Interactive module guide

Chi-Square Distribution Calculator & Guide

This chi-square distribution calculator lets you calculate right-tail probabilities and critical values and explore how degrees of freedom reshape the curve.

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The chi-square distribution is a continuous, right-skewed probability model supported on nonnegative values.

It arises by construction: the sum of the squares of k independent standard normal variables follows a chi-square distribution with k degrees of freedom, which makes it the standard reference distribution for count-based test statistics and for inference about a normal variance.

Use this page to examine the density, cumulative probability, quantiles, and tail areas implied by a chosen k; the page evaluates the reference model itself and does not run a chi-square test on data.

Is this the right module for my question?

Use it when

Use a chi-square model when a statistic is built, exactly or approximately, as a sum of squared standardized discrepancies. The classical exact case is variance inference for a normal population, where (n−1)S²/σ² follows a chi-square distribution with n−1 degrees of freedom, so chi-square quantiles set the endpoints of a confidence interval for σ².
It is also the standard large-sample reference for Pearson statistics in goodness-of-fit and independence problems. In that role the plotted variable is an idealized test statistic under the null hypothesis, not a raw measurement, so state the null model and the degrees-of-freedom calculation before interpreting any area under the curve.
Choose this page when the question asks for a critical value such as Q(0.95), a right-tail probability P(X>x), or an interval probability P(a<X≤b) at stated degrees of freedom. The quantile lookup, point calculator, and interval calculator connect those requests to the same plotted density and cumulative curve.

Choose another method when

Do not use a chi-square reference for a statistic that was not constructed as a sum of squared standardized terms, and do not apply it to variables that can be negative; the model places zero probability below zero. A statistic with estimated parameters or dependent contributions may follow a different distribution entirely.
The chi-square reference for count-based tests is an approximation that requires adequate expected counts. With small expected frequencies or sparse tables, the true sampling distribution of the statistic can depart from chi-square, and an exact or simulation-based method is the more defensible route.
Avoid it as a default model for raw skewed measurements. One parameter forces the variance to equal twice the mean, a constraint most data sets will not obey; the two-parameter gamma family, of which chi-square is a special case, usually represents such measurements more honestly.

Interactive tool

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Example preview

Right-tail chi-square threshold

Inputs: Chi-square with k=5; find the 95th percentile.
Representative result: Q(0.95) ≈ 11.0705, the usual 5% critical value for df = 5.

Illustrative only. Load the interactive tool to enter your own values and review assumptions.

Watch the explanation

3:03 min

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Read the complete transcript

The Chi-Square Distribution. Why do squared standardized discrepancies keep appearing in variance estimates and categorical test statistics? Square k independent standard normal variables and add them to construct a chi-square variable. Squaring removes every sign, so the resulting support begins at zero and extends rightward. Degrees of freedom count the independent squared contributions retained in this reference construction. Equivalently, chi-square is a gamma distribution with shape k over two and scale two. Independent chi-square variables add cleanly, with their degrees of freedom added as well.

At finite k the density is right-skewed, especially when few squared terms contribute. For k above two the mode is k minus two; otherwise it sits at zero. The mean equals k, so increasing degrees of freedom shifts the raw distribution rightward. Its variance equals two k and its standard deviation is the square root of two k. Skewness is square root of eight over k, shrinking as more contributions accumulate. Excess kurtosis is twelve over k, so tail excess also declines toward zero. Center by k and scale by square root two k before comparing with standard normal. Density height remains distinct from probability; probabilities always require area over a stated event.

Consider a chi-square reference with five degrees of freedom and an upper five-percent boundary. Its mean is five, variance ten, standard deviation square root ten, and mode three. An upper five-percent critical boundary is the quantile at cumulative probability zero point nine five. For k equals five, that exact quantile is approximately eleven point zero seven zero five. The survival area beyond eleven point zero seven zero five is therefore exactly five percent. The five and ninety-five percent quantiles enclose a central ninety percent, but asymmetrically. Exact chi-square quantiles preserve that asymmetry better than a casual symmetric normal approximation.

Goodness-of-fit and independence statistics use an upper tail because squared discrepancies only accumulate. Normal-variance intervals instead use separate lower and upper chi-square quantiles, not mirrored boundaries. The explorer supplies reference areas and quantiles; it does not calculate a statistic from data. Degrees of freedom must come from the design, including reductions for parameters estimated from data. Count-based procedures also require independent observations and expected counts adequate for the approximation. Report k, the exact event, and whether the result is density, area, or quantile. Explore chi-square tails, critical values, and changing degrees of freedom free at Distri Scope dot com.

How to read the result

Read the PDF as relative concentration, with area rather than curve height carrying probability; the point calculator reports the density, cumulative, and survival values separately at any x≥0. For every k the curve is right-skewed, so the mean k lies above the median, and the CDF rises from zero at the origin toward one without ever decreasing.

In testing applications the usual rejection region is the right tail, because squared standardized discrepancies can only accumulate: the worse the disagreement between observation and null model, the larger the statistic. Report the survival probability P(X>x) together with k, and remember that a small right-tail area describes the reference model, not the truth of any specific alternative.

Inputs and parameter meaning

The only parameter is the degrees of freedom k, set with a slider from 1 to 50 in integer steps with a default of 5. In the construction, k counts the independent squared standard normal terms; in testing applications it comes from the design, for example the number of categories minus one or (rows−1)(columns−1), not from the size of the data values.

The Key Statistics panel updates as the slider moves: the mean equals k, the variance equals 2k, the mode equals max(k−2, 0), the skewness equals √(8/k), and the excess kurtosis equals 12/k. The panel also reports the median, which has no simple closed form. Because one parameter controls everything, location and spread cannot be adjusted independently as they can for a normal model.

Shapes differ sharply at small k, and for k≤2 the density has no interior mode. At k=1 the density rises without bound as x approaches zero, at k=2 it coincides with an exponential distribution and decreases from its maximum at zero, and only from k=3 onward does an interior peak appear at k−2, after which the curve increasingly resembles a skewed bell.

Formula and available parameters

Density or probability mass

f(x) = \frac{1}{2^{k/2}\Gamma(k/2)}\, x^{k/2 - 1} e^{-x/2}, \quad x > 0

Mean

\mu = k

Variance

\sigma^2 = 2k

Degrees of freedom (k): interactive range 1 to 50; default 5.

What the method is doing

If each of k independent variables is standard normal, the sum of their squares is chi-square with k degrees of freedom. This construction explains the support on x≥0, gives the additivity property that independent chi-square variables sum to a chi-square variable with the degrees of freedom added, and identifies the family as the gamma distribution with shape k/2 and scale 2.

The skewness √(8/k) and excess kurtosis 12/k both shrink toward zero as k grows, so the standardized distribution approaches a normal shape, a direct instance of the central limit theorem applied to the k squared terms. The approach is slow in the tails, so for critical values the explorer computes the exact chi-square quantile rather than substituting a normal approximation.

Worked example: the 5% critical value at five degrees of freedom

Suppose an analyst has computed a goodness-of-fit statistic elsewhere for a design with five degrees of freedom and needs the conventional 5% critical value together with a picture of the reference distribution. Leave the slider at the default k=5, so the Key Statistics panel shows mean 5, variance 10, and mode 3.

1Use the quantile lookup with p=0.95. It returns Q(0.95)≈11.0705, so P(X>11.0705)≈0.05 under a chi-square model with five degrees of freedom; a correctly constructed statistic above 11.0705 falls in the conventional 5% right-tail rejection region. Confirm the round trip with the point calculator, whose survival value at x=11.0705 is approximately 0.05.
2Compare the centers reported in the panel: the median is about 4.3515, below the mean of 5, while the mode is 3. The ordering mode, then median, then mean is the numerical signature of right skew and matches the skewness √(8/5)≈1.265 shown for this k.
3Slide k upward and watch the skewness fall toward zero as the curve becomes more symmetric around its mean, the visual basis of the normal approximation for large degrees of freedom. Because the critical value changes with k, record the configuration by sharing the URL or exporting the chart as PNG or CSV for the analysis file.

Interpretation

The number 11.0705 is a property of the reference distribution, not of any data set. A statistic with five degrees of freedom exceeding it lands in a region that occurs about 5% of the time when the null model holds and the large-sample approximation is adequate. Whether the statistic was built correctly, whether expected counts support the approximation, and what a rejection means scientifically are questions for the testing procedure and study design, not for this page.

Common mistakes

The most consequential error is degrees-of-freedom bookkeeping: using the number of observations or the number of cells instead of the design-based count, such as categories minus one, reduced further when parameters are estimated from the data. A tail area computed at the wrong k is precisely wrong.
Do not double the right-tail area by habit. Fit and independence tests are one-sided in the right tail because discrepancies only accumulate, while a confidence interval for a normal variance uses both a lower and an upper chi-square quantile, so the convention depends on the task.
Do not read a statistic far below the mean as strong evidence of a good model. An implausibly small chi-square value can signal dependence between observations, overfitting, or agreement with expectations closer than random sampling permits, which deserves investigation rather than celebration.

Limits and independent validation

This explorer evaluates the chi-square reference distribution for a chosen k; it does not run chi-square tests of independence or goodness of fit, estimate degrees of freedom from data, or fit a distribution to observations. Compute test statistics in the hypothesis test calculator and use a separate estimation workflow when data are involved.

The slider covers integer degrees of freedom from 1 to 50 only. Fractional degrees of freedom, which arise in some corrected approximations, and very large k are out of range; for very large k a normal approximation with mean k and variance 2k becomes serviceable, though exact values from statistical software remain preferable for tail work.

Before using the result

Before relying on a chi-square reference for a test, verify that the statistic is a genuine sum of squared standardized discrepancies, that the observations are independent, and that the expected counts meet the usual adequacy guidance. Nothing in a tail area can certify these preconditions.
Recompute one quantile or one tail probability in an independent package or table and compare it with the explorer. Record k, the exact event, whether the number is a density, CDF, survival, or interval probability, and the endpoint convention, so another person can reproduce the result.

chi-square to normal convergence →Watch the right skew fade and the standardized chi-square shape approach a normal reference as degrees of freedom grow.chi-square goodness-of-fit test →Compute an observed-versus-expected test statistic there; this page supplies only the reference distribution and its critical values.gamma distribution explorer →Place chi-square in its parent family as a gamma distribution with shape k/2 and scale 2.

← Student’s t Distribution F Distribution →

See every option in the Probability Distribution Explorer or review the DistriScope methodology. Educational information; last reviewed 2026-08-09. Verify consequential calculations independently.