Grubbs’ test is a statistical method that can detect a single outlier in a dataset that is approximately normally distributed. It tests whether the most extreme value in the dataset is statistically different from the others. The test evaluates whether all the data points come from a single normal distribution, or whether one value deviates so strongly that it is unlikely to belong to the same population.
Grubbs’ test is useful in quality control, experimental data analysis, and any situation where identifying a potential outlier is important before continuing with further analysis.
When to Use Grubbs’ Test
- You want to test for one potential outlier in a continuous dataset.
- Your data appear to follow a normal distribution.
- You need an objective test to confirm whether an extreme value is significantly different from the rest of the data.
- You do not already know that the extreme value is caused by measurement error or an external factor.
Grubbs’ test only checks for one outlier. If you suspect more outliers exist, do not apply it iteratively because each removal changes the dataset and affects assumptions. Instead, use a different method designed for multiple outliers, such as Generalized ESD.
Learn more in-depth about 5 Ways to Find Outliers in Your Data.
Test Statistic and Hypotheses
The following is the Grubbs’ test statistic formula:
G = |xᵢ − x̄| / s
Where:
- xᵢ is the value suspected of being an outlier (either the maximum or minimum in the dataset),
- x̄ is the sample mean,
- s is the sample standard deviation.
The null and alternative hypotheses for Grubbs’ test are the following:
- Null hypothesis (H₀): All sample data are from a single normal distribution. There are no outliers in the data.
- Alternative hypothesis (H₁): One data point is not from the same normal distribution as the other values. The value being tested is an outlier.
After calculating the G statistic, you determine the p-value. If the p-value is less than your significance level (typically 0.05), you reject the null hypothesis and conclude that the extreme value is a statistically significant outlier.
If you find an outlier, learn more in-depth about the Guidelines for Removing and Handling Outliers.
Grubbs’ Test Example
Note: Use my free online Outlier Calculator to find outliers in your dataset! It perform the Grubbs test along with several others.
A researcher records the following values:
10.2, 10.5, 10.3, 10.4, 10.6, 12.5
The value 12.5 seems unusually high. Grubbs’ test is applied to evaluate whether it is a significant outlier. The test calculates a G statistic of 3.12 and produces a p-value of 0.019.
Because 0.019 is less than 0.05, the test result is statistically significant, and the researcher concludes that 12.5 is an outlier. This result suggests that it does not come from the same normally distributed population as the rest of the values.
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