Automatic Distribution Fit Analyzer
Free Automatic distribution fit Calculator for ai enhanced. Enter parameters to get optimized results with detailed breakdowns.
Formula
AD = -n - (1/n) * Sum[(2i-1)(ln F(Yi) + ln(1-F(Y(n+1-i))))]
The Anderson-Darling statistic measures the distance between the empirical and theoretical cumulative distribution functions. F(Yi) is the CDF of the fitted distribution evaluated at the i-th sorted observation. Lower AD values indicate a better fit. The test is computed for each candidate distribution and the one with the lowest AD statistic is selected as the best fit.
Worked Examples
Example 1: Manufacturing Quality Data — Normal Fit
Problem: Analyze 20 measurements of bolt diameters (mm): 12.3, 14.1, 11.8, 13.5, 12.9, 15.2, 11.2, 14.8, 13.1, 12.6, 13.9, 14.5, 12.1, 13.7, 11.9, 14.3, 13.0, 12.4, 15.0, 13.3
Solution: Mean = 13.18, StdDev = 1.08, Skewness = 0.12, Kurtosis = -0.94\nAnderson-Darling: Normal = 0.21, Log-Normal = 0.23, Uniform = 0.45\nBest fit: Normal distribution (lowest AD statistic)\nInterpretation: Near-zero skewness and mild negative kurtosis consistent with normal data.
Result: Best Fit: Normal (Mean = 13.18, StdDev = 1.08) | AD = 0.21
Example 2: Income Data — Log-Normal Fit
Problem: Analyze household incomes ($K): 35, 42, 48, 55, 58, 62, 67, 72, 85, 95, 110, 125, 150, 200, 350
Solution: Mean = 103.6, StdDev = 82.8, Skewness = 1.87, Kurtosis = 3.21\nAnderson-Darling: Normal = 1.42, Log-Normal = 0.31, Exponential = 0.89\nBest fit: Log-Normal distribution\nInterpretation: Strong positive skewness and high kurtosis indicate right-skewed data consistent with log-normal.
Result: Best Fit: Log-Normal (LogMean = 4.38, LogStd = 0.60) | AD = 0.31
Frequently Asked Questions
What is distribution fitting and why is it important in data analysis?
Distribution fitting is the process of selecting a probability distribution that best describes a given dataset based on statistical criteria. It is fundamentally important because many statistical methods, machine learning algorithms, and engineering reliability models assume the underlying data follows a specific distribution. Correctly identifying the distribution allows analysts to make accurate predictions, calculate probabilities of future events, perform hypothesis tests, construct confidence intervals, and run simulations. For example, quality control in manufacturing relies on knowing whether defect rates follow a Poisson or normal distribution to set proper control limits and predict failure rates accurately.
How does the Anderson-Darling test work for distribution fitting?
The Anderson-Darling (AD) test is a goodness-of-fit test that measures how well a sample dataset follows a specific theoretical distribution. It computes a test statistic by comparing the empirical cumulative distribution function of the data against the theoretical CDF. The AD test gives more weight to the tails of the distribution compared to other tests like Kolmogorov-Smirnov, making it more sensitive to deviations in extreme values. A lower AD statistic indicates a better fit. The formula involves summing weighted differences between observed and expected cumulative probabilities across all sorted data points. Critical values depend on the distribution being tested and the sample size.
What sample size is needed for reliable distribution fitting?
The reliability of distribution fitting increases substantially with sample size. As a general guideline, a minimum of 30 data points is recommended for basic distribution identification, though 50 to 100 observations provide more reliable results. For distinguishing between similar distributions (such as normal versus log-normal when skewness is mild), 100 or more observations may be necessary. With fewer than 20 data points, goodness-of-fit tests have low statistical power and may fail to reject incorrect distributions. The Anderson-Darling test performs reasonably well with samples as small as 8 to 10 observations for detecting major departures from normality, but subtle distributional differences require larger samples for confident identification.
How do skewness and kurtosis help identify the right distribution?
Skewness and kurtosis are key shape statistics that provide initial clues about which distribution family might fit the data. Skewness measures asymmetry: a value near zero suggests symmetry (normal, uniform), positive skewness suggests a right tail (log-normal, exponential, gamma), and negative skewness suggests a left tail (Weibull with certain parameters). Kurtosis measures tail heaviness relative to a normal distribution. Excess kurtosis near zero is consistent with normal data. Positive excess kurtosis indicates heavier tails (t-distribution, Laplace), while negative excess kurtosis indicates lighter tails (uniform, beta). Together they narrow down candidate distributions before formal goodness-of-fit testing. For example, high positive skewness combined with positive excess kurtosis strongly suggests exponential or log-normal.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.