Chi-Square Goodness of Fit Calculator

It is a formal statistical test that determines whether observed hydrological data — such as flood peaks, annual rainfall, or streamflow — follows a specified theoretical probability distribution. It quantifies the discrepancy between what was observed and what a theoretical model predicts, giving a statistically defensible basis for distribution selection in frequency analysis.

The chosen probability distribution directly determines design flood estimates for critical structures like dams, bridges, and spillways. An incorrect distribution can lead to either dangerous underestimation of design flows or costly over-engineering. The Chi-Square test provides a formal, reproducible criterion for accepting or rejecting a candidate distribution.

Degrees of freedom (df) = k − 1 − m, where k is the number of class intervals and m is the number of distribution parameters estimated from the data. Subtracting parameters accounts for the fact that fitting parameters uses some of the data's information, reducing independent variation. For example, a Normal distribution fit has m = 2 (mean and SD), so with 6 bins: df = 6 − 1 − 2 = 3.

Bins with E < 5 violate a core assumption of the chi-square approximation, causing the test statistic to be unreliable — typically inflating χ², leading to false rejections. Adjacent bins must be merged until all expected frequencies are ≥ 5. This calculator warns you when this condition is violated and includes an auto-merge button.

The Gumbel (Extreme Value Type I) distribution arises naturally as the limiting distribution of the maximum of a large number of independent identically distributed variables. Annual flood peaks are the maximum of daily flows — so by the Extreme Value Theorem, they tend toward a Gumbel distribution. It has a heavier upper tail than the Normal, reflecting the higher probability of extreme events.

LP-III is the log-transformed version of the Pearson Type III (Gamma) distribution. It can accommodate positive skewness common in flood data. The US Army Corps of Engineers mandates its use (Bulletin 17C) for flood frequency analysis. It has three parameters: mean of log(Q), standard deviation of log(Q), and skewness coefficient of log(Q), so m = 3.

The K-S test works on the continuous CDF without requiring data grouping. It is preferred for: small samples (n < 50), continuous distributions where natural bin boundaries are unclear, or when you want to test at every point in the distribution rather than just within bins. Chi-Square is better when data is naturally grouped or when you have n ≥ 50 with clear class intervals.

The p-value is the probability of obtaining a χ² statistic as large as the calculated value if the null hypothesis (good fit) is true. A small p-value (< α) indicates the observed data would be unlikely under the assumed distribution — evidence against it. p > α means the data is consistent with the assumed distribution. This calculator provides an approximation using a gamma function based method.

Significantly sensitive. Too few bins reduce power; too many create bins with low expected frequencies. A common rule of thumb is to use k = 1 + 3.3 × log₁₀(n) bins (Sturges' rule). Equal-probability bins (each bin has the same expected frequency) are generally preferred over equal-width bins as they distribute information evenly. Always try multiple binning strategies to check robustness.

For goodness-of-fit, only the upper tail matters — a very large χ² indicates a poor fit. Very small χ² (better than expected) can sometimes indicate data manipulation, but standard practice uses only the upper-tail critical value. Two-tailed versions are used in chi-square tests of independence (contingency tables), which is a different application.

Absolutely. The test is generic and applicable to any frequency data: structural load distributions, wind speed analysis, earthquake magnitude frequency, traffic volume, or any engineering variable where you wish to validate a theoretical model against observations. The principles remain the same regardless of the physical domain.

At minimum n = 50 observations are recommended, with each bin having E ≥ 5. Below n = 30, results are unreliable. For very small datasets (n < 30), consider the Anderson-Darling test, which is more powerful for small samples and works on continuous distributions without binning. Larger samples (n > 100) give the chi-square test high statistical power.

Expected frequency for bin i: Eᵢ = n × P(aᵢ ≤ X ≤ bᵢ), where P is calculated from the distribution's CDF. For Gumbel: P = exp(−exp(−y)) where y = (x − μ)/σ (reduced variate). For Normal: P = Φ((b − μ)/σ) − Φ((a − μ)/σ) where Φ is the standard normal CDF. Parameters μ and σ are estimated from the data using method of moments or maximum likelihood.

Probability paper plots are graphical methods — you plot data on specially scaled axes so that a fitted distribution appears as a straight line. The chi-square test is the formal numerical counterpart, providing a quantitative decision criterion. Best practice is to use both: plot to visually assess fit and identify outliers, then use chi-square to formally confirm or reject the distribution choice.

Ties (identical values) can create artificial bin boundaries. When multiple observations fall exactly on a bin boundary, assign them consistently to one bin (e.g., always to the upper bin using the convention aᵢ ≤ x < bᵢ₊₁). Document your convention. For continuous distributions, the probability of exactly tied values is theoretically zero, so ties often indicate measurement resolution issues.