What Is Distribution Identification and Why Does It Matter for Your Data?

Before you run a control chart, a capability analysis, or a reliability study, you need to know which probability distribution your data actually follows. Distribution identification (also called distribution fitting) is the process of testing your sample against several candidate distributions to find the best statistical match. This article explains how it works, how to interpret the key tests, and what to watch out for.

What It Is

Distribution identification is a formal, data-driven method to select the theoretical probability distribution (e.g., Normal, Lognormal, Weibull, Exponential, Gamma) that most likely generated your observed data. The goal is not to prove that your data is a certain distribution, but to determine which distribution is the least unreasonable, given the evidence.

The method relies on two complementary approaches:

• Goodness-of-fit (GOF) tests – quantify how well each candidate distribution fits the sample.
• Information criteria (AIC/BIC) – compare candidate models, penalizing extra parameters to avoid overfitting.

How It Works: The Core Steps and Statistics

### Step 1 – Fit candidate distributions
For each candidate distribution, estimate its parameters (e.g., mean and standard deviation for a Normal) using maximum likelihood estimation (MLE).

### Step 2 – Run goodness-of-fit tests
The two most common GOF tests are:

Anderson–Darling (A-D) test

• Statistic: \( A^2 = -n - \frac{1}{n}\sum_{i=1}^{n} (2i-1)\left[\ln F(x_i) + \ln(1-F(x_{n+1-i}))\right] \)
• Where \( F(x) \) is the cumulative distribution function (CDF) of the fitted distribution, and \( x_i \) are the ordered sample values.
• The A-D test gives more weight to the tails of the distribution, making it sensitive to deviations at the extremes.

Kolmogorov–Smirnov (K-S) test

• Statistic: \( D = \max_x |F_n(x) - F(x)| \)
• Where \( F_n(x) \) is the empirical CDF of your sample and \( F(x) \) is the fitted CDF.
• The K-S test measures the largest vertical distance between the two CDFs.

How to interpret the results:

• Each test produces a p-value. If the p-value is less than your chosen significance level (commonly 0.05), you reject the null hypothesis that the data follows that distribution. Higher p-values mean a better fit.
• Caution: With small samples, GOF tests have low power – they may fail to reject a poor fit. With very large samples, they may reject even trivial deviations.

### Step 3 – Compare AIC and BIC
The Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) balance fit against model complexity:

\[
AIC = -2\ln(L) + 2k
\]
\[
BIC = -2\ln(L) + k\ln(n)
\]

Where:

• \( L \) = maximized likelihood of the fitted model
• \( k \) = number of estimated parameters
• \( n \) = sample size

Rule: Lower AIC or BIC values indicate a better model. BIC penalizes extra parameters more heavily than AIC, especially for large samples. When two distributions have similar AIC/BIC values (difference less than about 2), treat them as statistically equivalent.

A Worked Illustrative Example

Example data (illustrative only): Suppose you have 50 measurements of a process cycle time (in minutes), and you suspect it is not Normal. You fit four candidates: Normal, Lognormal, Weibull, and Gamma.

Distribution | A-D statistic | A-D p-value | K-S statistic | K-S p-value | AIC | BIC

• Normal | 1.87 | <0.005 | 0.142 | 0.012 | 412.3 | 418.1
• Lognormal | 0.41 | 0.352 | 0.068 | 0.974 | 398.7 | 404.5
• Weibull | 0.52 | 0.241 | 0.075 | 0.891 | 401.2 | 407.0
• Gamma | 0.44 | 0.318 | 0.070 | 0.943 | 399.5 | 405.3

Interpretation:

• The Normal distribution is clearly rejected (p < 0.005 for A-D, p = 0.012 for K-S). Do not use Normal-based tools.
• Lognormal has the highest p-values (0.352 and 0.974) and the lowest AIC (398.7). It is the best fit.
• Gamma is a close second (AIC difference = 0.8, less than 2), so it is statistically equivalent to Lognormal for practical purposes.
• Weibull is acceptable but slightly worse.

Decision: Use Lognormal (or Gamma) for your capability or reliability analysis.

Common Pitfalls

1. Testing only the Normal distribution. Many real-world datasets (cycle times, failure data, costs) are skewed. Always test several candidates.
2. Ignoring the p-value direction. A high p-value is good for GOF tests – it means no evidence of a poor fit. This is the opposite of many other hypothesis tests.
3. Over-relying on one statistic. Use both A-D and K-S, plus AIC/BIC. If they disagree, investigate why (e.g., outliers, small sample).
4. Forgetting the practical impact. A statistically significant difference may be practically irrelevant. Check if the fitted distributions give meaningfully different predictions for your metric of interest.
5. Using GOF tests with very small samples (n < 20). Results are unreliable; consider graphical methods (probability plots) instead.

Try It Yourself

If you need to identify the best-fitting distribution for your data, you can use a free, interactive tool at:

https://www.6sq.com/tools/dist_fit/

Upload or paste your sample, select candidate distributions, and the tool will compute the A-D and K-S statistics, p-values, and AIC/BIC for each candidate – giving you a clear, defensible basis for your next analysis.