446 | Variance review in a lab process |
| 2 | 8 | 10.00 | 0.0877 | 0.7349 | 0.2651 | Quality control acceptance check |
| 3 | 12 | 18.00 | 0.0404 | 0.8847 | 0.1153 | Reliability spread comparison |
| 4 | 20 | 31.41 | 0.0138 | 0.9500 | 0.0500 | Upper critical limit estimation |
The chi-square density formula is:
f(x; k) = [1 / (2^(k/2) Γ(k/2))] × x^((k/2)-1) × e^(-x/2)
Here, x is the test value and k is the degrees of freedom.
The cumulative probability uses the regularized lower incomplete gamma function:
F(x; k) = P(k/2, x/2)
The right-tail probability is:
P(X > x) = 1 - F(x; k)
Critical values are solved numerically. This file uses a stable bisection search on the cumulative distribution.
The x² distribution, also called the chi-square distribution, is important in engineering statistics. It helps measure spread, variation, and fit. Engineers use it when process data follows a normal assumption and variance must be tested. It is common in quality control, reliability studies, sensor calibration, and tolerance analysis.
This distribution supports hypothesis testing for variance. It also helps compare observed and expected frequencies. In manufacturing, it can test whether a production line holds a target variation. In materials work, it can examine sample consistency. In electronics, it can support noise analysis and repeated measurement review.
This calculator handles three practical tasks. First, it evaluates an x value and returns the density, cumulative probability, and right-tail probability. Second, it finds a critical x value from a left-tail probability. Third, it finds a critical x value from a right-tail probability. These outputs are useful for acceptance limits, rejection regions, and engineering decisions.
The PDF shows relative density at a specific point. The CDF shows the probability that the random value is at or below x. The right-tail result shows the probability above x. A small right-tail value can signal an unusual result. That matters in process control and model checking.
Degrees of freedom control the shape of the curve. Low values create a strong right skew. Higher values make the curve more spread out and balanced. Always confirm that the statistical assumptions match your engineering data. The calculator is strong for analysis, but final decisions should still use proper sampling, domain knowledge, and validated test procedures.
It computes chi-square density, cumulative probability, right-tail area, and critical x values. It also shows mean, variance, standard deviation, mode, and an approximate median.
Degrees of freedom define the curve shape. Small values create more skew. Larger values spread the distribution and move the center to the right.
Use it when your test or acceptance rule depends on the probability above a chosen value. This is common in upper-limit variance tests and rejection-region work.
No. The chi-square variable starts at zero. Negative x values are not valid for this distribution.
Yes. Engineers often use chi-square methods to study process variance, check sample consistency, and support decisions in inspection and process validation.
The PDF gives the density at one point. The CDF gives the cumulative probability from zero up to that point.
The script uses a numerical search. It repeatedly narrows the range until the cumulative probability matches the requested probability closely.
Yes. It can support variance checks, repeated test analysis, and probability-based review in many reliability and engineering data workflows.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.