Cramér-von Mises Test

The Cramér-von Mises test is a goodness-of-fit test that sums squared differences between the empirical distribution function and a hypothesized distribution function across all observations.

Cramér-von Mises 검정은 경험분포함수와 가정한 분포함수의 차이를 제곱해 모든 관측값에 걸쳐 합한 통계량으로 적합도를 판정하는 검정이다.

Which normality test uses the whole distribution rather than only the largest gap?

가장 큰 차이 하나가 아니라 분포 전체를 보고 정규성을 판단하는 검정은 무엇인가요?

The Cramér-von Mises test integrates the squared gap between the empirical and normal distribution functions, so it reflects discrepancies across the whole range rather than only the maximum distance. When the mean and SD are estimated from the data, the statistic is modified (W*² = W²(1 + 0.5/n)) and judged against Stephens' tables or a parametric bootstrap rather than the known-parameter distribution.

When to use Cramér-von Mises Test

• You want an EDF-based normality test sensitive to deviations across the whole distribution.
• Mean and SD are estimated from the sample.
• You want an alternative to the Kolmogorov-Smirnov test with more power.

When to use something else

• Tail behavior is the main concern; use the Anderson-Darling test instead.
• The sample is small and maximum power is needed; use the Shapiro-Wilk test instead.

Assumptions and requirements

• Observations are independent and identically distributed.
• The data are continuous.
• P-values must account for estimated parameters.

What the analysis reports

• W² and modified W*² statistics
• P-value from Stephens' approximation
• Parametric bootstrap p-value

How to report it

A Cramér-von Mises test against a normal distribution with estimated parameters [did/did not] indicate a departure from normality, W*² = [value], p = [value].

Related methods

• Anderson-Darling Test
• Lilliefors Normality Test
• Shapiro-Wilk Test

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