Grubbs Test for Outliers

The Grubbs test (also known as the extreme studentized deviate test or ESD test) is a statistical test used to detect a single outlier in a univariate dataset that is assumed to follow an approximately normal distribution.

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What is the Grubbs Test?

The Grubbs test is designed to identify whether the most extreme value in your dataset is statistically significantly different from the rest of the data. It tests the null hypothesis that there are no outliers in the dataset against the alternative that there is exactly one outlier.

The G Statistic

The test statistic G is calculated as the maximum absolute z-score in the dataset:

G = max|xi - x̄| / s

Where x̄ is the sample mean and s is the sample standard deviation. The value with the largest absolute deviation from the mean is identified as the most extreme score.

Critical Value

The critical value Gcrit is calculated using the t-distribution:

Gcrit = (n - 1) × tcrit / √[n(n - 2 + tcrit²)]

Where tcrit is the critical value from the t-distribution with df = n - 2 at significance level α/(2n).

Decision Rule

  • If G > Gcrit: The most extreme value is considered a statistically significant outlier
  • If G ≤ Gcrit: No significant outlier detected at the chosen significance level

Requirements and Assumptions

Important Considerations:

  • Minimum 3 values required (at least 7 recommended for reliable results)
  • Normality assumption: Data should be approximately normally distributed
  • Single outlier only: This test detects only one outlier at a time
  • If multiple outliers are suspected, consider the Generalized ESD test

When to Use Grubbs Test

  • You suspect one value in your dataset might be an outlier
  • Your data is approximately normally distributed
  • You want a formal statistical test rather than just visual inspection
  • You need to document outlier detection for research purposes

Interpreting the Output

The calculator provides descriptive statistics to help assess normality:

  • Skewness: Measures asymmetry. Values near 0 suggest symmetry.
  • Kurtosis: Measures tail heaviness. Values near 0 suggest normal-like tails.
  • Significant departures from normality may invalidate Grubbs test results.

Limitations

  • Cannot detect multiple outliers simultaneously
  • May fail if two outliers are present (they can mask each other)
  • Assumption of normality is strict
  • Not suitable for small samples (n < 7)
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