Anderson-Darling Test

Tail sensitive test that gives more weight to extreme values in the distribution.

Test Overview & Tail-Sensitivity

What it measures:

The Anderson-Darling test is a modification of the Kolmogorov-Smirnov test that gives more weight to the tails of the distribution. It measures the weighted squared difference between the empirical and theoretical cumulative distribution functions, making it particularly sensitive to departures from normality in the extreme values.

Strengths:

• • High sensitivity to tail departures
• • Excellent outlier detection capabilities
• • Effective for detecting skewness
• • Good power for most sample sizes
• • Automated on the sixsigmacenter.com platform

Limitations:

• • Complex manual calculation
• • May be overly sensitive to minor deviations
• • Requires minimum sample size (n ≥ 8)
• • Less effective with very small samples

A² Statistic & Adjusted Critical Value

The Anderson-Darling statistic A² measures the weighted integral of the squared difference between the empirical and theoretical distribution functions, with the weighting function emphasizing the tails.

Anderson-Darling statistic

= sample size

F(Xi)

= hypothesized (normal) CDF evaluated at the i-th sorted observation

Sample-size adjustment

A2*

= adjusted statistic compared against the fixed critical value

Critical value at α = 0.05: A²* = 0.787.

Key Assumptions

• • Observations are independent and continuous
• • Sample is sorted in ascending order before computing F(Xi)
• • Minimum sample size of n ≥ 8 for the test to be reliable
• • Mean and standard deviation used in F(Xi) are estimated from the sample when testing composite normality

Reading the Anderson-Darling Fit Plot

Example Result & Decision Interpretation

Worked Example:

A sample of n = 15 measurements is sorted, standardized, and converted to CDF values. Summing the weighted log terms and applying the sample-size correction yields A²* = 0.62.

Decision Rule:

• A²* > 0.787: Reject normality

• A²* ≤ 0.787: Fail to reject normality

• Larger values indicate greater departure

Applying the Example:

• A²* = 0.62 ≤ 0.787

• Fail to reject normality

• Data are consistent with a normal distribution

sixsigmacenter.com Implementation Guide

Preparing Your Data for Upload

• • A single column of the continuous characteristic under test, one row per measured unit. Anderson-Darling needs the raw values — a histogram or frequency table cannot be tested.
• • No mixed populations in one column. Two machines, two cavities, or two shifts stacked together will fail normality for a reason that has nothing to do with the distribution.
• • Numeric cells only; strip units, flags, and "<LOD" censoring markers. Censored data needs a censored-data method, not this test.
• • Keep at least 8 observations, and prefer 25 or more — below 8 the test has almost no power to detect a departure.

Very large samples flag trivial, harmless departures as significant. Read the A-squared statistic and the probability plot together, never the p-value alone.

Platform Implementation:

The sixsigmacenter.com platform automatically performs accurate Anderson-Darling test calculations, eliminating manual computation errors and providing reliable normality assessment results.

Interpretation Guidelines & Best Practices

Decision Framework:

Normal Distribution (A²* ≤ 0.787):

• • Data follows normal distribution
• • Parametric tests appropriate
• • No extreme outliers detected
• • Symmetric distribution pattern

Non-Normal (A²* > 0.787):

• • Significant departure from normality
• • Consider non-parametric alternatives
• • Investigate outliers or skewness
• • May need data transformation

Sample Size Guidelines:

• Minimum: n ≥ 8 (test becomes unreliable below this)

• Small samples (8-50): Use with caution, consider Shapiro-Wilk instead

• Medium samples (50-300): Excellent performance, highly recommended

• Large samples (>300): May detect trivial departures, consider practical significance

When to Use Anderson-Darling:

• • When tail behavior is critical (outlier detection)
• • For decision making applications
• • When comparing distributions with known reference
• • For detecting heavy-tailed or light-tailed distributions

Example Data Layout for Anderson-Darling Test

Single column of continuous data for tail-sensitive normality testing. Sample Size Recommendation: Minimum 8 observations required, optimal performance with 50+ observations for reliable tail-sensitive normality assessment.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.